Nikiya Anton Bettey 2021: 09-20-2021-2253 - Gas Laws; Ideal gas law general gas equation

Tuesday, September 21, 2021

09-20-2021-2253 - Gas Laws; Ideal gas law general gas equation

The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases.

Boyle's law[edit]

In 1662 Robert Boyle studied the relationship between volume and pressure of a gas of fixed amount at constant temperature. He observed that volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. Boyle's law, published in 1662, states that, at constant temperature, the product of the pressure and volume of a given mass of an ideal gas in a closed system is always constant. It can be verified experimentally using a pressure gauge and a variable volume container. It can also be derived from the kinetic theory of gases: if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will strike a given area of the sides of the container per unit time, causing a greater pressure.

A statement of Boyle's law is as follows:

The volume of a given mass of a gas is inversely related to pressure when the temperature is constant.

The concept can be represented with these formulae:

V\propto {\frac {1}{P}}

, meaning "Volume is inversely proportional to Pressure", or

P\propto {\frac {1}{V}}

, meaning "Pressure is inversely proportional to Volume", or

PV=k_{1}

, or

P_{1}V_{1}=P_{2}V_{2}\,

where P is the pressure, and V is the volume of a gas, and k₁ is the constant in this equation (and is not the same as the proportionality constants in the other equations in this article).

Charles's law[edit]

Charles's law, or the law of volumes, was found in 1787 by Jacques Charles. It states that, for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, assuming in a closed system.

The statement of Charles's law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). As a mathematical equation, Charles's law is written as either:

V\propto T\,

, or

V/T=k_{2}

, or

V_{1}/T_{1}=V_{2}/T_{2}

where "V" is the volume of a gas, "T" is the absolute temperature and k₂ is a proportionality constant (which is not the same as the proportionality constants in the other equations in this article).

Gay-Lussac's law[edit]

Gay-Lussac's law, Amontons' law or the pressure law was found by Joseph Louis Gay-Lussac in 1808. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.

As a mathematical equation, Gay-Lussac's law is written as either:

P\propto T\,

, or

P/T=k

, or

P_{1}/T_{1}=P_{2}/T_{2}

where P is the pressure, T is the absolute temperature, and k is another proportionality constant.

Avogadro's law[edit]

Avogadro's law (hypothesized in 1811) states that the volume occupied by an ideal gas is directly proportional to the number of molecules of the gas present in the container. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. The relation is given by

{\frac {V_{1}}{n_{1}}}={\frac {V_{2}}{n_{2}}}\,

where n is equal to the number of molecules of gas (or the number of moles of gas).

Combined and ideal gas laws[edit]

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant k_B = RN_A = n RN (in each law, properties circled are variable and properties not circled are held constant)

The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas:

PV=k_{5}T\,

This can also be written as:

\qquad {\frac {P_{1}V_{1}}{T_{1}}}={\frac {P_{2}V_{2}}{T_{2}}}

With the addition of Avogadro's law, the combined gas law develops into the ideal gas law:

PV = nRT \,

where

P is pressure

V is volume

n is the number of moles

R is the universal gas constant

T is temperature (K)

where the proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K). An equivalent formulation of this law is:

PV=NkT\,

where

P is the pressure

V is the volume

N is the number of gas molecules

k is the Boltzmann constant (1.381×10⁻²³J·K⁻¹ in SI units)

T is the temperature (K)

These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:

If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Other gas laws[edit]

Graham's law: states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at constant temperature. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.
Dalton's law of partial pressures: states that the pressure of a mixture of gases simply is the sum of the partial pressures of the individual components. Dalton's law is as follows:; $P_{\textrm {total}}=P_{1}+P_{2}+P_{3}+\cdots +P_{n}\equiv \sum _{i=1}^{n}P_{i},$; and all component gases and the mixture are at the same temperature and volume; where P_total is the total pressure of the gas mixture; P_i is the partial pressure, or pressure of the component gas at the given volume and temperature.
Amagat's law of partial volumes: states that the volume of a mixture of gases (or the volume of the container) simply is the sum of the partial volumes of the individual components. Amagat's law is as follows:; $V_{\textrm {total}}=V_{1}+V_{2}+V_{3}+\cdots +V_{n}\equiv \sum _{i=1}^{n}V_{i},$; and all component gases and the mixture are at the same temperature and pressure; where V_total is the total volume of the gas mixture, or the volume of the container,; V_i is the partial volume, or volume of the component gas at the given pressure and temperature.
Henry's law: states that At constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.; $p=k_{\rm {H}}\,c$
Real gas law: formulated by Johannes Diderik van der Waals (1873).

References[edit]

Castka, Joseph F.; Metcalfe, H. Clark; Davis, Raymond E.; Williams, John E. (2002). Modern Chemistry. Holt, Rinehart and Winston. ISBN 0-03-056537-5.
Guch, Ian (2003). The Complete Idiot's Guide to Chemistry. Alpha, Penguin Group Inc. ISBN 1-59257-101-8.
Zumdahl, Steven S (1998). Chemical Principles. Houghton Mifflin Company. ISBN 0-395-83995-5.

https://en.wikipedia.org/wiki/Gas_laws

In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state; named after Dutch physicist Johannes Diderik van der Waals) is an equation of state that generalizes the ideal gas law based on plausible reasons that real gases do not act ideally. The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions(i.e. all collisions are perfectly elastic).^[1] The ideal gas law states that volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in kelvinsgiven by the following relationship, where R is the gas constant:

PV=nRT

To account for the volume that a real gas molecule takes up, the Van der Waals equation replaces V in the ideal gas law with $(V_{m}-b)$ , where V_m is the molar volume of the gas and b is the volume that is occupied by one mole of the molecules. This leads to:^[1]

P(V_{m}-b)=RT

Van der Waals equation on a wall in Leiden

The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact interact with each other (they usually experience attraction at low pressures and repulsion at high pressures) and that real gases therefore show different compressibility than ideal gases. Van der Waals provided for intermolecular interaction by adding to the observed pressure P in the equation of state a term $a/V_{m}^{2}$ , where a is a constant whose value depends on the gas. The Van der Waals equation is therefore written as:^[1]

\left(P+a{\frac {1}{V_{m}^{2}}}\right)(V_{m}-b)=RT

and, for n moles of gas, can also be written as the equation below:

\left(P+a{\frac {n^{2}}{V^{2}}}\right)(V-nb)=nRT

where V_m is the molar volume of the gas, R is the universal gas constant, T is temperature, P is pressure, and V is volume. When the molar volume V_m is large, bbecomes negligible in comparison with V_m, a/V_m² becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PV_m=RT.^[1]

It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified. It successfully approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behaviour, in particular that p is a constant function of V at given temperatures. As such, the Van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Corrections to address these predictive deficiencies have since been made, such as the equal area rule or the principle of corresponding states.

https://en.wikipedia.org/wiki/Van_der_Waals_equation

According to van der Waals, the theorem of corresponding states (or principle/law of corresponding states) indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree.^[1]^[2]

Material constants that vary for each type of material are eliminated, in a recast reduced form of a constitutive equation. The reduced variables are defined in terms of critical variables.

The principle originated with the work of Johannes Diderik van der Waals in about 1873^[3] when he used the critical temperature and critical pressure to characterize a fluid.

The most prominent example is the van der Waals equation of state, the reduced form of which applies to all fluids.

Compressibility factor at the critical point[edit]

The compressibility factor at the critical point, which is defined as $Z_{c}={\frac {P_{c}v_{c}\mu }{RT_{c}}}$ , where the subscript $c$ indicates the critical point, is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of $3/8=0.375$ .

Where:

$T_{c}$ : critical temperature [K]
$P_{c}$ : critical pressure [Pa]
$v_{c}$ : critical specific volume [m³⋅kg⁻¹]
$R$ : gas constant (8.314 J⋅K⁻¹⋅mol⁻¹)
$\mu$ : Molar mass [kg⋅mol⁻¹]

For example:

Substance	$P_{c}$ [Pa]	$T_{c}$ [K]	$v_{c}$ [m³/kg]	$Z_c$
H₂O	21.817×10⁶	647.3	3.154×10⁻³	0.23^[4]
⁴He	0.226×10⁶	5.2	14.43×10⁻³	0.31^[4]
He	0.226×10⁶	5.2	14.43×10⁻³	0.30^[5]
H₂	1.279×10⁶	33.2	32.3×10⁻³	0.30^[5]
Ne	2.73×10⁶	44.5	2.066×10⁻³	0.29^[5]
N₂	3.354×10⁶	126.2	3.2154×10⁻³	0.29^[5]
Ar	4.861×10⁶	150.7	1.883×10⁻³	0.29^[5]
Xe	5.87×10⁶	289.7	0.9049×10⁻³	0.29
O₂	5.014×10⁶	154.8	2.33×10⁻³	0.291
CO₂	7.290×10⁶	304.2	2.17×10⁻³	0.275
SO₂	7.88×10⁶	430.0	1.900×10⁻³	0.275
CH₄	4.58×10⁶	190.7	6.17×10⁻³	0.285
C₃H₈	4.21×10⁶	370.0	4.425×10⁻³	0.267

Overview[edit]

At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.

Another common use is in modeling the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluid equation of state used in cosmology.

Equations of state can also describe solids, including the transition of solids from one crystalline state to another.

In a practical context, equations of state are instrumental for PVT calculations in process engineering problems, such as petroleum gas/liquid equilibrium calculations. A successful PVT model based on a fitted equation of state can be helpful to determine the state of the flow regime, the parameters for handling the reservoir fluids, and pipe sizing.

Measurements of equation-of-state parameters, especially at high pressures, can be made using lasers.^[2]^[3]^[4]

Historical[edit]

Boyle's law (1662)[edit]

Boyle's Law was perhaps the first expression of an equation of state.^{[citation needed]} In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:

pV={\mathrm {constant}}.\,\!

The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.

Charles's law or Law of Charles and Gay-Lussac (1787)[edit]

In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature (Charles's Law):

{\frac {V_{1}}{T_{1}}}={\frac {V_{2}}{T_{2}}}.

Dalton's law of partial pressures (1801)[edit]

Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.

Mathematically, this can be represented for n species as:

p_{\text{total}}=p_{1}+p_{2}+\cdots +p_{n}=\sum _{i=1}^{n}p_{i}.

The ideal gas law (1834)[edit]

In 1834, Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially, the law was formulated as pV_m = R(T_C + 267) (with temperature expressed in degrees Celsius), where R is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:

pV_{m}=R\left(T_{C}+273.15\ {}^{\circ }{\text{C}}\right).

Van der Waals equation of state (1873)[edit]

In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules.^[5] His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state^[6] and the Soave modification of Redlich-Kwong.^[7]

General form of an equation of state[edit]

Thermodynamic systems are specified by an equation of state that constrains the values that the state variables may assume. For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form

f(p,V,T)=0

An equation used to model this relationship is called an equation of state. In the following sections major equations of state are described, and the variables used here are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to the use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.

\ p

, pressure (absolute)

\ V

, volume

\ n

, number of moles of a substance

\ V_{m}

{\frac {V}{n}}

, molar volume, the volume of 1 mole of gas or liquid

\ T

, absolute temperature

\ R

, ideal gas constant ≈ 8.3144621 J/mol·K

\ p_{c}

, pressure at the critical point

\ V_{c}

, molar volume at the critical point

\ T_{c}

, absolute temperature at the critical point

Classical ideal gas law[edit]

The classical ideal gas law may be written

pV=nRT.

In the form shown above, the equation of state is thus

f(p,V,T)=pV-nRT=0.

If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows

p=\rho (\gamma -1)e

where $\rho$ is the density, $\gamma =C_{p}/C_{v}$ is the (constant) adiabatic index (ratio of specific heats), $e=C_{v}T$ is the internal energy per unit mass (the "specific internal energy"), $C_{v}$ is the constant specific heat at constant volume, and $C_{p}$ is the constant specific heat at constant pressure.

Quantum ideal gas law[edit]

Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass $m$ and spin $s$ that takes into account of quantum effects. In the following, the upper sign will always correspond to Fermi-Dirac statistics and the lower sign to Bose–Einstein statistics. The equation of state of such gases with $N$ particles occupying a volume $V$ with temperature $T$ and pressure $p$ is given by^[8]

p={\frac {(2s+1){\sqrt {2m^{3}k_{B}^{5}T^{5}}}}{3\pi ^{2}\hbar ^{3}}}\int _{0}^{\infty }{\frac {z^{3/2}\,\mathrm {d} z}{e^{z-\mu /(k_{B}T)}\pm 1}}

where $k_{B}$ is the Boltzmann constant and $\mu (T,N/V)$ the chemical potential is given by the following implicit function

{\frac {N}{V}}={\frac {(2s+1)(mk_{B}T)^{3/2}}{{\sqrt {2}}\pi ^{2}\hbar ^{3}}}\int _{0}^{\infty }{\frac {z^{1/2}\,\mathrm {d} z}{e^{z-\mu /(k_{B}T)}\pm 1}}.

In the limiting case where $e^{\mu /(k_{B}T)}\ll 1$ , this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit $e^{\mu /(k_{B}T)}\ll 1$ reduces to

pV=Nk_{B}T\left[1\pm {\frac {\pi ^{3/2}}{2(2s+1)}}{\frac {N\hbar ^{3}}{V(mk_{B}T)^{3/2}}}+\cdots \right]

With a fixed number density $N/V$ , decreasing the temperature causes in Fermi gas, a increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in Bose gas, a decrease in pressure from its classical value implying an effective attraction.

Cubic equations of state[edit]

Cubic equations of state are called such because they can be rewritten as a cubic function of $V_{m}$ .

Van der Waals equation of state[edit]

The Van der Waals equation of state may be written:

\left(p+{\frac {a}{V_{m}^{2}}}\right)\left(V_{m}-b\right)=RT

where $V_{m}$ is molar volume. The substance-specific constants $a$ and $b$ can be calculated from the critical properties $p_{c}$ , $T_{c}$ , and $V_{c}$ (noting that $V_{c}$ is the molar volume at the critical point) as:

a=3p_{c}V_{c}^{2}

b = \frac{V_c}{3}.

Also written as

a={\frac {27(RT_{c})^{2}}{64p_{c}}}

b={\frac {RT_{c}}{8p_{c}}}.

