Real gas explained

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.

Models

See main article: Equation of state.

Van der Waals model

See main article: van der Waals equation. Real gases are often modeled by taking into account their molar weight and molar volume

RT = \left(p + (a)/(V ^{2} _{m}) \right)\left(V _{m} - b\right)

or alternatively:

p = (RT)/(V _{m} - b) - (a)/(V ^{2} _{m})

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:

\begin{align} a & = (27 R ^{2} T ^{2} _{c})/(64 p _{c}) \\ b & = (RT _{c})/(8 p _{c}) \end{align}

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

p _{c = (a)/(27 b ^{2})} , T _{c = (8 a)/(27 bR)} , V _{m,c} =3 b, Z _{c = (3)/(8)} p _{r} = (p)/(p _{c}) , V _{r} = (V _{m})/(V _{m,c}) , T _{r} = (T)/(T _{c}) the equation can be written in the reduced form:p _{r} = (8)/(3) (T _{r})/(V _{r} - (1)/(3)) - (3)/(V ^{2} _{r})

Redlich–Kwong model

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 = \left(p + (a)/(\sqrt{T) V _{m\left(V} _{m} + b\right)}\right)\left(V _{m} - b\right)

or alternatively:

p = (RT)/(V _{m} - b) - (a)/(\sqrt{T) V _{m\left(V} _{m} + b\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:

\begin{align} a & = 0 . 42748 (R ^{2 {T} _{c})/(^{(5)/(2)}) } \\ b &= 0.08664\, \frac\end

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

p ^{7 / 3} _{c = ((\sqrt[ 3 ]{ 2)/(-1 ))} }R^\frac, \quad T_c=3^ (\sqrt[3]-1)^ (\frac)^, \qquad V_=\frac, \qquad Z_c=\frac

Using

p _{r} = (p)/(p _{c}) , V _{r} = (V _{m})/(V _{m,c}) , T _{r} = (T)/(T _{c}) the equation of state can be written in the reduced form:p _{r} = (3 T _{r})/(V _{r} - b') - (1)/(b'\sqrt{T _{r}) V _{r} \left(V _{r} + b'\right)} with b' = \sqrt[ 3 ]{ 2 } - 1 ≈ 0 . 26

Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot)^[1] is very rarely used,

p = (RT)/(V _{m} - b) - (a)/(TV ^{2} _{m})

but the modified version is somewhat more accurate

p = (RT)/(V _{m}) \left[ 1 + (9 (p)/(p _{c}))/(128 (T)/(T _{c})) \left( 1 - (6)/((T ^{2})/(T ^{2} _{c})) \right)\right]

Dieterici model

This model (named after C. Dieterici^[2]) fell out of usage in recent years

p = (RT)/(\exp\left( - V _{m} - b) (a)/(V _{mRT}) \right)

with parameters a, b. These can be normalized by dividing with the critical point state:$$\backslash tilde\; p\; =\; p\; \backslash frac;\; \backslash quad\; \backslash tilde\; T\; =T\; \backslash frac;\; \backslash quad\; \backslash tilde\; V\_m\; =\; V\_m\; \backslash frac$$which casts the equation into the reduced form:^[3] $$\backslash tilde\; p(2\backslash tilde\; V\_m\; -1)\; =\; \backslash tilde\; T\; e^$$

Clausius model

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

RT = \left(p + (a)/(T(V _{m} + c) ^{2}) \right)\left(V _{m} - b\right)

or alternatively:

p = (RT)/(V _{m} - b) - (a)/(T\left(V _{m} + c\right) ^{2})

where

\begin{align} a & = (27 R ^{2} T ^{3} _{c})/(64 p _{c}) \\ b & = V _{c} - (RT _{c})/(4 p _{c}) \\ c & = (3 RT _{c})/(8 p _{c}) - V _{c \end{align}}

where V_c is critical volume.

Virial model

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

pV _{m} = RT\left[ 1 + (B(T))/(V _{m}) + (C(T))/(V ^{2} _{m}) + (D(T))/(V ^{3} _{m}) + \ldots\right]

or alternatively

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

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

Peng–Robinson model

Peng–Robinson equation of state (named after D.-Y. Peng and D. B. Robinson^[4]) has the interesting property being useful in modeling some liquids as well as real gases.

p = (RT)/(V _{m} - b) - (a(T))/(V _{m\left(V} _{m} + b\right) + b\left(V _{m} - b\right))

Wohl model

The Wohl equation (named after A. Wohl^[5]) 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 = (RT)/(V _{m} - b) - (a)/(TV _{m\left(V} _{m} - b\right)) + (c)/(T ^{2} V ^{3} _{m})

or:

\left(p - (c)/(T ^{2} V ^{3} _{m}) \right)\left(V _{m} - b\right) = RT - (a)/(TV _{m})

or, alternatively:

RT = \left(p + (a)/(TV _{m(V} _{m} - b)) - (c)/(T ^{2} V ^{3} _{m}) \right)\left(V _{m} - b\right)

where

a = 6 p _{c} T _{c} V ^{2} _{m,c} b = (V _{m,c})/(4) with V _{m,c} = (4)/(15) (RT _{c})/(p _{c}) c = 4 p _{c} T ^{2} _{c} V ^{3} _{m,c} , where V _{m,c,} p _{c,} T _{c} are (respectively) the molar volume, the pressure and the temperature at the critical point.p _{r} = (p)/(p _{c}) , V _{r} = (V _{m})/(V _{m,c}) , T _{r} = (T)/(T _{c}) one can write the first equation in the reduced form:p _{r} = (15)/(4) (T _{r})/(V _{r} - (1)/(4)) - (6)/(T _{r} V _{r\left(V} _{r} - (1)/(4) \right)) + (4)/(T ^{2} _{r} V ^{3} _{r})

Beattie–Bridgeman model

^[6] This equation is based on five experimentally determined constants. It is expressed as

p = (RT)/(v ^{2}) \left( 1 - (c)/(vT ^{3}) \right)(v + B) - (A)/(v ^{2})

where

\begin{align} A & = A _{0} \left( 1 - (a)/(v) \right) & B & = B _{0} \left( 1 - (b)/(v) \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 is in

(m ^{3})/(k mol) , T is in K and R = 8.314(kPa ⋅ m ^{3})/(k mol ⋅ K) ^[7]

Gas	A0	a	B0	b	c
Air	131.8441	0.01931	0.04611	−0.001101	4.34×10^4
Argon, Ar	130.7802	0.02328	0.03931	0.0	5.99×10^4
Carbon dioxide, CO2	507.2836	0.07132	0.10476	0.07235	6.60×10^5
Ethane, C2H6	595.791	0.05861	0.09400	0.01915	90.00×10^4
Helium, He	2.1886	0.05984	0.01400	0.0	40
Hydrogen, H2	20.0117	−0.00506	0.02096	−0.04359	504
Methane, CH4	230.7069	0.01855	0.05587	-0.01587	12.83×10^4
Nitrogen, N2	136.2315	0.02617	0.05046	−0.00691	4.20×10^4
Oxygen, O2	151.0857	0.02562	0.04624	0.004208	4.80×10^4

Benedict–Webb–Rubin model

See main article: Benedict–Webb–Rubin equation.

The BWR equation,

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

Thermodynamic expansion work

The expansion work of the real gas is different than that of the ideal gas by the quantity

\int ^{V _{f}} _{V _{i}} \left( (RT)/(V _{m}) - P _{real} \right)dV .

See also

• Compressibility factor
• Equation of state
• Ideal gas law

Boyle's law and Gay-Lussac's law

Further reading

• Book: Kondepudi . D. K. . Prigogine . I. . 1998 . Modern thermodynamics: From heat engines to dissipative structures . . 978-0-471-97393-5.
• Book: Hsieh . J. S. . 1993 . Engineering Thermodynamics . . 978-0-13-275702-7.
• Book: Walas . S. M. . 1985 . Fazovyje ravnovesija v chimiceskoj technologii v 2 castach . . 978-0-409-95162-2.
• Aznar . M. . Silva Telles . A. . 1997 . A Data Bank of Parameters for the Attractive Coefficient of the Peng-Robinson Equation of State . . 14 . 1 . 19–39 . 10.1590/S0104-66321997000100003. free .
• Book: Rao . Y. V. C . 2004 . 978-81-7371-461-0 . An introduction to thermodynamics . Universities Press.
• Book: Xiang . H. W. . 2005 . The Corresponding-States Principle and its Practice: Thermodynamic, Transport and Surface Properties of Fluids . . 978-0-08-045904-2.

External links

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

Notes and References

1. D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907)
2. C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)
3. Book: Pippard, Alfred B. . Elements of classical thermodynamics: for advanced students of physics . 1981 . Univ. Pr . 978-0-521-09101-5 . Repr . Cambridge . 74.
4. A New Two-Constant Equation of State . Industrial and Engineering Chemistry: Fundamentals . 15 . 1976 . 59–64 . Peng, D. Y. . Robinson, D. B. . amp . 10.1021/i160057a011. 98225845 .
5. A. Wohl . Investigation of the condition equation . Zeitschrift für Physikalische Chemie . 1914 . 87 . 1–39 . 10.1515/zpch-1914-8702 . 92940790 . en.
6. Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010,
7. Gordan J. Van Wylen and Richard E. Sonntage, Fundamental of Classical Thermodynamics, 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3