Redlich - Kwong equation of state

Equation of state

The article is part of the Thermodynamics series.
The ideal gas equation of state
Van der Waals equation
Berthelot's equation
Diterichi equation
Beatty-Bridgman Equation
Redlich - Kwong equation of state
Peng - Robinson equation of state
Barner-Adler equation of state
Sugi's equation of state - Liu
Equation of state Benedict - Webb - Rubin
The equation of state of Lee - Erbar - Edmister
Mi - Gruneisen equation of state
Sections of thermodynamics
The beginnings of thermodynamics
Equation of state
Thermodynamic quantities
Thermodynamic potentials
Thermodynamic cycles
Phase transitions

See also “Physical Portal”

The Redlich – Kwong equation of state is a two-parameter equation of state for a real gas obtained by O. Redlich ( Eng. O. Redlich ) and J. Kwong ( Eng. JNS Kwong ) in 1949 as an improvement of the Van der Waals equation ^[1] . At the same time, Otto Redlich in his article ^{[2] of} 1975 writes that the equation is not based on theoretical justification, but is essentially a successful empirical modification of previously known equations.

The equation has the form:

P={\frac {RT}{V-b}}-{\frac {a}{T^{0{,}5}V(V+b)}},

{\ displaystyle P = {\ frac {RT} {Vb}} - {\ frac {a} {T ^ {0 {,} 5} V (V + b)}},}

{\ displaystyle P = {\ frac {RT} {V-b}} - {\ frac {a} {T ^ {0 {,} 5} V (V + b)}},}

Where $P$ ${\ displaystyle P}$ $P$ - pressure , Pa;

$T$ ${\ displaystyle T}$ $T$ - absolute temperature , K;
$V$ ${\ displaystyle V}$ $V$ - molar volume , m³ / mol;
$R=8{,}31441\pm 0{,}00026$ ${\ displaystyle R = 8 {,} 31441 \ pm 0 {,} 00026}$ ${\ displaystyle R = 8 {,} 31441 \ pm 0 {,} 00026}$ - universal gas constant , J / (mol · K);
$a$ ${\ displaystyle a}$ $a$ and $b$ ${\ displaystyle b}$ $b$ - some constants depending on a specific substance.

From the conditions of thermodynamic stability at a critical point - $\left({\frac {dT}{dV}}\right)_{T_{\mathrm {k} }}=0$ ${\ displaystyle \ left ({\ frac {dT} {dV}} \ right) _ {T _ {\ mathrm {k}}} = 0}$ ${\ displaystyle \ left ({\ frac {dT} {dV}} \ right) _ {T _ {\ mathrm {k}}} = 0}$ and $\left({\frac {d^{2}T}{dV^{2}}}\right)_{T_{\mathrm {k} }}=0$ ${\ displaystyle \ left ({\ frac {d ^ {2} T} {dV ^ {2}}} \ right) _ {T _ {\ mathrm {k}}} = 0}$ ${\ displaystyle \ left ({\ frac {d ^ {2} T} {dV ^ {2}}} \ right) _ {T _ {\ mathrm {k}}} = 0}$ ( $T_{\mathrm {k} }$ ${\ displaystyle T _ {\ mathrm {k}}}$ ${\ displaystyle T _ {\ mathrm {k}}}$ - critical temperature ) - you can get that:

a={\frac {1}{9\cdot ({\sqrt[{3}]{2}}-1)}}{\frac {R^{2}T_{\mathrm {k} }^{2{,}5}}{P_{\mathrm {k} }}}\approx {\frac {0{,}42748R^{2}T_{\mathrm {k} }^{2{,}5}}{P_{\mathrm {k} }}},

{\ displaystyle a = {\ frac {1} {9 \ cdot ({\ sqrt [{3}] {2}} - 1)}} {\ frac {R ^ {2} T _ {\ mathrm {k}} ^ {2 {,} 5}} {P _ {\ mathrm {k}}}} approx {\ frac {0 {,} 42748R ^ {2} T _ {\ mathrm {k}} ^ {2 {,} 5 }} {P _ {\ mathrm {k}}}},}

{\ displaystyle a = {\ frac {1} {9 \ cdot ({\ sqrt [{3}] {2}} - 1)}} {\ frac {R ^ {2} T _ {\ mathrm {k}} ^ {2 {,} 5}} {P _ {\ mathrm {k}}}} approx {\ frac {0 {,} 42748R ^ {2} T _ {\ mathrm {k}} ^ {2 {,} 5 }} {P _ {\ mathrm {k}}}},}

b={\frac {{\sqrt[{3}]{2}}-1}{3}}{\frac {RT_{\mathrm {k} }}{P_{\mathrm {k} }}}\approx {\frac {0{,}08664RT_{\mathrm {k} }}{P_{\mathrm {k} }}},