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation $a$ is called the attraction parameter and $b$ the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.

The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:

Molecules are thought as particles with volume, not material points. Thus $V_{m}$ cannot be too little, less than some constant. So we get ( $V_m - b$ ) instead of $V_{m}$ .
While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write ( $p +$ something) instead of $p$ . To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~ $\rho$ , the force acting on the whole element is ~ $\rho ^{2}$ ~ $\frac{1}{V_m^2}$ .

With the reduced state variables, i.e. $V_{r}=V_{m}/V_{c}$ , $P_{r}=P/P_{c}$ and $T_{r}=T/T_{c}$ , the reduced form of the Van der Waals equation can be formulated:

\left(P_{r}+{\frac {3}{V_{r}^{2}}}\right)\left(3V_{r}-1\right)=8T_{r}

The benefit of this form is that for given $T_r$ and $P_{r}$ , the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:

V_{r}^{3}-\left({\frac {1}{3}}+{\frac {8T_{r}}{3P_{r}}}\right)V_{r}^{2}+{\frac {3V_{r}}{P_{r}}}-{\frac {1}{P_{r}}}=0

For $P_{r}<1$ and $T_{r}<1$ , the system is in a state of vapor–liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.

Redlich-Kwong equation of state^[6][edit]

{\begin{aligned}p&={\frac {R\,T}{V_{m}-b}}-{\frac {a}{{\sqrt {T}}\,V_{m}\left(V_{m}+b\right)}}\\[3pt]a&\approx 0.42748{\frac {R^{2}\,T_{c}^{\frac {5}{2}}}{p_{c}}}\\[3pt]b&\approx 0.08664{\frac {R\,T_{c}}{p_{c}}}\end{aligned}}

Introduced in 1949, the Redlich-Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor–liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.

The Redlich-Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):

\frac{p}{p_c} < \frac{T}{2T_c}.

Soave modification of Redlich-Kwong^[7][edit]

p={\frac {R\,T}{V_{m}-b}}-{\frac {a\alpha }{V_{m}\left(V_{m}+b\right)}}

a={\frac {0.42747\,R^{2}T_{c}^{2}}{P_{c}}}

b={\frac {0.08664\,RT_{c}}{P_{c}}}

\alpha =\left(1+\left(0.48508+1.55171\,\omega -0.15613\,\omega ^{2}\right)\left(1-T_{r}^{0.5}\right)\right)^{2}

T_r = \frac{T}{T_c}

Where ω is the acentric factor for the species.

This formulation for $\alpha$ is due to Graboski and Daubert. The original formulation from Soave is:

\alpha =\left(1+\left(0.480+1.574\,\omega -0.176\,\omega ^{2}\right)\left(1-T_{r}^{0.5}\right)\right)^{2}

for hydrogen:

\alpha = 1.202 \exp\left(-0.30288\,T_r\right).

We can also write it in the polynomial form, with:

A={\frac {a\alpha P}{R^{2}T^{2}}}

B={\frac {bP}{RT}}

then we have:

0=Z^{3}-Z^{2}+Z\left(A-B-B^{2}\right)-AB

where $R$ is the universal gas constant and Z=PV/(RT) is the compressibility factor.

In 1972 G. Soave^[9] replaced the 1/√T term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor (the resulting equation is also known as the Soave-Redlich-Kwong equation of state; SRK EOS). The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.

Note especially that this replacement changes the definition of a slightly, as the $T_{c}$ is now to the second power.

Volume translation of Peneloux et al. (1982)[edit]

The SRK EOS may be written as

p={\frac {R\,T}{V_{m,{\text{SRK}}}-b}}-{\frac {a}{V_{m,{\text{SRK}}}\left(V_{m,{\text{SRK}}}+b\right)}}

where

{\begin{aligned}a&=a_{c}\,\alpha \\a_{c}&\approx 0.42747{\frac {R^{2}\,T_{c}^{2}}{P_{c}}}\\b&\approx 0.08664{\frac {R\,T_{c}}{P_{c}}}\end{aligned}}

where $\alpha$ and other parts of the SRK EOS is defined in the SRK EOS section.

A downside of the SRK EOS, and other cubic EOS, is that the liquid molar volume is significantly less accurate than the gas molar volume. Peneloux et alios (1982)^[10] proposed a simple correction for this by introducing a volume translation

V_{m,{\text{SRK}}}=V_{m}+c

where $c$ is an additional fluid component parameter that translates the molar volume slightly. On the liquid branch of the EOS, a small change in molar volume corresponds to a large change in pressure. On the gas branch of the EOS, a small change in molar volume corresponds to a much smaller change in pressure than for the liquid branch. Thus, the perturbation of the molar gas volume is small. Unfortunately, there are two versions that occur in science and industry.

In the first version only $V_{m,{\text{SRK}}}$ is translated,^[11] ^[12] and the EOS becomes

p={\frac {R\,T}{V_{m}+c-b}}-{\frac {a}{\left(V_{m}+c\right)\left(V_{m}+c+b\right)}}

In the second version both $V_{m,{\text{SRK}}}$ and $b_{\text{SRK}}$ are translated, or the translation of $V_{m,{\text{SRK}}}$ is followed by a renaming of the composite parameter b − c.^[13] This gives

{\begin{aligned}b_{\text{SRK}}&=b+c\quad {\text{or}}\quad b-c\curvearrowright b\\p&={\frac {R\,T}{V_{m}-b}}-{\frac {a}{\left(V_{m}+c\right)\left(V_{m}+2c+b\right)}}\end{aligned}}

The c-parameter of a fluid mixture is calculated by

c=\sum _{i=1}^{n}z_{i}c_{i}

The c-parameter of the individual fluid components in a petroleum gas and oil can be estimated by the correlation

c_{i}\approx 0.40768\ {\frac {RT_{ci}}{P_{ci}}}\left(0.29441-Z_{{\text{RA}},i}\right)

where the Rackett compressibility factor $Z_{{\text{RA}},i}$ can be estimated by

Z_{{\text{RA}},i}\approx 0.29056-0.08775\ \omega _{i}

A nice feature with the volume translation method of Peneloux et al. (1982) is that it does not affect the vapor-liquid equilibrium calculations.^[14] This method of volume translation can also be applied to other cubic EOSs if the c-parameter correlation is adjusted to match the selected EOS.

Peng–Robinson equation of state[edit]

{\begin{aligned}p&={\frac {R\,T}{V_{m}-b}}-{\frac {a\,\alpha }{V_{m}^{2}+2bV_{m}-b^{2}}}\\[3pt]a&\approx 0.45724{\frac {R^{2}\,T_{c}^{2}}{p_{c}}}\\[3pt]b&\approx 0.07780{\frac {R\,T_{c}}{p_{c}}}\\[3pt]\alpha &=\left(1+\kappa \left(1-T_{r}^{\frac {1}{2}}\right)\right)^{2}\\[3pt]\kappa &\approx 0.37464+1.54226\,\omega -0.26992\,\omega ^{2}\\[3pt]T_{r}&={\frac {T}{T_{c}}}\end{aligned}}

In polynomial form:

A={\frac {\alpha ap}{R^{2}\,T^{2}}}

B={\frac {bp}{RT}}

Z^{3}-(1-B)Z^{2}+\left(A-2B-3B^{2}\right)Z-\left(AB-B^{2}-B^{3}\right)=0

where $\omega$ is the acentric factor of the species, $R$ is the universal gas constant and $Z=PV/nRT$ is compressibility factor.

The Peng–Robinson equation of state (PR EOS) was developed in 1976 at The University of Alberta by Ding-Yu Peng and Donald Robinson in order to satisfy the following goals:^[15]

The parameters should be expressible in terms of the critical properties and the acentric factor.
The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density.
The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature, pressure, and composition.
The equation should be applicable to all calculations of all fluid properties in natural gas processes.

For the most part the Peng–Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones.^[16] The departure functions of the Peng–Robinson equation are given on a separate article.

The analytic values of its characteristic constants are:

Z_{c}={\frac {1}{32}}\left(11-2{\sqrt {7}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {13}{7{\sqrt {7}}}}\right)\right)\right)\approx 0.307401

b'={\frac {b}{V_{m,c}}}={\frac {1}{3}}\left({\sqrt {8}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\sqrt {8}}\right)\right)-1\right)\approx 0.253077\approx {\frac {0.07780}{Z_{c}}}

{\frac {P_{c}V_{m,c}^{2}}{a\,b'}}={\frac {3}{8}}\left(1+\cosh \left({\frac {1}{3}}\operatorname {arcosh} (3)\right)\right)\approx 0.816619\approx {\frac {Z_{c}^{2}}{0.45724\,b'}}

Peng–Robinson-Stryjek-Vera equations of state[edit]

PRSV1[edit]

A modification to the attraction term in the Peng–Robinson equation of state published by Stryjek and Vera in 1986 (PRSV) significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factor.^[17]

The modification is:

{\begin{aligned}\kappa &=\kappa _{0}+\kappa _{1}\left(1+T_{r}^{\frac {1}{2}}\right)\left(0.7-T_{r}\right)\\\kappa _{0}&=0.378893+1.4897153\,\omega -0.17131848\,\omega ^{2}+0.0196554\,\omega ^{3}\end{aligned}}

where $\kappa_1$ is an adjustable pure component parameter. Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article. At reduced temperatures above 0.7, they recommend to set $\kappa _{1}=0$ and simply use $\kappa = \kappa_0$ . For alcohols and water the value of $\kappa _{1}$ may be used up to the critical temperature and set to zero at higher temperatures.^[17]

PRSV2[edit]

A subsequent modification published in 1986 (PRSV2) further improved the model's accuracy by introducing two additional pure component parameters to the previous attraction term modification.^[18]

The modification is:

{\begin{aligned}\kappa &=\kappa _{0}+\left[\kappa _{1}+\kappa _{2}\left(\kappa _{3}-T_{r}\right)\left(1-T_{r}^{\frac {1}{2}}\right)\right]\left(1+T_{r}^{\frac {1}{2}}\right)\left(0.7-T_{r}\right)\\\kappa _{0}&=0.378893+1.4897153\,\omega -0.17131848\,\omega ^{2}+0.0196554\,\omega ^{3}\end{aligned}}

where $\kappa_1$ , $\kappa_2$ , and $\kappa _{3}$ are adjustable pure component parameters.

PRSV2 is particularly advantageous for VLE calculations. While PRSV1 does offer an advantage over the Peng–Robinson model for describing thermodynamic behavior, it is still not accurate enough, in general, for phase equilibrium calculations.^[17] The highly non-linear behavior of phase-equilibrium calculation methods tends to amplify what would otherwise be acceptably small errors. It is therefore recommended that PRSV2 be used for equilibrium calculations when applying these models to a design. However, once the equilibrium state has been determined, the phase specific thermodynamic values at equilibrium may be determined by one of several simpler models with a reasonable degree of accuracy.^[18]

One thing to note is that in the PRSV equation, the parameter fit is done in a particular temperature range which is usually below the critical temperature. Above the critical temperature, the PRSV alpha function tends to diverge and become arbitrarily large instead of tending towards 0. Because of this, alternate equations for alpha should be employed above the critical point. This is especially important for systems containing hydrogen which is often found at temperatures far above its critical point. Several alternate formulations have been proposed. Some well known ones are by Twu et al or by Mathias and Copeman.

Peng-Robinson-Babalola equation of state (PRB)[edit]

Babalola ^[19] modified the Peng–Robinson Equation of state as:

$P=\left({\frac {RT}{v-b}}\right)-\left[{\frac {(a_{1}P+a_{2})\alpha }{v(v+b)+b(v-b)}}\right]$

The attractive force parameter ‘a’, which was considered to be a constant with respect to pressure in Peng–Robinson EOS. The modification, in which parameter ‘a’ was treated as a variable with respect to pressure for multicomponent multi-phase high density reservoir systems was to improve accuracy in the prediction of properties of complex reservoir fluids for PVT modeling. The variation was represented with a linear equation where a₁ and a₂ represent the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.

This modification increases the accuracy of Peng–Robinson equation of state for heavier fluids particularly at pressure ranges (>30MPa) and eliminates the need for tuning the original Peng-Robinson equation of state. Values for a

Elliott, Suresh, Donohue equation of state[edit]

The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990.^[20] The equation seeks to correct a shortcoming in the Peng–Robinson EOS in that there was an inaccuracy in the van der Waals repulsive term. The EOS accounts for the effect of the shape of a non-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding.

\frac{p V_m}{RT}=Z=1 + Z^{\rm{rep}} + Z^{\rm{att}}

where:

Z^{\rm{rep}} = \frac{4 c \eta}{1-1.9 \eta}

Z^{\rm{att}} = -\frac{z_m q \eta Y}{1+ k_1 \eta Y}

and

c

is a "shape factor", with

c=1

for spherical molecules

For non-spherical molecules, the following relation is suggested

c=1+3.535\omega+0.533\omega^2

where

\omega

is the acentric factor.

The reduced number density

\eta

is defined as

\eta=\frac{v^* n}{V}

where

v^{*}

is the characteristic size parameter

n

is the number of molecules

V

is the volume of the container

The characteristic size parameter is related to the shape parameter $c$ through

v^*=\frac{kT_c}{P_c}\Phi

where

\Phi=\frac{0.0312+0.087(c-1)+0.008(c-1)^2}{1.000+2.455(c-1)+0.732(c-1)^2}

and

k

is Boltzmann's constant.

Noting the relationships between Boltzmann's constant and the Universal gas constant, and observing that the number of molecules can be expressed in terms of Avogadro's number and the molar mass, the reduced number density $\eta$ can be expressed in terms of the molar volume as

\eta=\frac{R T_c}{P_c}\Phi\frac{1}{V_m}.

The shape parameter $q$ appearing in the Attraction term and the term $Y$ are given by

q=1+k_3(c-1)

(and is hence also equal to 1 for spherical molecules).

Y=\exp\left(\frac{\epsilon}{kT}\right) - k_2

where $\epsilon$ is the depth of the square-well potential and is given by

\frac{\epsilon}{k} =\frac{1.000+0.945(c-1)+0.134(c-1)^2}{1.023+2.225(c-1)+0.478(c-1)^2}

z_{m}

k_{1}

k_{2}

and

k_{3}

are constants in the equation of state:

z_m = 9.49

for spherical molecules (c=1)

k_1 = 1.7745

for spherical molecules (c=1)

k_2 = 1.0617

for spherical molecules (c=1)

k_3 = 1.90476.

