Real gas#Berthelot and modified Berthelot model

File:Real Gas Isotherms.svg of real gas

Dark blue curves – isotherms below the critical temperature. Green sections – metastable states.

The section to the left of point F – normal liquid.

Point F – boiling point.

Line FG – equilibrium of liquid and gaseous phases.

Section FA – superheated liquid.

Section F′A – stretched liquid (p<0).

Section AC – analytic continuation of isotherm, physically impossible.

Section CG – supercooled vapor.

Point G – dew point.

The plot to the right of point G – normal gas.

Areas FAB and GCB are equal.

Red curve – Critical isotherm.

Point K – critical point.

Light blue curves – supercritical isotherms]]

{{Main|Equation of state}}

{{Main|van der Waals equation}}

Real gases are often modeled by taking into account their molar weight and molar volume

$$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{V\_\backslash text\{m\}^2\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively:

$$p\; =\; \backslash frac\{RT\}\{V\_m\; -\; b\}\; -\; \backslash frac\{a\}\{V\_m^2\}$$

Where p is the pressure, T is the temperature, R the ideal gas constant, and V_m the molar volume. a and b are parameters that are determined empirically for each gas, but are sometimes estimated from their critical temperature (T_c) and critical pressure (p_c) using these relations:

$$\backslash begin\{align\}$$

a &= \frac{27R^2 T_\text{c}^2}{64p_\text{c}}, &

b &= \frac{RT_\text{c}}{8p_\text{c}}

\end{align}

The constants at critical point can be expressed as functions of the parameters a, b:

\begin{align}

p_c &= \frac{a}{27b^2}, &

V_{m,c} &= 3b, \\[2pt]

T_c &= \frac{8a}{27bR}, &

Z_c &= \frac{3}{8}

\end{align}

With the reduced properties $$

p_r = p / p_\text{c}

, $$

V_r = V_\text{m} / V_\text{m,c}

, $$

T_r = T / T_\text{c}

the equation can be written in the reduced form:

$$p\_r\; =\; \backslash frac\{8\}\{3\}\backslash frac\{T\_r\}\{V\_r\; -\; \backslash frac\{1\}\{3\}\}\; -\; \backslash frac\{3\}\{V\_r^2\}$$

File:Critical isotherm Redlich-Kwong model.png

The Redlich–Kwong equation is another two-parameter equation that is used to model real gases. It is almost always more accurate than the van der Waals equation, and often more accurate than some equations with more than two parameters. The equation is

$$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{\backslash sqrt\{T\}V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively:

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{\backslash sqrt\{T\}V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\}$$

where a and b are two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:

$$\backslash begin\{align\}$$

a &= 0.42748\, \frac{R^2{T_\text{c}}^\frac{5}{2}}{p_\text{c}}, \\[2pt]

b &= 0.08664\, \frac{RT_\text{c}}{p_\text{c}}

\end{align}

The constants at critical point can be expressed as functions of the parameters a, b:

\begin{align}

p_c &= {\left[\frac{(\sqrt[3]{2} - 1)^7}{3} \, R \, \frac{a^2}{b^5}\right]}^{1/3}, &

V_{m,c} &= \frac{b}{\sqrt[3]{2}-1}, \\[4pt]

T_c &= {\left[3 {\left(\sqrt[3]{2} - 1\right)}^2 \frac{a}{bR}\right]}^{2/3}, &

Z_c &= \frac{1}{3}

\end{align}

Using $p\_r\; =\; p/p\_\backslash text\{c\}$, $V\_r\; =\; V\_\backslash text\{m\}/V\_\backslash text\{m,c\}$, $T\_r\; =\; T\; /\; T\_\backslash text\{c\}$

the equation of state can be written in the reduced form:

$$p\_r\; =\; \backslash frac\{3\; T\_r\}\{V\_r\; -\; b\text{'}\}\; -\; \backslash frac\{1\}\{b\text{'}\backslash sqrt\{T\_r\}\; V\_r\; \backslash left(V\_r\; +\; b\text{'}\backslash right)\}$$ with $b\text{'}\; =\; \backslash sqrt[3]\{2\}\; -\; 1\; \backslash approx\; 0.26$

The Berthelot equation (named after D. Berthelot)D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907) is very rarely used,

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}^2\}$$

but the modified version is somewhat more accurate

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\}\; \backslash left[1\; +\; \backslash frac\{9\}\{128\}\; \backslash cdot\; \backslash frac\{p\}\{p\_c\}\; \backslash cdot\; \backslash frac\{T\_c\}\{T\}\; \backslash left(1\; -\; 6\; \backslash frac\{T\_\backslash text\{c\}^2\}\{T^2\}\backslash right)\backslash right]$$

This model (named after C. DietericiC. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)) fell out of usage in recent years

