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Real gas

From Wikipedia, the free encyclopedia
Non-hypothetical gases whose molecules occupy space and have interactions
Thermodynamics
The classicalCarnot heat engine

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

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

Models

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Isotherms 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 article:Equation of state

Van der Waals model

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Main article:van der Waals equation

Real gases are often modeled by taking into account their molar weight and molar volumeRT=(p+aVm2)(Vm−b){\displaystyle RT=\left(p+{\frac {a}{V_{\text{m}}^{2}}}\right)\left(V_{\text{m}}-b\right)}

or alternatively:p=RTVm−b−aVm2{\displaystyle p={\frac {RT}{V_{m}-b}}-{\frac {a}{V_{m}^{2}}}}

Wherep is the pressure,T is the temperature,R the ideal gas constant, andVm themolar volume.a andb are parameters that are determined empirically for each gas, but are sometimes estimated from theircritical temperature (Tc) andcritical pressure (pc) using these relations:a=27R2Tc264pc,b=RTc8pc{\displaystyle {\begin{aligned}a&={\frac {27R^{2}T_{\text{c}}^{2}}{64p_{\text{c}}}},&b&={\frac {RT_{\text{c}}}{8p_{\text{c}}}}\end{aligned}}}

The constants at critical point can be expressed as functions of the parameters a, b:pc=a27b2,Vm,c=3b,Tc=8a27bR,Zc=38{\displaystyle {\begin{aligned}p_{c}&={\frac {a}{27b^{2}}},&V_{m,c}&=3b,\\[2pt]T_{c}&={\frac {8a}{27bR}},&Z_{c}&={\frac {3}{8}}\end{aligned}}}

With thereduced propertiespr=p/pc{\displaystyle p_{r}=p/p_{\text{c}}},Vr=Vm/Vm,c{\displaystyle V_{r}=V_{\text{m}}/V_{\text{m,c}}},Tr=T/Tc{\displaystyle T_{r}=T/T_{\text{c}}} the equation can bewritten in thereduced form:pr=83TrVr−13−3Vr2{\displaystyle p_{r}={\frac {8}{3}}{\frac {T_{r}}{V_{r}-{\frac {1}{3}}}}-{\frac {3}{V_{r}^{2}}}}

Redlich–Kwong model

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Critical isotherm for Redlich-Kwong model in comparison to van-der-Waals model and ideal gas (with V0=RTc/pc)

TheRedlich–Kwong equation is another two-parameter equation that is used to model real gases. It is almost always more accurate than thevan der Waals equation, and often more accurate than some equations with more than two parameters. The equation isRT=(p+aTVm(Vm+b))(Vm−b){\displaystyle RT=\left(p+{\frac {a}{{\sqrt {T}}V_{\text{m}}\left(V_{\text{m}}+b\right)}}\right)\left(V_{\text{m}}-b\right)}

or alternatively:p=RTVm−b−aTVm(Vm+b){\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{{\sqrt {T}}V_{\text{m}}\left(V_{\text{m}}+b\right)}}}

wherea andb are two empirical parameters that arenot the same parameters as in the van der Waals equation. These parameters can be determined:a=0.42748R2Tc52pc,b=0.08664RTcpc{\displaystyle {\begin{aligned}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{aligned}}}

The constants at critical point can be expressed as functions of the parametersa,b:pc=[(23−1)73Ra2b5]1/3,Vm,c=b23−1,Tc=[3(23−1)2abR]2/3,Zc=13{\displaystyle {\begin{aligned}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{aligned}}}

Usingpr=p/pc{\displaystyle p_{r}=p/p_{\text{c}}},Vr=Vm/Vm,c{\displaystyle V_{r}=V_{\text{m}}/V_{\text{m,c}}},Tr=T/Tc{\displaystyle T_{r}=T/T_{\text{c}}} the equation of state can be written in thereduced form:pr=3TrVr−b′−1b′TrVr(Vr+b′){\displaystyle p_{r}={\frac {3T_{r}}{V_{r}-b'}}-{\frac {1}{b'{\sqrt {T_{r}}}V_{r}\left(V_{r}+b'\right)}}} withb′=23−1≈0.26{\displaystyle b'={\sqrt[{3}]{2}}-1\approx 0.26}

Berthelot and modified Berthelot model

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The Berthelot equation (named after D. Berthelot)[1] is very rarely used,p=RTVm−b−aTVm2{\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{TV_{\text{m}}^{2}}}}

but the modified version is somewhat more accuratep=RTVm[1+9128⋅ppc⋅TcT(1−6Tc2T2)]{\displaystyle p={\frac {RT}{V_{\text{m}}}}\left[1+{\frac {9}{128}}\cdot {\frac {p}{p_{c}}}\cdot {\frac {T_{c}}{T}}\left(1-6{\frac {T_{\text{c}}^{2}}{T^{2}}}\right)\right]}