The model can be extended to associating components and mixtures of nonassociating components. Details are in the paper by J.R. Elliott, Jr. et al. (1990).^[20]

Cubic-Plus-Association[edit]

The Cubic-Plus-Association (CPA) equation of state combines the Soave-Redlich-Kwong equation with the association term from SAFT^[21]^[22] based on Chapman's extensions and simplifications of a theory of associating molecules due to Michael Wertheim.^[23] The development of the equation began in 1995 as a research project that was funded by Shell, and in 1996 an article was published which presented the CPA equation of state.^[23]^[24]

P={\frac {RT}{(V-b)}}-{\frac {a}{V(V+b)}}+{\frac {RT}{V}}\rho \sum _{A}\left[{\frac {1}{X^{A}}}-{\frac {1}{2}}\right]{\frac {\partial X^{A}}{\partial \rho }}

In the association term $X^{A}$ is the mole fraction of molecules not bonded at site A.

Non-cubic equations of state[edit]

Dieterici equation of state[edit]

p(V-b)=RTe^{-{\frac {a}{RTV}}}

where a is associated with the interaction between molecules and b takes into account the finite size of the molecules, similar to the Van der Waals equation.

The reduced coordinates are:

T_{c}={\frac {a}{4Rb}},\ p_{c}={\frac {a}{4b^{2}e^{2}}},\ V_{c}=2b.

Virial equations of state[edit]

Virial equation of state[edit]

{\frac {pV_{m}}{RT}}=A+{\frac {B}{V_{m}}}+{\frac {C}{V_{m}^{2}}}+{\frac {D}{V_{m}^{3}}}+\cdots

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. A is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only.

One of the most accurate equations of state is that from Benedict-Webb-Rubin-Starling^[25] shown next. It was very close to a virial equation of state. If the exponential term in it is expanded to two Taylor terms, a virial equation can be derived:

p=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}+{\frac {c}{T^{2}}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}

Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.

The BWR equation of state[edit]

p=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)

where

p is pressure

ρ is molar density

Values of the various parameters can be found in reference materials.^[26]

Lee-Kesler equation of state[edit]

The Lee-Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.^[27]

P={\frac {RT}{V}}\left(1+{\frac {B}{V_{r}}}+{\frac {C}{V_{r}^{2}}}+{\frac {D}{V_{r}^{5}}}+{\frac {c_{4}}{T_{r}^{3}V_{r}^{2}}}\left(\beta +{\frac {\gamma }{V_{r}^{2}}}\right)\exp \left({\frac {-\gamma }{V_{r}^{2}}}\right)\right)

SAFT equations of state[edit]

Statistical associating fluid theory (SAFT) equations of state predict the effect of molecular size and shape and hydrogen bonding on fluid properties and phase behavior. The SAFT equation of state was developed using statistical mechanical methods (in particular perturbation theory) to describe the interactions between molecules in a system.^[21]^[22]^[28] The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989.^[21]^[22]^[28] Many different versions of the SAFT equation of state have been proposed, but all use the same chain and association terms derived by Chapman.^[21]^[29]^[30] SAFT equations of state represent molecules as chains of typically spherical particles that interact with one another through short range repulsion, long range attraction, and hydrogen bonding between specific sites.^[28] One popular version of the SAFT equation of state includes the effect of chain length on the shielding of the dispersion interactions between molecules (PC-SAFT).^[31] In general, SAFT equations give more accurate results than traditional cubic equations of state, especially for systems containing liquids or solids.^[32]^[33]

Multiparameter equations of state[edit]

Helmholtz Function form[edit]

Multiparameter equations of state (MEOS) can be used to represent pure fluids with high accuracy, in both the liquid and gaseous states. MEOS's represent the Helmholtz function of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in reduced temperature and reduced density - thus:

{\frac {a(T,\rho )}{RT}}={\frac {a^{o}(T,\rho )+a^{r}(T,\rho )}{RT}}=\alpha ^{o}(\tau ,\delta )+\alpha ^{r}(\tau ,\delta )

where:

\tau ={\frac {T_{r}}{T}},\delta ={\frac {\rho }{\rho _{r}}}

The reduced density and temperature are typically, though not always, the critical values for the pure fluid.

Other thermodynamic functions can be derived from the MEOS by using appropriate derivatives of the Helmholtz function; hence, because integration of the MEOS is not required, there are few restrictions as to the functional form of the ideal or residual terms.^[34]^[35] Typical MEOS use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. MEOS are available currently for about 50 of the most common industrial fluids including refrigerants. The IAPWS95 reference equation of state for water is also an MEOS.^[36] Mixture models for MEOS exist, as well.

One example of such an equation of state is the form proposed by Span and Wagner.^[34]

a^{res}=\sum _{i=1}^{8}\sum _{j=-8}^{12}n_{i,j}\delta ^{i}\tau ^{j/8}+\sum _{i=1}^{5}\sum _{j=-8}^{24}n_{i,j}\delta ^{i}\tau ^{j/8}\exp \left(-\delta \right)+\sum _{i=1}^{5}\sum _{j=16}^{56}n_{i,j}\delta ^{i}\tau ^{j/8}\exp \left(-\delta ^{2}\right)+\sum _{i=2}^{4}\sum _{j=24}^{38}n_{i,j}\delta ^{i}\tau ^{j/2}\exp \left(-\delta ^{3}\right)

This is a somewhat simpler form that is intended to be used more in technical applications.^[34] Reference equations of state require a higher accuracy and use a more complicated form with more terms.^[36]^[35]

Other equations of state of interest[edit]

Stiffened equation of state[edit]

When considering water under very high pressures, in situations such as underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence, the stiffened equation of state^[37] is often used:

p=\rho (\gamma -1)e-\gamma p^{0}\,

where $e$ is the internal energy per unit mass, $\gamma$ is an empirically determined constant typically taken to be about 6.1, and $p^0$ is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).

The equation is stated in this form because the speed of sound in water is given by $c^{2}=\gamma \left(p+p^{0}\right)/\rho$ .

Thus water behaves as though it is an ideal gas that is already under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).

This equation mispredicts the specific heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.

Ultrarelativistic equation of state[edit]

An ultrarelativistic fluid has equation of state

p=\rho _{m}c_{s}^{2}

where $p$ is the pressure, $\rho_m$ is the mass density, and $c_{s}$ is the speed of sound.

Ideal Bose equation of state[edit]

The equation of state for an ideal Bose gas is

pV_{m}=RT~{\frac {{\text{Li}}_{\alpha +1}(z)}{\zeta (\alpha )}}\left({\frac {T}{T_{c}}}\right)^{\alpha }

where α is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), z is exp(μ/kT) where μ is the chemical potential, Li is the polylogarithm, ζ is the Riemann zeta function, and T_c is the critical temperature at which a Bose–Einstein condensate begins to form.

Jones–Wilkins–Lee equation of state for explosives (JWL equation)[edit]

The equation of state from Jones–Wilkins–Lee is used to describe the detonation products of explosives.

p=A\left(1-{\frac {\omega }{R_{1}V}}\right)\exp(-R_{1}V)+B\left(1-{\frac {\omega }{R_{2}V}}\right)\exp \left(-R_{2}V\right)+{\frac {\omega e_{0}}{V}}

The ratio $V=\rho _{e}/\rho$ is defined by using $\rho _{e}$ = density of the explosive (solid part) and $\rho$ = density of the detonation products. The parameters $A$ , $B$ , $R_1$ , $R_2$ and $\omega$ are given by several references.^[38] In addition, the initial density (solid part) $\rho_0$ , speed of detonation $V_D$ , Chapman–Jouguet pressure $P_{CJ}$ and the chemical energy of the explosive $e_0$ are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results. Typical parameters for some explosives are listed in the table below.

Material	$\rho_0\,$ (g/cm³)	$v_{D}\,$ (m/s)	$p_{CJ}\,$ (GPa)	$A\,$ (GPa)	$B\,$ (GPa)	$R_1\,$	$R_2\,$	$\omega\,$	$e_0\,$ (GPa)
TNT	1.630	6930	21.0	373.8	3.747	4.15	0.90	0.35	6.00
Composition B	1.717	7980	29.5	524.2	7.678	4.20	1.10	0.35	8.50
PBX 9501^[39]	1.844		36.3	852.4	18.02	4.55	1.3	0.38	10.2

Equations of state for solids and liquids[edit]

Common abbreviations: $\eta =\left({\frac {V}{V_{0}}}\right)^{\frac {1}{3}}~,~~K_{0}^{\prime }={\frac {dK_{0}}{dp}}$

Tait equation for water and other liquids. Several equations are referred to as the Tait equation.
Murnaghan equation of state

p(V)={\frac {K_{0}}{K_{0}'}}\left[\eta ^{-3K_{0}'}-1\right]

Birch–Murnaghan equation of state

p(V)={\frac {3K_{0}}{2}}\left({\frac {1-\eta ^{2}}{\eta ^{7}}}\right)\left\{1+{\frac {3}{4}}\left(K_{0}'-4\right)\left({\frac {1-\eta ^{2}}{\eta ^{2}}}\right)\right\}

Stacey-Brennan-Irvine equation of state^[40] (falsely often refer to Rose-Vinet equation of state)

p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{2}}}\right)\exp \left[{\frac {3}{2}}\left(K_{0}'-1\right)(1-\eta )\right]

Modified Rydberg equation of state^[41]^[42]^[43] (more reasonable form for strong compression)

p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[{\frac {3}{2}}\left(K_{0}'-3\right)(1-\eta )\right]

Adapted Polynomial equation of state^[44] (second order form = AP2, adapted for extreme compression)

p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[c_{0}(1-\eta )\right]\left\{1+c_{2}\eta (1-\eta )\right\}

with

c_{0}=-\ln \left({\frac {3K_{0}}{p_{\text{FG0}}}}\right)~,~~p_{\text{FG0}}=a_{0}\left({\frac {Z}{V_{0}}}\right)^{\frac {5}{3}}~,~~c_{2}={\frac {3}{2}}\left(K_{0}'-3\right)-c_{0}

where

a_{0}

= 0.02337 GPa.nm⁵. The total number of electrons

Z

in the initial volume

V_{0}

determines the Fermi gas pressure

p_{\text{FG0}}

, which provides for the correct behavior at extreme compression. So far there are no known "simple" solids that require higher order terms.

Adapted polynomial equation of state^[44] (third order form = AP3)

p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[c_{0}(1-\eta )\right]\left\{1+c_{2}\eta (1-\eta )+c_{3}\eta (1-\eta )^{2}\right\}

Johnson–Holmquist equation of state

p(V)={\begin{cases}k_{1}~\xi +k_{2}~\xi ^{2}+k_{3}~\xi ^{3}+\Delta p&\qquad {\text{Compression}}\\k_{1}~\xi &\qquad {\text{Tension}}\end{cases}}~;~~\xi :={\cfrac {V_{0}}{V}}-1

Mie–Grüneisen equation of state (for a more detailed discussion see refs.^[45] ^[46])

p(V)-p_{0}={\frac {\Gamma }{V}}\left(e-e_{0}\right)

Anton-Schmidt equation of state

p(V) = - \beta \left(\frac{V}{V_0}\right)^n \ln\left(\frac{V}{V_0}\right)

where

\beta = K_0

is the bulk modulus at equilibrium volume

V_0

and

n=-{\frac {K'_{0}}{2}}

typically about −2 is often related to the Grüneisen parameter by

n = -\frac{1}{6} - \gamma_G

Equation[edit]

Molecular collisions within a closed container (the propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: as the temperature is increased, the pressure of the propane increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.^[4]

Common forms[edit]

The most frequently introduced forms are:

pV=nRT=nk_{\text{B}}N_{\text{A}}T,

where:

$p$ is the pressure of the gas,
$V$ is the volume of the gas,
$n$ is the amount of substance of gas (also known as number of moles),
$R$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
$k_{\text{B}}$ is the Boltzmann constant
$N_{{A}}$ is the Avogadro constant
$T$ is the absolute temperature of the gas.

In SI units, p is measured in pascals, V is measured in cubic metres, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0.00 K = −273.15 °C, the lowest possible temperature). R has the value 8.314 J/(K⋅mol) ≈ 2 cal/(K⋅mol), or 0.0821 L⋅atm/(mol⋅K).

Molar form[edit]

How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount (n) (in moles) is equal to total mass of the gas (m) (in kilograms) divided by the molar mass (M) (in kilograms per mole):

n={\frac {m}{M}}.

By replacing n with m/M and subsequently introducing density ρ = m/V, we get:

pV={\frac {m}{M}}RT

p={\frac {m}{V}}{\frac {RT}{M}}

p=\rho {\frac {R}{M}}T

Defining the specific gas constant R_specific(r) as the ratio R/M,

p=\rho R_{\text{specific}}T

This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as

pv=R_{\text{specific}}T.

It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In such cases, the universalgas constant is usually given a different symbol such as ${\bar {R}}$ or $R^{*}$ to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.^[5]

Statistical mechanics[edit]

In statistical mechanics the following molecular equation is derived from first principles

P=nk_{\text{B}}T,

where $P$ is the absolute pressure of the gas, $n$ is the number density of the molecules (given by the ratio $n = N / V$ , in contrast to the previous formulation in which $n$ is the number of moles), $T$ is the absolute temperature, and $k B$ is the Boltzmann constant relating temperature and energy, given by:

k_{\text{B}}={\frac {R}{N_{\text{A}}}}

where $N A$ is the Avogadro constant.

From this we notice that for a gas of mass $m$ , with an average particle mass of $μ$ times the atomic mass constant, $m u$ , (i.e., the mass is $μ$ u) the number of molecules will be given by

N={\frac {m}{\mu m_{\text{u}}}},

and since $ρ = m / V = nμm u$ , we find that the ideal gas law can be rewritten as

P={\frac {1}{V}}{\frac {m}{\mu m_{\text{u}}}}k_{\text{B}}T={\frac {k_{\text{B}}}{\mu m_{\text{u}}}}\rho T.