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; \backslash exp\backslash left(-\backslash frac\{a\}\{V\_\backslash text\{m\}RT\}\backslash right)$$

with parameters a, b. These can be normalized by dividing with the critical point state{{NoteTag|The critical state can be calculated by starting with $p\; =\; \backslash frac\{RT\}\{\{(V\_m-b)\}\; e^\{\backslash frac\{a\}\{RTV\_m\}\}\}$, and taking the derivative with respect to $V\_m$. The equation $(\backslash partial\_\{V\_m\}p)\_T\; =\; 0$ is a quadratic equation in $V\_m$, and it has a double root precisely when $V\_m\; =\; V\_c;\; T=T\_c$.}}:$$\backslash tilde\; p\; =\; p\; \backslash frac\{(2be)^2\}\{a\};\; \backslash quad\; \backslash tilde\; T\; =T\; \backslash frac\{4bR\}\{a\};\; \backslash quad\; \backslash tilde\; V\_m\; =\; V\_m\; \backslash frac\{1\}\{2b\}$$which casts the equation into the reduced form:{{Cite book |last=Pippard |first=Alfred B. |title=Elements of classical thermodynamics: for advanced students of physics |date=1981 |publisher=Univ. Pr |isbn=978-0-521-09101-5 |edition=Repr |location=Cambridge |pages=74}}$$\backslash tilde\; p\; \backslash left(2\backslash tilde\; V\_m\; -\; 1\backslash right)\; =\; \backslash tilde\; T\; \backslash exp\backslash left(2\; -\; \backslash frac\{2\}\{\backslash tilde\; T\; \backslash tilde\; V\_m\}\backslash right)$$

The Clausius equation (named after Rudolf Clausius) is a very simple three-parameter equation used to model gases.

$$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{T\; \{\backslash left(V\_\backslash text\{m\}\; +\; c\backslash right)\}^2\}\backslash right)\; \backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively:

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{T\backslash left(V\_\backslash text\{m\}\; +\; c\backslash right)^2\}$$

where

$$\backslash begin\{align\}$$

a &= \frac{27R^2 T_\text{c}^3}{64p_\text{c}}, \\[4pt]

b &= V_\text{c} - \frac{RT_\text{c}}{4p_\text{c}}, \\[4pt]

c &= \frac{3RT_\text{c}}{8p_\text{c}} - V_\text{c}

\end{align}

where V_c is critical volume.

The Virial equation derives from a perturbative treatment of statistical mechanics.

$$pV\_\backslash text\{m\}\; =\; RT\backslash left[1\; +\; \backslash frac\{B(T)\}\{V\_\backslash text\{m\}\}\; +\; \backslash frac\{C(T)\}\{V\_\backslash text\{m\}^2\}\; +\; \backslash frac\{D(T)\}\{V\_\backslash text\{m\}^3\}\; +\; \backslash cdots\backslash right]$$

or alternatively

$$pV\_\backslash text\{m\}\; =\; RT\; \backslash left[1\; +\; B\text{'}(T)\; p\; +\; C\text{'}(T)\; p^2\; +\; D\text{'}(T)\; p^3\; +\; \backslash cdots\backslash right]$$

where A, B, C, A′, B′, and C′ are temperature dependent constants.

Peng–Robinson equation of state (named after D.-Y. Peng and D. B. Robinson{{cite journal |title= A New Two-Constant Equation of State |journal= Industrial and Engineering Chemistry: Fundamentals |volume= 15 |year= 1976 |pages= 59–64 |author1=Peng, D. Y. |author2=Robinson, D. B. |name-list-style=amp |doi= 10.1021/i160057a011|s2cid= 98225845 }}) has the interesting property being useful in modeling some liquids as well as real gases.

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a(T)\}\{V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\; +\; b\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\}$$

File:Isotherm wohl model.png

File:ZPhCh87p9f.png

The Wohl equation (named after A. Wohl{{cite journal |author1=A. Wohl |title=Investigation of the condition equation |journal=Zeitschrift für Physikalische Chemie |date=1914 |volume=87 |pages=1–39 |doi=10.1515/zpch-1914-8702 |s2cid=92940790 |language=en}}) is formulated in terms of critical values, making it useful when real gas constants are not available, but it cannot be used for high densities, as for example the critical isotherm shows a drastic decrease of pressure when the volume is contracted beyond the critical volume.

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\}\; +\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash quad$$

or:

$$\backslash left(p\; -\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\; =\; RT\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}\}$$

or, alternatively:

$$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{TV\_\backslash text\{m\}(V\_\backslash text\{m\}\; -\; b)\}\; -\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

where

$$\backslash begin\{align\}$$

a &= 6p_\text{c} T_\text{c} V_\text{m,c}^2, &

b &= \frac{V_\text{m,c}}{4}, \\[2pt]

c &= 4p_\text{c} T_\text{c}^2 V_\text{m,c}^3

\end{align} where $V\_\backslash text\{m,c\}\; =\; \backslash frac\{4\}\{15\}\backslash frac\{RT\_c\}\{p\_c\}$, $$

p_\text{c}

, $T\_c$ are (respectively) the molar volume, the pressure and the temperature at the critical point.