Dieterici model

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This model (named after C. Dieterici[2]) fell out of usage in recent yearsp=RTVm−bexp⁡(−aVmRT){\displaystyle p={\frac {RT}{V_{\text{m}}-b}}\exp \left(-{\frac {a}{V_{\text{m}}RT}}\right)}

with parameters a, b. These can be normalized by dividing with the critical point state[note 1]:p~=p(2be)2a;T~=T4bRa;V~m=Vm12b{\displaystyle {\tilde {p}}=p{\frac {(2be)^{2}}{a}};\quad {\tilde {T}}=T{\frac {4bR}{a}};\quad {\tilde {V}}_{m}=V_{m}{\frac {1}{2b}}}which casts the equation into thereduced form:[3]p~(2V~m−1)=T~exp⁡(2−2T~V~m){\displaystyle {\tilde {p}}\left(2{\tilde {V}}_{m}-1\right)={\tilde {T}}\exp \left(2-{\frac {2}{{\tilde {T}}{\tilde {V}}_{m}}}\right)}

Clausius model

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The Clausius equation (named afterRudolf Clausius) is a very simple three-parameter equation used to model gases.RT=(p+aT(Vm+c)2)(Vm−b){\displaystyle RT=\left(p+{\frac {a}{T{\left(V_{\text{m}}+c\right)}^{2}}}\right)\left(V_{\text{m}}-b\right)}

or alternatively:p=RTVm−b−aT(Vm+c)2{\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{T\left(V_{\text{m}}+c\right)^{2}}}}

wherea=27R2Tc364pc,b=Vc−RTc4pc,c=3RTc8pc−Vc{\displaystyle {\begin{aligned}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{aligned}}}

whereVc is critical volume.

Virial model

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TheVirial equation derives from aperturbative treatment of statistical mechanics.pVm=RT[1+B(T)Vm+C(T)Vm2+D(T)Vm3+⋯]{\displaystyle pV_{\text{m}}=RT\left[1+{\frac {B(T)}{V_{\text{m}}}}+{\frac {C(T)}{V_{\text{m}}^{2}}}+{\frac {D(T)}{V_{\text{m}}^{3}}}+\cdots \right]}

or alternativelypVm=RT[1+B′(T)p+C′(T)p2+D′(T)p3+⋯]{\displaystyle pV_{\text{m}}=RT\left[1+B'(T)p+C'(T)p^{2}+D'(T)p^{3}+\cdots \right]}

whereA,B,C,A′,B′, andC′ are temperature dependent constants.

Peng–Robinson model

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Peng–Robinson equation of state (named afterD.-Y. Peng and D. B. Robinson[4]) has the interesting property being useful in modeling some liquids as well as real gases.p=RTVm−b−a(T)Vm(Vm+b)+b(Vm−b){\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a(T)}{V_{\text{m}}\left(V_{\text{m}}+b\right)+b\left(V_{\text{m}}-b\right)}}}

Wohl model

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Isotherm (V/V0->p_r) at critical temperature for Wohl model, van der Waals model and ideal gas model (with V0=RTc/pc)
Untersuchungen über die Zustandsgleichung, pp. 9,10,Zeitschr. f. Physikal. Chemie 87

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 drasticdecrease of pressure when the volume is contracted beyond the critical volume.p=RTVm−b−aTVm(Vm−b)+cT2Vm3{\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{TV_{\text{m}}\left(V_{\text{m}}-b\right)}}+{\frac {c}{T^{2}V_{\text{m}}^{3}}}\quad }

or:(p−cT2Vm3)(Vm−b)=RT−aTVm{\displaystyle \left(p-{\frac {c}{T^{2}V_{\text{m}}^{3}}}\right)\left(V_{\text{m}}-b\right)=RT-{\frac {a}{TV_{\text{m}}}}}

or, alternatively:RT=(p+aTVm(Vm−b)−cT2Vm3)(Vm−b){\displaystyle RT=\left(p+{\frac {a}{TV_{\text{m}}(V_{\text{m}}-b)}}-{\frac {c}{T^{2}V_{\text{m}}^{3}}}\right)\left(V_{\text{m}}-b\right)}

wherea=6pcTcVm,c2,b=Vm,c4,c=4pcTc2Vm,c3{\displaystyle {\begin{aligned}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{aligned}}} whereVm,c=415RTcpc{\displaystyle V_{\text{m,c}}={\frac {4}{15}}{\frac {RT_{c}}{p_{c}}}},pc{\displaystyle p_{\text{c}}},Tc{\displaystyle T_{c}} are (respectively) the molar volume, the pressure and the temperature at thecritical point.