In SI units, $P$ is measured in pascals, $V$ in cubic metres, $T$ in kelvins, and $k B = 1.38 \times 10 -23 J\cdotK -1$ in SI units.

Combined gas law[edit]

Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law save that the number of moles is unspecified, and the ratio of $PV$ to $T$ is simply taken as a constant:^[6]

{\frac {PV}{T}}=k,

where $P$ is the pressure of the gas, $V$ is the volume of the gas, $T$ is the absolute temperature of the gas, and $k$ is a constant. When comparing the same substance under two different sets of conditions, the law can be written as

{\frac {P_{1}V_{1}}{T_{1}}}={\frac {P_{2}V_{2}}{T_{2}}}.

Energy associated with a gas[edit]

According to assumptions of the kinetic theory of ideal gases, we assume that there are no intermolecular attractions between the molecules of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is in the kinetic energy of the molecules of the gas.

E={\frac {3}{2}}nRT

This is the kinetic energy of n moles of a monatomic gas having 3 degrees of freedom; x, y, z.

Energy of gas	Mathematical expression
energy associated with one mole of a monatomic gas	$E={\frac {3}{2}}RT$
energy associated with one gram of a monatomic gas	$E={\frac {3}{2}}rT$
energy associated with one molecule (or atom) of a monatomic gas	$E={\frac {3}{2}}k_{B}T$

Applications to thermodynamic processes[edit]

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.

A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, S, or H) is constant throughout the process.

For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).

In the final three columns, the properties (p, V, or T) at state 2 can be calculated from the properties at state 1 using the equations listed.

Process	Constant	Known ratio or delta	p₂	V₂	T₂
Isobaric process	Pressure	V₂/V₁	p₂ = p₁	V₂ = V₁(V₂/V₁)	T₂ = T₁(V₂/V₁)
Isobaric process	Pressure	T₂/T₁	p₂ = p₁	V₂ = V₁(T₂/T₁)	T₂ = T₁(T₂/T₁)
Isochoric process (Isovolumetric process) (Isometric process)	Volume	p₂/p₁	p₂ = p₁(p₂/p₁)	V₂ = V₁	T₂ = T₁(p₂/p₁)
	Volume	T₂/T₁	p₂ = p₁(T₂/T₁)	V₂ = V₁	T₂ = T₁(T₂/T₁)
Isothermal process	Temperature	p₂/p₁	p₂ = p₁(p₂/p₁)	V₂ = V₁/(p₂/p₁)	T₂ = T₁
Isothermal process	Temperature	V₂/V₁	p₂ = p₁/(V₂/V₁)	V₂ = V₁(V₂/V₁)	T₂ = T₁
Isentropic process (Reversible adiabatic process)	Entropy^[a]	p₂/p₁	p₂ = p₁(p₂/p₁)	V₂ = V₁(p₂/p₁)^(−1/γ)	T₂ = T₁(p₂/p₁)^{(γ − 1)/γ}
		V₂/V₁	p₂ = p₁(V₂/V₁)^−γ	V₂ = V₁(V₂/V₁)	T₂ = T₁(V₂/V₁)^{(1 − γ)}
		T₂/T₁	p₂ = p₁(T₂/T₁)^{γ/(γ − 1)}	V₂ = V₁(T₂/T₁)^{1/(1 − γ)}	T₂ = T₁(T₂/T₁)
Polytropic process	P Vⁿ	p₂/p₁	p₂ = p₁(p₂/p₁)	V₂ = V₁(p₂/p₁)^(−1/n)	T₂ = T₁(p₂/p₁)^{(n − 1)/n}
		V₂/V₁	p₂ = p₁(V₂/V₁)⁻ⁿ	V₂ = V₁(V₂/V₁)	T₂ = T₁(V₂/V₁)^{(1 − n)}
		T₂/T₁	p₂ = p₁(T₂/T₁)^{n/(n − 1)}	V₂ = V₁(T₂/T₁)^{1/(1 − n)}	T₂ = T₁(T₂/T₁)
Isenthalpic process (Irreversible adiabatic process)	Enthalpy^[b]	p₂ − p₁	p₂ = p₁ + (p₂ − p₁)		T₂ = T₁ + μ_JT(p₂ − p₁)
		T₂ − T₁	p₂ = p₁ + (T₂ − T₁)/μ_JT		T₂ = T₁ + (T₂ − T₁)

^{^} a. In an isentropic process, system entropy (S) is constant. Under these conditions, p₁V₁^γ = p₂V₂^γ, where γ is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.

^{^} b. In an isenthalpic process, system enthalpy (H) is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μ_JT for air at room temperature and sea level is 0.22 °C/bar.^[7]

Deviations from ideal behavior of real gases[edit]

The equation of state given here (PV = nRT) applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and inter molecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.

Derivations[edit]

Empirical[edit]

The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.

All the possible gas laws that could have been discovered with this kind of setup are:

PV=C_{1}

P_{1}V_{1}=P_{2}V_{2}

(1) known as Boyle's law

{\frac {V}{T}}=C_{2}

{\frac {V_{1}}{T_{1}}}={\frac {V_{2}}{T_{2}}}

(2) known as Charles's law

{\frac {V}{N}}=C_{3}

{\frac {V_{1}}{N_{1}}}={\frac {V_{2}}{N_{2}}}

(3) known as Avogadro's law

{\frac {P}{T}}=C_{4}

{\frac {P_{1}}{T_{1}}}={\frac {P_{2}}{T_{2}}}

(4) known as Gay-Lussac's law

NT=C_{5}

N_{1}T_{1}=N_{2}T_{2}

(5)

{\frac {P}{N}}=C_{6}

{\frac {P_{1}}{N_{1}}}={\frac {P_{2}}{N_{2}}}

(6)

where P stands for pressure, V for volume, N for number of particles in the gas and T for temperature; where $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ are not actual constants but are in this context because of each equation requiring only the parameters explicitly noted in it changing.

To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.

Since each formula only holds when only the state variables involved in said formula change while the others remain constant, we cannot simply use algebra and directly combine them all. I.e. Boyle did his experiments while keeping N and Tconstant and this must be taken into account.

Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time. The derivation using 4 formulas can look like this:

at first the gas has parameters $P_{1},V_{1},N_{1},T_{1}$

Say, starting to change only pressure and volume, according to Boyle's law, then:

P_{1}V_{1}=P_{2}V_{2}

(7)

After this process, the gas has parameters $P_{2},V_{2},N_{1},T_{1}$

Using then equation (5) to change the number of particles in the gas and the temperature,

N_{1}T_{1}=N_{2}T_{2}

(8)

After this process, the gas has parameters $P_{2},V_{2},N_{2},T_{2}$

Using then equation (6) to change the pressure and the number of particles,

{\frac {P_{2}}{N_{2}}}={\frac {P_{3}}{N_{3}}}

(9)

After this process, the gas has parameters $P_{3},V_{2},N_{3},T_{2}$

Using then Charles's law to change the volume and temperature of the gas,

{\frac {V_{2}}{T_{2}}}={\frac {V_{3}}{T_{3}}}

(10)

After this process, the gas has parameters $P_{3},V_{3},N_{3},T_{3}$

Using simple algebra on equations (7), (8), (9) and (10) yields the result:

{\frac {P_{1}V_{1}}{N_{1}T_{1}}}={\frac {P_{3}V_{3}}{N_{3}T_{3}}}

{\frac {PV}{NT}}=k_{\text{B}},

where $k_{\text{B}}$ stands for Boltzmann's constant.

Another equivalent result, using the fact that $nR=Nk_{\text{B}}$ , where n is the number of moles in the gas and R is the universal gas constant, is:

PV=nRT,

which is known as the ideal gas law.

If three of the six equations are known, it may be possible to derive the remaining three using the same method. However, because each formula has two variables, this is possible only for certain groups of three. For example, if you were to have equations (1), (2) and (4) you would not be able to get any more because combining any two of them will only give you the third. However, if you had equations (1), (2) and (3) you would be able to get all six equations because combining (1) and (2) will yield (4), then (1) and (3) will yield (6), then (4) and (6) will yield (5), as well as would the combination of (2) and (3) as is explained in the following visual relation:

Relationship between the 6 gas laws

where the numbers represent the gas laws numbered above.

If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third.

For example:

Change only pressure and volume first:

P_{1}V_{1}=P_{2}V_{2}

(1')

then only volume and temperature:

{\frac {V_{2}}{T_{1}}}={\frac {V_{3}}{T_{2}}}

(2')

then as we can choose any value for $V_{3}$ , if we set $V_{1}=V_{3}$ , equation (2') becomes:

{\frac {V_{2}}{T_{1}}}={\frac {V_{1}}{T_{2}}}

(3')

combining equations (1') and (3') yields ${\frac {P_{1}}{T_{1}}}={\frac {P_{2}}{T_{2}}}$ , which is equation (4), of which we had no prior knowledge until this derivation.

Theoretical[edit]

Kinetic theory[edit]

The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

The fundamental assumptions of the kinetic theory of gases imply that

PV={\frac {1}{3}}Nmv_{\text{rms}}^{2}.

Using the Maxwell–Boltzmann distribution, the fraction of molecules that have a speed in the range $v$ to $v+dv$ is $f(v)\,dv$ , where

f(v)=4\pi \left({\frac {m}{2\pi kT}}\right)^{\!{\frac {3}{2}}}v^{2}e^{-{\frac {mv^{2}}{2kT}}}

and $k$ denotes the Boltzmann constant. The root-mean-square speed can be calculated by

v_{\text{rms}}^{2}=\int _{0}^{\infty }v^{2}f(v)\,dv=4\pi \left({\frac {m}{2\pi kT}}\right)^{\frac {3}{2}}\int _{0}^{\infty }v^{4}e^{-{\frac {mv^{2}}{2kT}}}\,dv.

Using the integration formula

\int _{0}^{\infty }x^{2n}e^{-{\frac {x^{2}}{a^{2}}}}\,dx={\sqrt {\pi }}\,{\frac {(2n)!}{n!}}\left({\frac {a}{2}}\right)^{2n+1},

it follows that

v_{\text{rms}}^{2}=4\pi \left({\frac {m}{2\pi kT}}\right)^{\!{\frac {3}{2}}}{\sqrt {\pi }}\,{\frac {4!}{2!}}\left({\frac {\sqrt {\frac {2kT}{m}}}{2}}\right)^{\!5}={\frac {3kT}{m}},

from which we get the ideal gas law:

PV={\frac {1}{3}}Nm\left({\frac {3kT}{m}}\right)=NkT.

Statistical mechanics[edit]

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time-averaged kinetic energy of the particle is:

{\begin{aligned}\langle \mathbf {q} \cdot \mathbf {F} \rangle &=\left\langle q_{x}{\frac {dp_{x}}{dt}}\right\rangle +\left\langle q_{y}{\frac {dp_{y}}{dt}}\right\rangle +\left\langle q_{z}{\frac {dp_{z}}{dt}}\right\rangle \\&=-\left\langle q_{x}{\frac {\partial H}{\partial q_{x}}}\right\rangle -\left\langle q_{y}{\frac {\partial H}{\partial q_{y}}}\right\rangle -\left\langle q_{z}{\frac {\partial H}{\partial q_{z}}}\right\rangle =-3k_{\text{B}}T,\end{aligned}}

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of Nparticles yields

3Nk_{B}T=-\left\langle \sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}\right\rangle .

By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure P of the gas. Hence

-\left\langle \sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}\right\rangle =P\oint _{\text{surface}}\mathbf {q} \cdot d\mathbf {S} ,

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

\nabla \cdot \mathbf {q} ={\frac {\partial q_{x}}{\partial q_{x}}}+{\frac {\partial q_{y}}{\partial q_{y}}}+{\frac {\partial q_{z}}{\partial q_{z}}}=3,

the divergence theorem implies that

P\oint _{\text{surface}}\mathbf {q} \cdot d\mathbf {S} =P\int _{\text{volume}}\left(\nabla \cdot \mathbf {q} \right)dV=3PV,

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

3Nk_{\text{B}}T=-\left\langle \sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}\right\rangle =3PV,

which immediately implies the ideal gas law for N particles:

PV=Nk_{B}T=nRT,

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.

Other dimensions[edit]

For a d-dimensional system, the ideal gas pressure is:^[8]

P^{(d)}={\frac {Nk_{B}T}{L^{d}}},

where $L^{d}$ is the volume of the d-dimensional domain in which the gas exists. Note that the dimensions of the pressure changes with dimensionality.

Examples[edit]

In mechanical systems, the position coordinates and velocities of mechanical parts are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.
In thermodynamics, a state variable is an independent variable of a state function like internal energy, enthalpy, and entropy. Examples include temperature, pressure, and volume. Heat and work are not state functions, but process functions.
In electronic/electrical circuits, the voltages of the nodes and the currents through components in the circuit are usually the state variables. In any electrical circuit, the number of state variables are equal to the number of storage elements, which are inductors and capacitors. The state variable for an inductor is the current through the inductor, while that for a capacitor is the voltage across the capacitor.
In ecosystem models, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.

Control systems engineering[edit]

In control engineering and other areas of science and engineering, state variables are used to represent the states of a general system. The set of possible combinations of state variable values is called the state space of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As shown below, the state equations and output equations for a linear time invariant system can be expressed using coefficient matrices: A, B, C, and D

A\in \mathbb {R} ^{N\times N},\quad B\in \mathbb {R} ^{N\times L},\quad C\in \mathbb {R} ^{M\times N},\quad D\in \mathbb {R} ^{M\times L},

where N, L and M are the dimensions of the vectors describing the state, input and output, respectively.

Discrete-time systems[edit]

The state vector (vector of state variables) representing the current state of a discrete-time system (i.e. digital system) is $x[n]$ , where n is the discrete point in time at which the system is being evaluated. The discrete-time state equations are

x[n+1]=Ax[n]+Bu[n],

which describes the next state of the system (x[n+1]) with respect to current state and inputs u[n] of the system. The output equations are

y[n]=Cx[n]+Du[n],

which describes the output y[n] with respect to current states and inputs u[n] to the system.