And with the reduced properties $p\_r\; =\; p/p\_\backslash text\{c\}$, $V\_r\; =\; V\_\backslash text\{m\}\; /\; V\_\backslash text\{m,c\}$, $T\_r\; =\; T\; /\; T\_\backslash text\{c\}$

one can write the first equation in the reduced form:

$$p\_r\; =\; \backslash frac\{15\}\{4\}\backslash frac\{T\_r\}\{V\_r\; -\; \backslash frac\{1\}\{4\}\}\; -\; \backslash frac\{6\}\{T\_r\; V\_r\backslash left(V\_r\; -\; \backslash frac\{1\}\{4\}\backslash right)\}\; +\; \backslash frac\{4\}\{T\_r^2\; V\_r^3\}$$

Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010, {{ISBN|007-352932-X}} This equation is based on five experimentally determined constants. It is expressed as

$$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}^2\}\backslash left(1\; -\; \backslash frac\{c\}\{V\_\backslash text\{m\}T^3\}\backslash right)(V\_\backslash text\{m\}\; +\; B)\; -\; \backslash frac\{A\}\{V\_\backslash text\{m\}^2\}$$

where

$$\backslash begin\{align\}$$

A &= A_0 \left(1 - \frac{a}{V_\text{m}}\right), &

B &= B_0 \left(1 - \frac{b}{V_\text{m}}\right)

\end{align}

This equation is known to be reasonably accurate for densities up to about 0.8 ρ_cr, where ρ_cr is the density of the substance at its critical point. The constants appearing in the above equation are available in the following table when p is in kPa, V_m is in $\backslash frac\{\backslash text\{m\}^3\}\{\backslash text\{k\}\backslash ,\backslash text\{mol\}\}$, T is in K and $R\; =\; 8.314\; \backslash mathrm\{\backslash frac\{kPa\; \backslash cdot\; m^3\}\{kmol\; \backslash cdot\; K\}\}$Gordan J. Van Wylen and Richard E. Sonntage, Fundamental of Classical Thermodynamics, 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3

{{Main|Benedict–Webb–Rubin equation}}

The BWR equation,

$$p\; =\; RTd\; +\; d^2\backslash left(RT(B\; +\; bd)\; -\; \backslash left(A\; +\; ad\; -\; a\backslash alpha\; d^4\backslash right)\; -\; \backslash frac\{1\}\{T^2\}\backslash left[C\; -\; cd\backslash left(1\; +\; \backslash gamma\; d^2\backslash right)\; \backslash exp\backslash left(-\backslash gamma\; d^2\backslash right)\backslash right]\backslash right)$$

where d is the molar density and where a, b, c, A, B, C, α, and γ are empirical constants. Note that the γ constant is a derivative of constant α and therefore almost identical to 1.

The expansion work of the real gas is different than that of the ideal gas by the quantity $\backslash int\_\{V\_i\}^\{V\_f\}\; \backslash left(\backslash frac\{RT\}\{V\_m\}\; -\; P\_\backslash text\{real\}\backslash right)\; dV$.

• Compressibility factor
• Equation of state
• Ideal gas law: Boyle's law and Gay-Lussac's law

{{Reflist}}{{reflist|group=note}}

• {{cite book

|last1=Kondepudi |first1=D. K.

|last2=Prigogine |first2=I.

|year=1998

|title=Modern thermodynamics: From heat engines to dissipative structures

|publisher=John Wiley & Sons

|isbn=978-0-471-97393-5

}}

• {{cite book

|last1=Hsieh |first1=J. S.

|year=1993

|title=Engineering Thermodynamics

|publisher=Prentice-Hall

|isbn=978-0-13-275702-7

}}

• {{cite book

|last1=Walas |first1=S. M.

|year=1985

|title=Fazovyje ravnovesija v chimiceskoj technologii v 2 castach

|publisher=Butterworth Publishers

|isbn=978-0-409-95162-2

}}

• {{cite journal

|last1=Aznar |first1=M.

|last2=Silva Telles |first2=A.

|year=1997

|title=A Data Bank of Parameters for the Attractive Coefficient of the Peng-Robinson Equation of State

|journal=Brazilian Journal of Chemical Engineering

|volume=14 |issue=1 |pages=19–39

|doi=10.1590/S0104-66321997000100003

|doi-access=free

}}

• {{cite book

|last1=Rao |first1=Y. V. C

|year=2004

|isbn=978-81-7371-461-0

|title=An introduction to thermodynamics

|publisher=Universities Press

}}

• {{cite book

|last1=Xiang |first1=H. W.

|year=2005

|title=The Corresponding-States Principle and its Practice: Thermodynamic, Transport and Surface Properties of Fluids

|publisher=Elsevier

|isbn=978-0-08-045904-2

}}

• http://www.ccl.net/cca/documents/dyoung/topics-orig/eq_state.html

Category:Gases