And with thereduced propertiespr=p/pc{\displaystyle p_{r}=p/p_{\text{c}}},Vr=Vm/Vm,c{\displaystyle V_{r}=V_{\text{m}}/V_{\text{m,c}}},Tr=T/Tc{\displaystyle T_{r}=T/T_{\text{c}}} one can write the first equation in thereduced form:pr=154TrVr−14−6TrVr(Vr−14)+4Tr2Vr3{\displaystyle p_{r}={\frac {15}{4}}{\frac {T_{r}}{V_{r}-{\frac {1}{4}}}}-{\frac {6}{T_{r}V_{r}\left(V_{r}-{\frac {1}{4}}\right)}}+{\frac {4}{T_{r}^{2}V_{r}^{3}}}}

Beattie–Bridgeman model

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[6] This equation is based on five experimentally determined constants. It is expressed asp=RTVm2(1−cVmT3)(Vm+B)−AVm2{\displaystyle p={\frac {RT}{V_{\text{m}}^{2}}}\left(1-{\frac {c}{V_{\text{m}}T^{3}}}\right)(V_{\text{m}}+B)-{\frac {A}{V_{\text{m}}^{2}}}}

whereA=A0(1−aVm),B=B0(1−bVm){\displaystyle {\begin{aligned}A&=A_{0}\left(1-{\frac {a}{V_{\text{m}}}}\right),&B&=B_{0}\left(1-{\frac {b}{V_{\text{m}}}}\right)\end{aligned}}}

This equation is known to be reasonably accurate for densities up to about 0.8 ρcr, whereρcr is the density of the substance at its critical point. The constants appearing in the above equation are available in the following table whenp is in kPa,Vm is inm3kmol{\displaystyle {\frac {{\text{m}}^{3}}{{\text{k}}\,{\text{mol}}}}},T is in K andR=8.314kPa⋅m3kmol⋅K{\displaystyle R=8.314\mathrm {\frac {kPa\cdot m^{3}}{kmol\cdot K}} }[7]

GasA0aB0bc
Air131.84410.019310.04611−0.0011014.34×104
Argon, Ar130.78020.023280.039310.05.99×104
Carbon dioxide, CO2507.28360.071320.104760.072356.60×105
Ethane, C2H6595.7910.058610.094000.0191590.00×104
Helium, He2.18860.059840.014000.040
Hydrogen, H220.0117−0.005060.02096−0.04359504
Methane, CH4230.70690.018550.05587-0.0158712.83×104
Nitrogen, N2136.23150.026170.05046−0.006914.20×104
Oxygen, O2151.08570.025620.046240.0042084.80×104

Benedict–Webb–Rubin model

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Main article:Benedict–Webb–Rubin equation

The BWR equation,p=RTd+d2(RT(B+bd)−(A+ad−aαd4)−1T2[C−cd(1+γd2)exp⁡(−γd2)]){\displaystyle p=RTd+d^{2}\left(RT(B+bd)-\left(A+ad-a\alpha d^{4}\right)-{\frac {1}{T^{2}}}\left[C-cd\left(1+\gamma d^{2}\right)\exp \left(-\gamma d^{2}\right)\right]\right)}

whered is the molar density and wherea,b,c,A,B,C,α, andγ are empirical constants. Note that theγ constant is a derivative of constantα and therefore almost identical to 1.

Thermodynamic expansion work

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The expansion work of the real gas is different than that of the ideal gas by the quantity∫ViVf(RTVm−Preal)dV{\displaystyle \int _{V_{i}}^{V_{f}}\left({\frac {RT}{V_{m}}}-P_{\text{real}}\right)dV}.

See also

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References

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  1. ^D. Berthelot inTravaux 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. ^Pippard, Alfred B. (1981).Elements of classical thermodynamics: for advanced students of physics (Repr ed.). Cambridge: Univ. Pr. p. 74.ISBN 978-0-521-09101-5.
  4. ^Peng, D. Y. & Robinson, D. B. (1976). "A New Two-Constant Equation of State".Industrial and Engineering Chemistry: Fundamentals.15:59–64.doi:10.1021/i160057a011.S2CID 98225845.
  5. ^A. Wohl (1914). "Investigation of the condition equation".Zeitschrift für Physikalische Chemie.87:1–39.doi:10.1515/zpch-1914-8702.S2CID 92940790.
  6. ^Yunus A. Cengel and Michael A. Boles,Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010,ISBN 007-352932-X
  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
  1. ^The critical state can be calculated by starting withp=RT(Vm−b)eaRTVm{\displaystyle p={\frac {RT}{{(V_{m}-b)}e^{\frac {a}{RTV_{m}}}}}}, and taking the derivative with respect toVm{\displaystyle V_{m}}. The equation(∂Vmp)T=0{\displaystyle (\partial _{V_{m}}p)_{T}=0} is a quadratic equation inVm{\displaystyle V_{m}}, and it has a double root precisely whenVm=Vc;T=Tc{\displaystyle V_{m}=V_{c};T=T_{c}}.

Further reading

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External links

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