Continuous time systems[edit]

The state vector representing the current state of a continuous-time system (i.e. analog system) is $x(t)$ , and the continuous-time state equations giving the evolution of the state vector are

{\frac {dx(t)}{dt}}=Ax(t)+Bu(t),

which describes the continuous rate of change ${\textstyle {\frac {dx(t)}{dt}}}$ of the state of the system with respect to current state x(t) and inputs u(t) of the system. The output equations are

y(t)=Cx(t)+Du(t),

which describes the output y(t) with respect to current states x(t) and inputs u(t) to the system.

Liquid–vapor critical point[edit]

Overview[edit]

The liquid–vapor critical point in a pressure–temperature phase diagram is at the high-temperature extreme of the liquid–gas phase boundary. The dotted green line shows the anomalous behavior of water.

For simplicity and clarity, the generic notion of critical point is best introduced by discussing a specific example, the vapor-liquid critical point. This was the first critical point to be discovered, and it is still the best known and most studied one.

The figure to the right shows the schematic PT diagram of a pure substance (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). The commonly known phases solid, liquid and vapor are separated by phase boundaries, i.e. pressure–temperature combinations where two phases can coexist. At the triple point, all three phases can coexist. However, the liquid–vapor boundary terminates in an endpoint at some critical temperature T_c and critical pressure p_c. This is the critical point.

In water, the critical point occurs at 647.096 K (373.946 °C; 705.103 °F) and 22.064 megapascals (3,200.1 psi; 217.75 atm).^[2]

In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming ever more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, all these properties change into the exact opposite: water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and prefers to mix with nonpolar gases and organic molecules.^[3]

At the critical point, only one phase exists. The heat of vaporization is zero. There is a stationary inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point:^[4]^[5]^[6]

\left({\frac {\partial p}{\partial V}}\right)_{T}=0,

\left({\frac {\partial ^{2}p}{\partial V^{2}}}\right)_{T}=0.

The critical isotherm with the critical point K

Above the critical point there exists a state of matter that is continuously connected with (can be transformed without phase transition into) both the liquid and the gaseous state. It is called supercritical fluid. The common textbook knowledge that all distinction between liquid and vapor disappears beyond the critical point has been challenged by Fisher and Widom,^[7] who identified a p–T line that separates states with different asymptotic statistical properties (Fisher–Widom line).

Sometimes^[ambiguous] the critical point does not manifest in most thermodynamic or mechanical properties, but is "hidden" and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets, and a sudden enhancement in defect pair concentration.^[8]

History[edit]

Critical carbon dioxide exuding fogwhile cooling from supercritical to critical temperature.

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822^[9]^[10] and named by Dmitri Mendeleev in 1860^[11]^[12] and Thomas Andrews in 1869.^[13] Cagniard showed that CO₂ could be liquefied at 31 °C at a pressure of 73 atm, but not at a slightly higher temperature, even under pressures as high as 3000 atm.

Theory[edit]

Solving the above condition $(\partial p/\partial V)_{T}=0$ for the van der Waals equation, one can compute the critical point as

T_{\text{c}}={\frac {8a}{27Rb}},\quad V_{\text{c}}=3nb,\quad p_{\text{c}}={\frac {a}{27b^{2}}}.

However, the van der Waals equation, based on a mean-field theory, does not hold near the critical point. In particular, it predicts wrong scaling laws.

To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties^[14]

T_{\text{r}}={\frac {T}{T_{\text{c}}}},\quad p_{\text{r}}={\frac {p}{p_{\text{c}}}},\quad V_{\text{r}}={\frac {V}{RT_{\text{c}}/p_{\text{c}}}}.

The principle of corresponding states indicates that substances at equal reduced pressures and temperatures have equal reduced volumes. This relationship is approximately true for many substances, but becomes increasingly inaccurate for large values of p_r.

For some gases, there is an additional correction factor, called Newton's correction, added to the critical temperature and critical pressure calculated in this manner. These are empirically derived values and vary with the pressure range of interest.^[15]

Table of liquid–vapor critical temperature and pressure for selected substances[edit]

Substance^[16]^[17]	Critical temperature	Critical pressure (absolute)
Argon	−122.4 °C (150.8 K)	48.1 atm (4,870 kPa)
Ammonia (NH₃)^[18]	132.4 °C (405.5 K)	111.3 atm (11,280 kPa)
R-134a	101.06 °C (374.21 K)	40.06 atm (4,059 kPa)
R-410A	72.8 °C (345.9 K)	47.08 atm (4,770 kPa)
Bromine	310.8 °C (584.0 K)	102 atm (10,300 kPa)
Caesium	1,664.85 °C (1,938.00 K)	94 atm (9,500 kPa)
Chlorine	143.8 °C (416.9 K)	76.0 atm (7,700 kPa)
Ethanol (C₂H₅OH)	241 °C (514 K)	62.18 atm (6,300 kPa)
Fluorine	−128.85 °C (144.30 K)	51.5 atm (5,220 kPa)
Helium	−267.96 °C (5.19 K)	2.24 atm (227 kPa)
Hydrogen	−239.95 °C (33.20 K)	12.8 atm (1,300 kPa)
Krypton	−63.8 °C (209.3 K)	54.3 atm (5,500 kPa)
Methane (CH₄)	−82.3 °C (190.8 K)	45.79 atm (4,640 kPa)
Neon	−228.75 °C (44.40 K)	27.2 atm (2,760 kPa)
Nitrogen	−146.9 °C (126.2 K)	33.5 atm (3,390 kPa)
Oxygen (O₂)	−118.6 °C (154.6 K)	49.8 atm (5,050 kPa)
Carbon dioxide (CO₂)	31.04 °C (304.19 K)	72.8 atm (7,380 kPa)
Nitrous oxide (N₂O)	36.4 °C (309.5 K)	71.5 atm (7,240 kPa)
Sulfuric acid (H₂SO₄)	654 °C (927 K)	45.4 atm (4,600 kPa)
Xenon	16.6 °C (289.8 K)	57.6 atm (5,840 kPa)
Lithium	2,950 °C (3,220 K)	652 atm (66,100 kPa)
Mercury	1,476.9 °C (1,750.1 K)	1,720 atm (174,000 kPa)
Sulfur	1,040.85 °C (1,314.00 K)	207 atm (21,000 kPa)
Iron	8,227 °C (8,500 K)
Gold	6,977 °C (7,250 K)	5,000 atm (510,000 kPa)
Aluminium	7,577 °C (7,850 K)
Water (H₂O)^[2]^[19]	373.946 °C (647.096 K)	217.7 atm (22,060 kPa)

Mixtures: liquid–liquid critical point[edit]

A plot of typical polymer solution phase behavior including two critical points: a LCST and an UCST

The liquid–liquid critical point of a solution, which occurs at the critical solution temperature, occurs at the limit of the two-phase region of the phase diagram. In other words, it is the point at which an infinitesimal change in some thermodynamic variable (such as temperature or pressure) leads to separation of the mixture into two distinct liquid phases, as shown in the polymer–solvent phase diagram to the right. Two types of liquid–liquid critical points are the upper critical solution temperature(UCST), which is the hottest point at which cooling induces phase separation, and the lower critical solution temperature (LCST), which is the coldest point at which heating induces phase separation.

Mathematical definition[edit]

From a theoretical standpoint, the liquid–liquid critical point represents the temperature–concentration extremum of the spinodal curve (as can be seen in the figure to the right). Thus, the liquid–liquid critical point in a two-component system must satisfy two conditions: the condition of the spinodal curve (the second derivative of the free energy with respect to concentration must equal zero), and the extremum condition (the third derivative of the free energy with respect to concentration must also equal zero or the derivative of the spinodal temperature with respect to concentration must equal zero).

External links[edit]

"Critical points for some common solvents". ProSciTech. Archived from the original on 2008-01-31.
"Critical Temperature and Pressure". Department of Chemistry. Purdue University. Retrieved 2006-12-03.

https://en.wikipedia.org/wiki/Compressibility_factor

Supercritical fluid extraction (SFE) is the process of separating one component (the extractant) from another (the matrix) using supercritical fluids as the extracting solvent. Extraction is usually from a solid matrix, but can also be from liquids. SFE can be used as a sample preparation step for analytical purposes, or on a larger scale to either strip unwanted material from a product (e.g. decaffeination) or collect a desired product (e.g. essential oils). These essential oils can include limonene and other straight solvents. Carbon dioxide (CO₂) is the most used supercritical fluid, sometimes modified by co-solvents such as ethanol or methanol. Extraction conditions for supercritical carbon dioxide are above the critical temperature of 31 °C and critical pressure of 74 bar. Addition of modifiers may slightly alter this. The discussion below will mainly refer to extraction with CO₂, except where specified.

Advantages[edit]

Selectivity[edit]

The properties of the supercritical fluid can be altered by varying the pressure and temperature, allowing selective extraction. For example, volatile oils can be extracted from a plant with low pressures (100 bar), whereas liquid extraction would also remove lipids. Lipids can be removed using pure CO₂ at higher pressures, and then phospholipids can be removed by adding ethanol to the solvent.^[1] The same principle can be used to extract polyphenols and unsaturated fatty acids separately from wine wastes.^[2]

Speed[edit]

Extraction is a diffusion-based process, in which the solvent is required to diffuse into the matrix and the extracted material to diffuse out of the matrix into the solvent. Diffusivities are much faster in supercritical fluids than in liquids, and therefore extraction can occur faster. In addition, due to the lack of surface tensionand negligible viscosities compared to liquids, the solvent can penetrate more into the matrix inaccessible to liquids. An extraction using an organic liquid may take several hours, whereas supercritical fluid extraction can be completed in 10 to 60 minutes.^[3]

Limitations[edit]

The requirement for high pressures increases the cost compared to conventional liquid extraction, so SFE will only be used where there are significant advantages. Carbon dioxide itself is non-polar, and has somewhat limited dissolving power, so cannot always be used as a solvent on its own, particularly for polar solutes. The use of modifiers increases the range of materials which can be extracted. Food grade modifiers such as ethanol can often be used, and can also help in the collection of the extracted material, but reduces some of the benefits of using a solvent which is gaseous at room temperature.

Procedure[edit]

The system must contain a pump for the CO₂, a pressure cell to contain the sample, a means of maintaining pressure in the system and a collecting vessel. The liquid is pumped to a heating zone, where it is heated to supercritical conditions. It then passes into the extraction vessel, where it rapidly diffuses into the solid matrix and dissolves the material to be extracted. The dissolved material is swept from the extraction cell into a separator at lower pressure, and the extracted material settles out. The CO₂ can then be cooled, re-compressed and recycled, or discharged to atmosphere.

Figure 1. Schematic diagram of SFE apparatus

Pumps[edit]

Carbon dioxide (CO
2) is usually pumped as a liquid, usually below 5 °C (41 °F) and a pressure of about 50 bar. The solvent is pumped as a liquid as it is then almost incompressible; if it were pumped as a supercritical fluid, much of the pump stroke would be "used up" in compressing the fluid, rather than pumping it. For small scale extractions (up to a few grams / minute), reciprocating CO
2 pumps or syringe pumps are often used. For larger scale extractions, diaphragm pumps are most common. The pump heads will usually require cooling, and the CO₂ will also be cooled before entering the pump.

Pressure vessels[edit]

Pressure vessels can range from simple tubing to more sophisticated purpose built vessels with quick release fittings. The pressure requirement is at least 74 bar, and most extractions are conducted at under 350 bar. However, sometimes higher pressures will be needed, such as extraction of vegetable oils, where pressures of 800 bar are sometimes required for complete miscibility of the two phases.^[4]

The vessel must be equipped with a means of heating. It can be placed inside an oven for small vessels, or an oil or electrically heated jacket for larger vessels. Care must be taken if rubber seals are used on the vessel, as the supercritical carbon dioxide may dissolve in the rubber, causing swelling, and the rubber will rupture on depressurization.^{[citation needed]}

Pressure maintenance[edit]

The pressure in the system must be maintained from the pump right through the pressure vessel. In smaller systems (up to about 10 mL / min) a simple restrictor can be used. This can be either a capillary tube cut to length, or a needle valve which can be adjusted to maintain pressure at different flow rates. In larger systems a back pressure regulator will be used, which maintains pressure upstream of the regulator by means of a spring, compressed air, or electronically driven valve. Whichever is used, heating must be supplied, as the adiabatic expansion of the CO₂ results in significant cooling. This is problematic if water or other extracted material is present in the sample, as this may freeze in the restrictor or valve and cause blockages.

Collection[edit]

The supercritical solvent is passed into a vessel at lower pressure than the extraction vessel. The density, and hence dissolving power, of supercritical fluids varies sharply with pressure, and hence the solubility in the lower density CO₂ is much lower, and the material precipitates for collection. It is possible to fractionate the dissolved material using a series of vessels at reducing pressure. The CO₂ can be recycled or depressurized to atmospheric pressure and vented. For analytical SFE, the pressure is usually dropped to atmospheric, and the now gaseous carbon dioxide bubbled through a solvent to trap the precipitated components.

Heating and cooling[edit]

This is an important aspect. The fluid is cooled before pumping to maintain liquid conditions, then heated after pressurization. As the fluid is expanded into the separator, heat must be provided to prevent excessive cooling. For small scale extractions, such as for analytical purposes, it is usually sufficient to pre-heat the fluid in a length of tubing inside the oven containing the extraction cell. The restrictor can be electrically heated, or even heated with a hairdryer. For larger systems, the energy required during each stage of the process can be calculated using the thermodynamic properties of the supercritical fluid.^[5]

Simple model of SFE[edit]

Figure 2. Concentration profiles during a typical SFE extraction

There are two essential steps to SFE, transport (by diffusion or otherwise) of the solid particles to the surface, and dissolution in the supercritical fluid. Other factors, such as diffusion into the particle by the SF and reversible release such as desorption from an active site are sometimes significant, but not dealt with in detail here. Figure 2 shows the stages during extraction from a spherical particle where at the start of the extraction the level of extractant is equal across the whole sphere (Fig. 2a). As extraction commences, material is initially extracted from the edge of the sphere, and the concentration in the center is unchanged (Fig 2b). As the extraction progresses, the concentration in the center of the sphere drops as the extractant diffuses towards the edge of the sphere (Figure 2c).^[6]

Figure 3. Concentration profiles for (a) diffusion limited and (b) solubility limited extraction

The relative rates of diffusion and dissolution are illustrated by two extreme cases in Figure 3. Figure 3a shows a case where dissolution is fast relative to diffusion. The material is carried away from the edge faster than it can diffuse from the center, so the concentration at the edge drops to zero. The material is carried away as fast as it arrives at the surface, and the extraction is completely diffusion limited. Here the rate of extraction can be increased by increasing diffusion rate, for example raising the temperature, but not by increasing the flow rate of the solvent. Figure 3b shows a case where solubility is low relative to diffusion. The extractant is able to diffuse to the edge faster than it can be carried away by the solvent, and the concentration profile is flat. In this case, the extraction rate can be increased by increasing the rate of dissolution, for example by increasing flow rate of the solvent.

Figure 4. Extraction Profile for Different Types of Extraction

The extraction curve of % recovery against time can be used to elucidate the type of extraction occurring. Figure 4(a) shows a typical diffusion controlled curve. The extraction is initially rapid, until the concentration at the surface drops to zero, and the rate then becomes much slower. The % extracted eventually approaches 100%. Figure 4(b) shows a curve for a solubility limited extraction. The extraction rate is almost constant, and only flattens off towards the end of the extraction. Figure 4(c) shows a curve where there are significant matrix effects, where there is some sort of reversible interaction with the matrix, such as desorption from an active site. The recovery flattens off, and if the 100% value is not known, then it is hard to tell that extraction is less than complete.

Optimization[edit]

The optimum will depend on the purpose of the extraction. For an analytical extraction to determine, say, antioxidant content of a polymer, then the essential factors are complete extraction in the shortest time. However, for production of an essential oil extract from a plant, then quantity of CO₂ used will be a significant cost, and "complete" extraction not required, a yield of 70 - 80% perhaps being sufficient to provide economic returns. In another case, the selectivity may be more important, and a reduced rate of extraction will be preferable if it provides greater discrimination. Therefore, few comments can be made which are universally applicable. However, some general principles are outlined below.

Maximizing diffusion[edit]

This can be achieved by increasing the temperature, swelling the matrix, or reducing the particle size. Matrix swelling can sometimes be increased by increasing the pressure of the solvent, and by adding modifiers to the solvent. Some polymers and elastomers in particular are swelled dramatically by CO₂, with diffusion being increased by several orders of magnitude in some cases.^[7]

Maximizing solubility[edit]

Generally, higher pressure will increase solubility. The effect of temperature is less certain, as close to the critical point, increasing the temperature causes decreases in density, and hence dissolving power. At pressures well above the critical pressure, solubility is likely to increase with temperature.^[8] Addition of low levels of modifiers (sometimes called entrainers), such as methanol and ethanol, can also significantly increase solubility, particularly of more polar compounds.

Optimizing flow rate[edit]

The flow rate of supercritical carbon dioxide should be measured in terms of mass flow rather than by volume because the density of the CO
2 changes according to the temperature both before entering the pump heads and during compression. Coriolis flow meters are best used to achieve such flow confirmation. To maximize the rate of extraction, the flow rate should be high enough for the extraction to be completely diffusion limited (but this will be very wasteful of solvent). However, to minimize the amount of solvent used, the extraction should be completely solubility limited (which will take a very long time). Flow rate must therefore be determined depending on the competing factors of time and solvent costs, and also capital costs of pumps, heaters and heat exchangers. The optimum flow rate will probably be somewhere in the region where both solubility and diffusion are significant factors.

Non-classical states

Glass

Atoms of Si and O; each atom has the same number of bonds, but the overall arrangement of the atoms is random.

Schematic representation of a random-network glassy form (left) and ordered crystalline lattice (right) of identical chemical composition.

Glass is a non-crystalline or amorphous solid material that exhibits a glass transition when heated towards the liquid state. Glasses can be made of quite different classes of materials: inorganic networks (such as window glass, made of silicate plus additives), metallic alloys, ionic melts, aqueous solutions, molecular liquids, and polymers. Thermodynamically, a glass is in a metastable state with respect to its crystalline counterpart. The conversion rate, however, is practically zero.

Crystals with some degree of disorder

A plastic crystal is a molecular solid with long-range positional order but with constituent molecules retaining rotational freedom; in an orientational glass this degree of freedom is frozen in a quenched disordered state.

Similarly, in a spin glass magnetic disorder is frozen.

Liquid crystal states

Liquid crystal states have properties intermediate between mobile liquids and ordered solids. Generally, they are able to flow like a liquid, but exhibiting long-range order. For example, the nematic phase consists of long rod-like molecules such as para-azoxyanisole, which is nematic in the temperature range 118–136 °C (244–277 °F).^[7] In this state the molecules flow as in a liquid, but they all point in the same direction (within each domain) and cannot rotate freely. Like a crystalline solid, but unlike a liquid, liquid crystals react to polarized light.

Other types of liquid crystals are described in the main article on these states. Several types have technological importance, for example, in liquid crystal displays.

Magnetically ordered

Transition metal atoms often have magnetic moments due to the net spin of electrons that remain unpaired and do not form chemical bonds. In some solids the magnetic moments on different atoms are ordered and can form a ferromagnet, an antiferromagnet or a ferrimagnet.

In a ferromagnet—for instance, solid iron—the magnetic moment on each atom is aligned in the same direction (within a magnetic domain). If the domains are also aligned, the solid is a permanent magnet, which is magnetic even in the absence of an external magnetic field. The magnetization disappears when the magnet is heated to the Curie point, which for iron is 768 °C (1,414 °F).

An antiferromagnet has two networks of equal and opposite magnetic moments, which cancel each other out so that the net magnetization is zero. For example, in nickel(II) oxide (NiO), half the nickel atoms have moments aligned in one direction and half in the opposite direction.

In a ferrimagnet, the two networks of magnetic moments are opposite but unequal, so that cancellation is incomplete and there is a non-zero net magnetization. An example is magnetite (Fe₃O₄), which contains Fe²⁺ and Fe³⁺ ions with different magnetic moments.

A quantum spin liquid (QSL) is a disordered state in a system of interacting quantum spins which preserves its disorder to very low temperatures, unlike other disordered states. It is not a liquid in physical sense, but a solid whose magnetic order is inherently disordered. The name "liquid" is due to an analogy with the molecular disorder in a conventional liquid. A QSL is neither a ferromagnet, where magnetic domains are parallel, nor an antiferromagnet, where the magnetic domains are antiparallel; instead, the magnetic domains are randomly oriented. This can be realized e.g. by geometrically frustrated magnetic moments that cannot point uniformly parallel or antiparallel. When cooling down and settling to a state, the domain must "choose" an orientation, but if the possible states are similar in energy, one will be chosen randomly. Consequently, despite strong short-range order, there is no long-range magnetic order.

Microphase-separated

SBS block copolymer in TEM

Copolymers can undergo microphase separation to form a diverse array of periodic nanostructures, as shown in the example of the styrene-butadiene-styrene block copolymer shown at right. Microphase separation can be understood by analogy to the phase separation between oil and water. Due to chemical incompatibility between the blocks, block copolymers undergo a similar phase separation. However, because the blocks are covalently bonded to each other, they cannot demix macroscopically as water and oil can, and so instead the blocks form nanometre-sized structures. Depending on the relative lengths of each block and the overall block topology of the polymer, many morphologies can be obtained, each its own phase of matter.

Ionic liquids also display microphase separation. The anion and cation are not necessarily compatible and would demix otherwise, but electric charge attraction prevents them from separating. Their anions and cations appear to diffuse within compartmentalized layers or micelles instead of freely as in a uniform liquid.^[8]

Low-temperature states

Superconductor

Superconductors are materials which have zero electrical resistivity, and therefore perfect conductivity. This is a distinct physical state which exists at low temperature, and the resistivity increases discontinuously to a finite value at a sharply-defined transition temperature for each superconductor.^[9]

A superconductor also excludes all magnetic fields from its interior, a phenomenon known as the Meissner effect or perfect diamagnetism.^[9] Superconducting magnets are used as electromagnets in magnetic resonance imaging machines.

The phenomenon of superconductivity was discovered in 1911, and for 75 years was only known in some metals and metallic alloys at temperatures below 30 K. In 1986 so-called high-temperature superconductivity was discovered in certain ceramic oxides, and has now been observed in temperatures as high as 164 K.^[10]

Superfluid

Liquid helium in a superfluid phase creeps up on the walls of the cup in a Rollin film, eventually dripping out from the cup.

Close to absolute zero, some liquids form a second liquid state described as superfluid because it has zero viscosity (or infinite fluidity; i.e., flowing without friction). This was discovered in 1937 for helium, which forms a superfluid below the lambda temperature of 2.17 K (−270.98 °C; −455.76 °F). In this state it will attempt to "climb" out of its container.^[11] It also has infinite thermal conductivity so that no temperature gradient can form in a superfluid. Placing a superfluid in a spinning container will result in quantized vortices.

These properties are explained by the theory that the common isotope helium-4 forms a Bose–Einstein condensate (see next section) in the superfluid state. More recently, Fermionic condensate superfluids have been formed at even lower temperatures by the rare isotope helium-3 and by lithium-6.^[12]

Bose–Einstein condensate

Velocity in a gas of rubidium as it is cooled: the starting material is on the left, and Bose–Einstein condensate is on the right.

In 1924, Albert Einstein and Satyendra Nath Bose predicted the "Bose–Einstein condensate" (BEC), sometimes referred to as the fifth state of matter. In a BEC, matter stops behaving as independent particles, and collapses into a single quantum state that can be described with a single, uniform wavefunction.

In the gas phase, the Bose–Einstein condensate remained an unverified theoretical prediction for many years. In 1995, the research groups of Eric Cornell and Carl Wieman, of JILA at the University of Colorado at Boulder, produced the first such condensate experimentally. A Bose–Einstein condensate is "colder" than a solid. It may occur when atoms have very similar (or the same) quantum levels, at temperatures very close to absolute zero, −273.15 °C (−459.67 °F).

Fermionic condensate

A fermionic condensate is similar to the Bose–Einstein condensate but composed of fermions. The Pauli exclusion principle prevents fermions from entering the same quantum state, but a pair of fermions can behave as a boson, and multiple such pairs can then enter the same quantum state without restriction.

Rydberg molecule

One of the metastable states of strongly non-ideal plasma are condensates of excited atoms. These atoms can also turn into ions and electrons if they reach a certain temperature. In April 2009, Nature reported the creation of Rydberg molecules from a Rydberg atom and a ground state atom,^[13] confirming that such a state of matter could exist.^[14] The experiment was performed using ultracold rubidium atoms.

Quantum Hall state

A quantum Hall state gives rise to quantized Hall voltage measured in the direction perpendicular to the current flow. A quantum spin Hall state is a theoretical phase that may pave the way for the development of electronic devices that dissipate less energy and generate less heat. This is a derivation of the Quantum Hall state of matter.

Photonic matter

Photonic matter is a phenomenon where photons interacting with a gas develop apparent mass, and can interact with each other, even forming photonic "molecules". The source of mass is the gas, which is massive. This is in contrast to photons moving in empty space, which have no rest mass, and cannot interact.

Dropleton

A "quantum fog" of electrons and holes that flow around each other and even ripple like a liquid, rather than existing as discrete pairs.^[15]

High-energy states

Degenerate matter

Under extremely high pressure, as in the cores of dead stars, ordinary matter undergoes a transition to a series of exotic states of matter collectively known as degenerate matter, which are supported mainly by quantum mechanical effects. In physics, "degenerate" refers to two states that have the same energy and are thus interchangeable. Degenerate matter is supported by the Pauli exclusion principle, which prevents two fermionic particles from occupying the same quantum state. Unlike regular plasma, degenerate plasma expands little when heated, because there are simply no momentum states left. Consequently, degenerate stars collapse into very high densities. More massive degenerate stars are smaller, because the gravitational force increases, but pressure does not increase proportionally.

Electron-degenerate matter is found inside white dwarf stars. Electrons remain bound to atoms but are able to transfer to adjacent atoms. Neutron-degenerate matter is found in neutron stars. Vast gravitational pressure compresses atoms so strongly that the electrons are forced to combine with protons via inverse beta-decay, resulting in a superdense conglomeration of neutrons. Normally free neutrons outside an atomic nucleus will decay with a half life of approximately 10 minutes, but in a neutron star, the decay is overtaken by inverse decay. Cold degenerate matter is also present in planets such as Jupiter and in the even more massive brown dwarfs, which are expected to have a core with metallic hydrogen. Because of the degeneracy, more massive brown dwarfs are not significantly larger. In metals, the electrons can be modeled as a degenerate gas moving in a lattice of non-degenerate positive ions.

Quark matter

In regular cold matter, quarks, fundamental particles of nuclear matter, are confined by the strong force into hadrons that consist of 2–4 quarks, such as protons and neutrons. Quark matter or quantum chromodynamical (QCD) matter is a group of phases where the strong force is overcome and quarks are deconfined and free to move. Quark matter phases occur at extremely high densities or temperatures, and there are no known ways to produce them in equilibrium in the laboratory; in ordinary conditions, any quark matter formed immediately undergoes radioactive decay.

Strange matter is a type of quark matter that is suspected to exist inside some neutron stars close to the Tolman–Oppenheimer–Volkoff limit (approximately 2–3 solar masses), although there is no direct evidence of its existence. In strange matter, part of the energy available manifests as strange quarks, a heavier analogue of the common down quark. It may be stable at lower energy states once formed, although this is not known.

Quark–gluon plasma is a very high-temperature phase in which quarks become free and able to move independently, rather than being perpetually bound into particles, in a sea of gluons, subatomic particles that transmit the strong force that binds quarks together. This is analogous to the liberation of electrons from atoms in a plasma. This state is briefly attainable in extremely high-energy heavy ion collisions in particle accelerators, and allows scientists to observe the properties of individual quarks, and not just theorize. Quark–gluon plasma was discovered at CERN in 2000. Unlike plasma, which flows like a gas, interactions within QGP are strong and it flows like a liquid.

At high densities but relatively low temperatures, quarks are theorized to form a quark liquid whose nature is presently unknown. It forms a distinct color-flavor locked (CFL) phase at even higher densities. This phase is superconductive for color charge. These phases may occur in neutron stars but they are presently theoretical.

Color-glass condensate

Color-glass condensate is a type of matter theorized to exist in atomic nuclei traveling near the speed of light. According to Einstein's theory of relativity, a high-energy nucleus appears length contracted, or compressed, along its direction of motion. As a result, the gluons inside the nucleus appear to a stationary observer as a "gluonic wall" traveling near the speed of light. At very high energies, the density of the gluons in this wall is seen to increase greatly. Unlike the quark–gluon plasma produced in the collision of such walls, the color-glass condensate describes the walls themselves, and is an intrinsic property of the particles that can only be observed under high-energy conditions such as those at RHIC and possibly at the Large Hadron Collider as well.

Very high energy states

Various theories predict new states of matter at very high energies. An unknown state has created the baryon asymmetry in the universe, but little is known about it. In string theory, a Hagedorn temperature is predicted for superstrings at about 10³⁰ K, where superstrings are copiously produced. At Planck temperature(10³² K), gravity becomes a significant force between individual particles. No current theory can describe these states and they cannot be produced with any foreseeable experiment. However, these states are important in cosmology because the universe may have passed through these states in the Big Bang.

The gravitational singularity predicted by general relativity to exist at the center of a black hole is not a phase of matter; it is not a material object at all (although the mass-energy of matter contributed to its creation) but rather a property of spacetime. Because spacetime breaks down there, the singularity should not be thought of as a localized structure, but as a global, topological feature of spacetime.^[16] It has been argued that elementary particles are fundamentally not material, either, but are localized properties of spacetime.^[17] In quantum gravity, singularities may in fact mark transitions to a new phase of matter.^[18]

Other proposed states

Supersolid

A supersolid is a spatially ordered material (that is, a solid or crystal) with superfluid properties. Similar to a superfluid, a supersolid is able to move without friction but retains a rigid shape. Although a supersolid is a solid, it exhibits so many characteristic properties different from other solids that many argue it is another state of matter.^[19]

String-net liquid

In a string-net liquid, atoms have apparently unstable arrangement, like a liquid, but are still consistent in overall pattern, like a solid. When in a normal solid state, the atoms of matter align themselves in a grid pattern, so that the spin of any electron is the opposite of the spin of all electrons touching it. But in a string-net liquid, atoms are arranged in some pattern that requires some electrons to have neighbors with the same spin. This gives rise to curious properties, as well as supporting some unusual proposals about the fundamental conditions of the universe itself.

Superglass

A superglass is a phase of matter characterized, at the same time, by superfluidity and a frozen amorphous structure.

Arbitrary definition

Although multiple attempts have been made to create a unified account, ultimately the definitions of what states of matter exist and the point at which states change are arbitrary.^[20]^[21]^[22] Some authors have suggested that states of matter are better thought of as a spectrum between a solid and plasma instead of being rigidly defined.^[23]

Phase transitions of matter (
v
t
e
)
To From	Solid	Liquid	Gas	Plasma
Solid		Melting	Sublimation
Liquid	Freezing		Vaporization
Gas	Deposition	Condensation		Ionization
Plasma			Recombination

A photoexcited transition to an H state[edit]

A hypothetical schematic diagram for the transition to an H state by photo excitation is shown in the Figure (After ^[4]). An absorbed photon causes an electron from the ground state G to an excited state E (red arrow). State E rapidly relaxes via Frank-Condon relaxation to an intermediate locally reordered state I. Through interactions with others of its kind, this state collectively orders to form a macroscopically ordered metastable state H, further lowering its energy as a result. The new state has a broken symmetry with respect to the G or E state, and may also involve further relaxation compared to the I state. The barrier E_Bprevents state H from reverting to the ground state G. If the barrier is sufficiently large compared to thermal energy k_BT, where k_B is the Boltzmann constant, the H state can be stable indefinitely.

https://en.wikipedia.org/wiki/Hidden_states_of_matter

A cooling curve is a line graph that represents the change of phase of matter, typically from a gas to a solid or a liquid to a solid. The independent variable (X-axis) is time and the dependent variable (Y-axis) is temperature.^[1] Below is an example of a cooling curve used in castings.

States of matter (list)
State	Solid Liquid Gas / Vapor Plasma
Low energy	Bose–Einstein condensate Fermionic condensate Degenerate matter Quantum Hall Rydberg matter Rydberg polaron Strange matter Superfluid Supersolid Photonic molecule
High energy	QCD matter Lattice QCD Quark–gluon plasma Color-glass condensate Supercritical fluid
Other states	Colloid Glass Crystal Liquid crystal Time crystal Quantum spin liquid Exotic matter Programmable matter Dark matter Antimatter Magnetically ordered Antiferromagnet Ferrimagnet Ferromagnet String-net liquid Superglass
Transitions	Boiling Boiling point Condensation Critical line Critical point Crystallization Deposition Evaporation Flash evaporation Freezing Chemical ionization Ionization Lambda point Melting Melting point Recombination Regelation Saturated fluid Sublimation Supercooling Triple point Vaporization Vitrification
Quantities	Enthalpy of fusion Enthalpy of sublimation Enthalpy of vaporization Latent heat Latent internal energy Trouton's rule Volatility
Concepts	Baryonic matter Binodal Compressed fluid Cooling curve Equation of state Leidenfrost effect Macroscopic quantum phenomena Mpemba effect Order and disorder (physics) Spinodal Superconductivity Superheated vapor Superheating Thermo-dielectric effect

Categories:

https://en.wikipedia.org/wiki/Cooling_curve

A pressure–volume diagram (or PV diagram, or volume–pressure loop)^[1] is used to describe corresponding changes in volume and pressure in a system. They are commonly used in thermodynamics, cardiovascular physiology, and respiratory physiology.

PV diagrams, originally called indicator diagrams, were developed in the 18th century as tools for understanding the efficiency of steam engines.

https://en.wikipedia.org/wiki/Pressure–volume_diagram

In condensed matter physics, dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at which three-phase coexistence terminates.^[1] This definition is clearly parallel to the definition of an ordinary critical point as the point at which two-phase coexistence terminates.

https://en.wikipedia.org/wiki/Tricritical_point

Critical exponents describe the behavior of physical quantities near continuous phase transitions.

https://en.wikipedia.org/wiki/Critical_exponent

https://en.wikipedia.org/wiki/Critical_points_of_the_elements_(data_page)

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.

https://en.wikipedia.org/wiki/Conformal_field_theory

Pages in category "Phase transitions"

The following 112 pages are in this category, out of 112 total. This list may not reflect recent changes (learn more).

Phase transition

A

B

D

E

F

G

H

I

J

K

L

M

P

Q

Quantum phase transition

R

S

T

V

Z

Zero-curtain effect

https://en.wikipedia.org/wiki/Category:Phase_transitions

Pages in category "Renormalization group"

The following 36 pages are in this category, out of 36 total. This list may not reflect recent changes (learn more).

https://en.wikipedia.org/wiki/Category:Renormalization_group

Pages in category "Threshold temperatures"

The following 27 pages are in this category, out of 27 total. This list may not reflect recent changes (learn more).

A

B

Boiling point

C

D

E

F

G

Glass transition

H

L

M

O

P

Pour point

R

Relative thermal index

S

T

categories:

https://en.wikipedia.org/wiki/Category:Threshold_temperatures

A eutectic system (/juːˈtɛktɪk/ yoo-TEK-tik)^[1] from the Greek "εύ" (eu = well) and "τήξις" (tēxis = melting) is a heterogeneous mixture of substances that melts or solidifies at a single temperature that is lower than the melting point of any of the constituents.^[2] This temperature is known as the eutectic temperature, and is the lowest possible melting temperature over all of the mixing ratios for the involved component species. On a phase diagram, the eutectic temperature is seen as the eutectic point (see plot on the right).^[3]

Non-eutectic mixture ratios would have different melting temperatures for their different constituents, since one component's lattice will melt at a lower temperature than the other's. Conversely, as a non-eutectic mixture cools down, each of its components would solidify (form a lattice) at a different temperature, until the entire mass is solid.

Not all binary alloys have eutectic points, since the valence electrons of the component species are not always compatible,^{[clarification needed]} in any mixing ratio, to form a new type of joint crystal lattice. For example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) "meet at the pure element endpoints of the atomic ratio axis while slightly separating in the mixture region of this axis".^[4]

The term eutectic was coined in 1884 by British physicist and chemist Frederick Guthrie (1833–1886).^[5]

https://en.wikipedia.org/wiki/Eutectic_system

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex(concave upward), or vice versa.

For the graph of a function of differentiability class $C 2$ (f, its first derivative f', and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vise versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive).^[1] A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.

https://en.wikipedia.org/wiki/Inflection_point

In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.^[1] It is that temperature and pressure at which the sublimation curve, fusion curve and the vaporisation curve meet. For example, the triple point of mercury occurs at a temperature of −38.83440 °C (−37.90192 °F) and a pressure of 0.165 m Pa.

In addition to the triple point for solid, liquid, and gas phases, a triple point may involve more than one solid phase, for substances with multiple polymorphs. Helium-4 is a special case that presents a triple point involving two different fluid phases (lambda point).^[1]

The triple point of water was used to define the kelvin, the base unit of thermodynamic temperature in the International System of Units (SI).^[2] The value of the triple point of water was fixed by definition, rather than measured, but that changed with the 2019 redefinition of SI base units. The triple points of several substances are used to define points in the ITS-90 international temperature scale, ranging from the triple point of hydrogen (13.8033 K) to the triple point of water (273.16 K, 0.01 °C, or 32.018 °F).

The term "triple point" was coined in 1873 by James Thomson, brother of Lord Kelvin.^[3]

https://en.wikipedia.org/wiki/Triple_point

In thermodynamics, the limit of local stability with respect to small fluctuations is clearly defined by the condition that the second derivative of Gibbs free energy is zero. The locus of these points (the inflection point within a G-x or G-c curve, Gibbs free energy as a function of composition) is known as the spinodal curve.^[1]^[2]^[3] For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via spinodal decomposition. Outside of the curve, the solution will be at least metastable with respect to fluctuations.^[3] In other words, outside the spinodal curve some careful process may obtain a single phase system.^[3] Inside it, only processes far from thermodynamic equilibrium, such as physical vapor deposition, will enable one to prepare single phase compositions.^[4] The local points of coexisting compositions, defined by the common tangent construction, are known as a binodal (coexistence) curve, which denotes the minimum-energy equilibrium state of the system. Increasing temperature results in a decreasing difference between mixing entropy and mixing enthalpy, and thus the coexisting compositions come closer. The binodal curve forms the basis for the miscibility gap in a phase diagram. The free energy of mixing changes with temperature and concentration, and the binodal and spinodal meet at the critical or consolute temperature and composition.^[5]

https://en.wikipedia.org/wiki/Spinodal

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.

The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.

In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.
Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.
In general, dimensionless quantities are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.

https://en.wikipedia.org/wiki/Scale_invariance

https://en.wikipedia.org/wiki/Compressibility_factor

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.^[1]

https://en.wikipedia.org/wiki/Viscosity

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in concentration.

https://en.wikipedia.org/wiki/Diffusion

Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law. To understand the behaviour of real gases, the following must be taken into account:

compressibility effects;
variable specific heat capacity;
van der Waals forces;
non-equilibrium thermodynamic effects;
issues with molecular dissociation and elementary reactions with variable composition

For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, real-gas models have to be used near the condensation point of gases, near critical points, at very high pressures, to explain the Joule–Thomson effect and in other less usual cases. The deviation from ideality can be described by the compressibility factor Z.

https://en.wikipedia.org/wiki/Real_gas

In science, an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment and observation but not necessarily supported by theory.^[1]

Analytical solutions without a theory[edit]

An empirical relationship is supported by confirmatory data irrespective of theoretical basis such as first principles. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the relationships are no longer considered empirical. An example was the Rydberg formula to predict the wavelengths of hydrogen spectral lines. Proposed in 1876, it perfectly predicted the wavelengths of the Lyman series, but lacked a theoretical basis until Niels Bohr produced his Bohr model of the atom in 1925.^[2]

On occasion, what was thought to be an empirical factor is later deemed to be a fundamental physical constant.^{[citation needed]}

Approximations[edit]

Some empirical relationships are merely approximations, often equivalent to the first few terms of the Taylor series of the analytical solution describing the phenomenon.^{[citation needed]} Other relationships only hold under certain specific conditions, reducing them to special cases of more general relationship.^[2] Some approximations, in particular phenomenological models, may even contradict theory; they are employed because they are more mathematically tractable than some theories, and are able to yield results.^[3]

Gas composition of air[edit]

To give a familiar example, air has a composition of:^[1]

Pure Gas Name	Symbol	Percent by Volume
Nitrogen	N₂	78.084
Oxygen	O₂	20.9476
Argon	Ar	0.934
Carbon Dioxide	CO₂	0.0314
Neon	Ne	0.001818
Methane	CH₄	0.0002
Helium	He	0.000524
Krypton	Kr	0.000114
Hydrogen	H₂	0.00005
Xenon	Xe	0.0000087

Standard Dry Air is the agreed-upon gas composition for air from which all water vapour has been removed. There are various standards bodies which publish documents that define a dry air gas composition. Each standard provides a list of constituent concentrations, a gas density at standard conditions and a molar mass.

It is extremely unlikely that the actual composition of any specific sample of air will completely agree with any definition for standard dry air. While the various definitions for standard dry air all attempt to provide realistic information about the constituents of air, the definitions are important in and of themselves because they establish a standard which can be cited in legal contracts and publications documenting measurement calculation methodologies or equations of state.

The standards below are two examples of commonly used and cited publications that provide a composition for standard dry air:

ISO TR 29922-2017 provides a definition for standard dry air which specifies an air molar mass of 28,965 46 ± 0,000 17 kg·kmol-1.^[2]

GPA 2145:2009 is published by the Gas Processors Association. It provides a molar mass for air of 28.9625 g/mol, and provides a composition for standard dry air as a footnote.^[3]

https://en.wikipedia.org/wiki/Gas_composition

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy or pressure and volume. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power.

For a mechanical system, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" that, when unbalanced, cause certain generalized "displacements", and the product of the two is the energy transferred as a result. These forces and their associated displacements are called conjugate variables. The thermodynamic force is always an intensive variable and the displacement is always an extensive variable, yielding an extensive energy transfer. The intensive (force) variable is the derivative of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.

The thermodynamic square can be used as a tool to recall and derive some of the thermodynamic potentials based on conjugate variables.

In the above description, the product of two conjugate variables yields an energy. In other words, the conjugate pairs are conjugate with respect to energy. In general, conjugate pairs can be defined with respect to any thermodynamic state function. Conjugate pairs with respect to entropy are often used, in which the product of the conjugate pairs yields an entropy. Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of the Onsager reciprocal relations.

https://en.wikipedia.org/wiki/Conjugate_variables_(thermodynamics)

In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas.^[1]

Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless fugacity coefficient $φ$ .^[1]

\varphi ={\frac {f}{P}}\,

For an ideal gas, fugacity and pressure are equal and so $φ = 1$ . Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to $RT ln φ$ .

The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and normally chosen as 1 atmosphere or 1 bar.

Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities.

For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to that of the vapor, and therefore the fugacity is equal to the fugacity of the vapor. This fugacity is approximately equal to the vapor pressure when the vapor pressure is not too high.

https://en.wikipedia.org/wiki/Fugacity

Pressure/volume and stress/strain pairs[edit]

As an example, consider the $pV$ conjugate pair. The pressure acts as a generalized force – pressure differences force a change in volume, and their product is the energy lost by the system due to mechanical work. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.

The above holds true only for non-viscous fluids. In the case of viscous fluids, plastic and elastic solids, the pressure force is generalized to the stress tensor, and changes in volume are generalized to the volume multiplied by the strain tensor.^[2] These then form a conjugate pair. If $\sigma _{ij}$ is the ij component of the stress tensor, and $\varepsilon _{ij}$ is the ij component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain $\mathrm {\varepsilon } _{ij}$ is:

\delta w=V\sum _{ij}\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}

or, using Einstein notation for the tensors, in which repeated indices are assumed to be summed:

\delta w=V\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}

In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times the unit tensor so that

\delta w=V\,(-p\delta _{ij})\,\mathrm {d} \varepsilon _{ij}=-\sum _{k}pV\,\mathrm {d} \varepsilon _{kk}

The trace of the strain tensor ( $\varepsilon _{{kk}}$ ) is the fractional change in volume so that the above reduces to $\delta w=-p\mathrm {d} V$ as it should.

Temperature/entropy pair[edit]

In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heating. Temperature is the driving force, entropy is the associated displacement, and the two form a pair of conjugate variables. The temperature/entropy pair of conjugate variables is the only heat term; the other terms are essentially all various forms of work.

Chemical potential/particle number pair[edit]

The chemical potential is like a force which pushes an increase in particle number. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds water and water vapor, there will be a chemical potential (which is negative) for the liquid, pushing water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate is equilibrium obtained.

Regarding work and heat[edit]

Work and heat are not thermodynamic properties, but rather process quantities: flows of energy across a system boundary. Systems do not contain work, but can perform work, and likewise, in formal thermodynamics, systems do not contain heat, but can transfer heat. Informally, however, a difference in the energy of a system that occurs solely because of a difference in its temperature is commonly called heat, and the energy that flows across a boundary as a result of a temperature difference is "heat".

Altitude (or elevation) is usually not a thermodynamic property. Altitude can help specify the location of a system, but that does not describe the state of the system. An exception would be if the effect of gravity need to be considered in order to describe a state, in which case altitude could indeed be a thermodynamic property.

Thermodynamic properties and their characteristics
Property	Symbol	Units	Conjugate	Potential?
Activity	$a$	–
Chemical potential	$μ i$	kJ/mol	Particle number $N i$
Compressibility (adiabatic)	$β S$ , $κ$	Pa⁻¹
Compressibility (isothermal)	$β T$ , $κ$	Pa⁻¹
Cryoscopic constant^[1]	$K f$	K·kg/mol
Density	$ρ$	kg/m³
Ebullioscopic constant	$K b$	K·kg/mol
Enthalpy	$H$	J
Specific enthalpy	$h$	J/kg
Entropy	$S$	J/K	Temperature $T$	(entropic)
Specific entropy	$s$	J/(kg K)
Fugacity	$f$	N/m²
Gibbs free energy	$G$	J
Specific Gibbs free energy	$g$	J/kg
Gibbs free entropy	$Ξ$	J/K		(entropic)
Grand / Landau potential	$Ω$	J
Heat capacity (constant pressure)	$C p$	J/K
Specific heat capacity (constant pressure)	$c p$	J/(kg·K)
Heat capacity (constant volume)	$C v$	J/K
Specific heat capacity (constant volume)	$c v$	J/(kg·K)
Helmholtz free energy	$A$ , $F$	J
Helmholtz free entropy	$Φ$	J/K		(entropic)
Internal energy	$U$	J
Specific internal energy	$u$	J/kg
Internal pressure	$π T$	Pa
Mass	$m$	kg
Particle number	$N i$	–	Chemical potential $μ i$
Pressure	$p$	Pa	Volume $V$
Temperature	$T$	K	Entropy $S$
Thermal conductivity	$k$	W/(m·K)
Thermal diffusivity	$α$	m²/s
Thermal expansion (linear)	$α L$	K⁻¹
Thermal expansion (area)	$α A$	K⁻¹
Thermal expansion (volumetric)	$α V$	K⁻¹
Vapor quality^[2]	$χ$	–
Volume	$V$	m³	Pressure $P$
Specific volume	$ν$	m³/kg

In crystals[edit]

Atomic diffusion across a 4-coordinated lattice. Note that the atoms often block each other from moving to adjacent sites. As per Fick’s law, the net flux (or movement of atoms) is always in the opposite direction of the concentration gradient.

In the crystal solid state, diffusion within the crystal lattice occurs by either interstitial or substitutional mechanisms and is referred to as lattice diffusion.^[1] In interstitial lattice diffusion, a diffusant (such as C in an iron alloy), will diffuse in between the lattice structure of another crystalline element. In substitutional lattice diffusion (self-diffusionfor example), the atom can only move by substituting place with another atom. Substitutional lattice diffusion is often contingent upon the availability of point vacancies throughout the crystal lattice. Diffusing particles migrate from point vacancy to point vacancy by the rapid, essentially random jumping about (jump diffusion).

Since the prevalence of point vacancies increases in accordance with the Arrhenius equation, the rate of crystal solid state diffusion increases with temperature.

For a single atom in a defect-free crystal, the movement can be described by the "random walk" model. In 3-dimensions it can be shown that after $n$ jumps of length $\alpha$ the atom will have moved, on average, a distance of:

r=\alpha {\sqrt {n}}.

If the jump frequency is given by $T$ (in jumps per second) and time is given by $t$ , then $r$ is proportional to the square root of $Tt$ :

r\sim\sqrt{Tt}.

Diffusion in polycrystalline materials can involve short circuit diffusion mechanisms. For example, along the grain boundaries and certain crystalline defects such as dislocations there is more open space, thereby allowing for a lower activation energy for diffusion. Atomic diffusion in polycrystalline materials is therefore often modeled using an effective diffusion coefficient, which is a combination of lattice, and grain boundary diffusion coefficients. In general, surface diffusionoccurs much faster than grain boundary diffusion, and grain boundary diffusion occurs much faster than lattice diffusion.

https://en.wikipedia.org/wiki/Atomic_diffusion

https://en.wikipedia.org/wiki/Grain_boundary_diffusion_coefficient

https://en.wikipedia.org/wiki/Drag_(physics)

https://en.wikipedia.org/wiki/Aerodynamic_force

https://en.wikipedia.org/wiki/Laminar_flow

https://en.wikipedia.org/wiki/Fluid_dynamics

https://en.wikipedia.org/wiki/Parasitic_drag#Form_drag

https://en.wikipedia.org/wiki/Parasitic_drag

https://en.wikipedia.org/wiki/Drag_(physics)#Very_low_Reynolds_numbers:_Stokes'_drag

https://en.wikipedia.org/wiki/Shear_force

https://en.wikipedia.org/wiki/Pressure

https://en.wikipedia.org/wiki/Viscosity

https://en.wikipedia.org/wiki/Center_of_pressure_(fluid_mechanics)

https://en.wikipedia.org/wiki/Propeller_(aeronautics)

https://en.wikipedia.org/wiki/Body_force

https://en.wikipedia.org/wiki/Archimedes%27_principle

https://en.wikipedia.org/wiki/Eötvös_number

https://en.wikipedia.org/wiki/Stall_(fluid_dynamics)

https://en.wikipedia.org/wiki/Boundary_layer

https://en.wikipedia.org/wiki/Stokes%27_law

https://en.wikipedia.org/wiki/Cross_section_(geometry)

https://en.wikipedia.org/wiki/Drag_equation

https://en.wikipedia.org/wiki/Square–cube_law

https://en.wikipedia.org/wiki/Drag_equation

https://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity

https://en.wikipedia.org/wiki/Stokes_radius

https://en.wikipedia.org/wiki/Freestream

https://en.wikipedia.org/wiki/Shock_wave

https://en.wikipedia.org/wiki/Hyperbolic_functions

https://en.wikipedia.org/wiki/Hydrodynamic_radius

https://en.wikipedia.org/wiki/Size-exclusion_chromatography

https://en.wikipedia.org/wiki/Radius_of_gyration

https://en.wikipedia.org/wiki/Centroid

https://en.wikipedia.org/wiki/Barycenter

https://en.wikipedia.org/wiki/Chemical_structure

https://en.wikipedia.org/wiki/Ensemble_average

https://en.wikipedia.org/wiki/Contour_length

https://en.wikipedia.org/wiki/Colloid

https://en.wikipedia.org/wiki/Born_equation

https://en.wikipedia.org/wiki/Capillary_electrophoresis

https://en.wikipedia.org/wiki/Gradient

https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity

https://en.wikipedia.org/wiki/PH

https://en.wikipedia.org/wiki/Electro-osmosis

A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise

https://en.wikipedia.org/wiki/Viscosity

https://en.wikipedia.org/wiki/Capillary_pressure

https://en.wikipedia.org/wiki/Buoyancy

https://en.wikipedia.org/wiki/Continuum_mechanics

https://en.wikipedia.org/wiki/Constitutive_equation

https://en.wikipedia.org/wiki/Strain_energy_density_function

https://en.wikipedia.org/wiki/Finite_strain_theory#Deformation_gradient_tensor

https://en.wikipedia.org/wiki/Linear_map

https://en.wikipedia.org/wiki/Inelastic_collision

https://en.wikipedia.org/wiki/Parameter

https://en.wikipedia.org/wiki/Lambda_calculus

https://en.wikipedia.org/wiki/Combinatory_logic

Computer programming[edit]

In computer programming, two notions of parameter are commonly used, and are referred to as parameters and arguments—or more formally as a formal parameter and an actual parameter.

For example, in the definition of a function such as

y = f(x) = x + 2,

x is the formal parameter (the parameter) of the defined function.

When the function is evaluated for a given value, as in

f(3): or, y = f(3) = 3 + 2 = 5,

3 is the actual parameter (the argument) for evaluation by the defined function; it is a given value (actual value) that is substituted for the formal parameter of the defined function. (In casual usage the terms parameter and argument might inadvertently be interchanged, and thereby used incorrectly.)

These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic. Terminology varies between languages; some computer languages such as C define parameter and argument as given here, while Eiffel uses an alternative convention.

https://en.wikipedia.org/wiki/Parameter

https://en.wikipedia.org/wiki/Drag_(physics)

https://en.wikipedia.org/wiki/Ludwig_Prandtl

Monday, September 20, 202109-20-2021-0835 - Distributive Lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.

Blog Archive

Tuesday, September 21, 2021

09-20-2021-2253 - Gas Laws; Ideal gas law general gas equation

Boyle's law[edit]

Charles's law[edit]

Gay-Lussac's law[edit]

Avogadro's law[edit]

Combined and ideal gas laws[edit]

Other gas laws[edit]

References[edit]

Compressibility factor at the critical point[edit]

See also[edit]

Overview[edit]

Historical[edit]

Boyle's law (1662)[edit]

Charles's law or Law of Charles and Gay-Lussac (1787)[edit]

Dalton's law of partial pressures (1801)[edit]

The ideal gas law (1834)[edit]

Van der Waals equation of state (1873)[edit]

General form of an equation of state[edit]

Classical ideal gas law[edit]

Quantum ideal gas law[edit]

Cubic equations of state[edit]

Van der Waals equation of state[edit]

Redlich-Kwong equation of state[6][edit]

Soave modification of Redlich-Kwong[7][edit]

Volume translation of Peneloux et al. (1982)[edit]

Peng–Robinson equation of state[edit]

Peng–Robinson-Stryjek-Vera equations of state[edit]

PRSV1[edit]

PRSV2[edit]

Peng-Robinson-Babalola equation of state (PRB)[edit]

Elliott, Suresh, Donohue equation of state[edit]

Cubic-Plus-Association[edit]

Non-cubic equations of state[edit]

Dieterici equation of state[edit]

Virial equations of state[edit]

Virial equation of state[edit]

The BWR equation of state[edit]

Lee-Kesler equation of state[edit]

SAFT equations of state[edit]

Multiparameter equations of state[edit]

Helmholtz Function form[edit]

Other equations of state of interest[edit]

Stiffened equation of state[edit]

Ultrarelativistic equation of state[edit]

Ideal Bose equation of state[edit]

Jones–Wilkins–Lee equation of state for explosives (JWL equation)[edit]

Equations of state for solids and liquids[edit]

See also[edit]

Equation[edit]

Common forms[edit]

Molar form[edit]

Statistical mechanics[edit]

Combined gas law[edit]

Energy associated with a gas[edit]

Applications to thermodynamic processes[edit]

Deviations from ideal behavior of real gases[edit]

Derivations[edit]

Empirical[edit]

Theoretical[edit]

Kinetic theory[edit]

Statistical mechanics[edit]

Other dimensions[edit]

See also[edit]

Examples[edit]

Control systems engineering[edit]

Discrete-time systems[edit]

Continuous time systems[edit]

See also[edit]

Liquid–vapor critical point[edit]

Overview[edit]

History[edit]

Theory[edit]

Table of liquid–vapor critical temperature and pressure for selected substances[edit]

Mixtures: liquid–liquid critical point[edit]

Mathematical definition[edit]

See also[edit]

External links[edit]

Advantages[edit]

Redlich-Kwong equation of state^[6][edit]

Soave modification of Redlich-Kwong^[7][edit]