{\ displaystyle b = {\ frac {{\ sqrt [{3}] {2}} - 1} {3}} {\ frac {RT _ {\ mathrm {k}}} {P _ {\ mathrm {k}} }} \ approx {\ frac {0 {,} 08664RT _ {\ mathrm {k}}} {P _ {\ mathrm {k}}}},}

{\ displaystyle b = {\ frac {{\ sqrt [{3}] {2}} - 1} {3}} {\ frac {RT _ {\ mathrm {k}}} {P _ {\ mathrm {k}} }} \ approx {\ frac {0 {,} 08664RT _ {\ mathrm {k}}} {P _ {\ mathrm {k}}}},}

Where $P_{\mathrm {k} }$ ${\ displaystyle P _ {\ mathrm {k}}}$ ${\ displaystyle P _ {\ mathrm {k}}}$ - critical pressure .

It is of interest to solve the Redlich - Kwong equation with respect to the compressibility coefficient $Z={\frac {PV}{RT}}$ ${\ displaystyle Z = {\ frac {PV} {RT}}}$ ${\ displaystyle Z = {\ frac {PV} {RT}}}$ . In this case, we have the cubic equation:

Z^{3}-Z^{2}+(A-B^{2}-B)Z-AB=0,

{\ displaystyle Z ^ {3} -Z ^ {2} + (AB ^ {2} -B) Z-AB = 0,}

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

Where $A={\frac {aP}{R^{2}T^{2{,}5}}},\;B={\frac {bP}{RT}}$ ${\ displaystyle A = {\ frac {aP} {R ^ {2} T ^ {2 {,} 5}}}, \; B = {\ frac {bP} {RT}}}$ ${\ displaystyle A = {\ frac {aP} {R ^ {2} T ^ {2 {,} 5}}}, \; B = {\ frac {bP} {RT}}}$ .

The Redlich - Kwong equation is applicable if the condition ${\frac {P}{P_{\mathrm {k} }}}<0{,}5{\frac {T}{T_{\mathrm {k} }}}$ ${\ displaystyle {\ frac {P} {P _ {\ mathrm {k}}}} <0 {,} 5 {\ frac {T} {T _ {\ mathrm {k}}}}}$ ${\ displaystyle {\ frac {P} {P _ {\ mathrm {k}}}} <0 {,} 5 {\ frac {T} {T _ {\ mathrm {k}}}}}$ .

After 1949, several generalizations and modifications of the Redlich – Kwong equation were obtained (see below), however, as A. Bjerre and T. Bak showed ^{[3], the} original equation more accurately describes the behavior of gases.

Gray - Rent - Zudkevich Modification

R. Gray ( RD Gray, Jr. ), N. Rent ( NH Rent ) and D. Zudkevich proposed ^{[4] to} adjust the compressibility factor $Z_{\mathrm {RK} }$ ${\ displaystyle Z _ {\ mathrm {RK}}}$ obtained from the cubic Redlich - Kwong equation by introducing a correction term $\Delta Z$ ${\ displaystyle \ Delta Z}$ :

Z'=Z_{\mathrm {RK} }+\Delta Z,

{\ displaystyle Z '= Z _ {\ mathrm {RK}} + \ Delta Z,}

Where $Z^{'}$ ${\ displaystyle Z '}$ - modified compressibility factor;

\Delta Z=-0{,}04666626T_{\mathrm {r} }^{2}P_{\mathrm {r} }^{2}\exp[-7000(1-T_{\mathrm {r} })^{2}-770(1{,}02-P_{\mathrm {r} })^{2}]\,-

{\ displaystyle \ Delta Z = -0 {,} 04666626T _ {\ mathrm {r}} ^ {2} P _ {\ mathrm {r}} ^ {2} \ exp [-7000 (1-T _ {\ mathrm {r }}) ^ {2} -770 (1 {,} 02-P _ {\ mathrm {r}}) ^ {2}] \, -}

-\,\omega (0{,}464419-0{,}424568T_{\mathrm {r} }^{2}){\frac {P_{\mathrm {r} }}{T_{\mathrm {r} }^{4}+P_{\mathrm {r} }^{4}}}\,-\,\omega (41{,}76451266-40{,}47298767T_{\mathrm {r} }){\frac {P_{\mathrm {r} }^{2}}{(1+T_{\mathrm {r} })^{4}+P_{\mathrm {r} }^{4}}}\,-

{\ displaystyle - \, \ omega (0 {,} 464419-0 {,} 424568T _ {\ mathrm {r}} ^ {2}) {\ frac {P _ {\ mathrm {r}}} {T _ {\ mathrm {r}} ^ {4} + P _ {\ mathrm {r}} ^ {4}}} \, - \, \ omega (41 {,} 76451266-40 {,} 47298767T _ {\ mathrm {r}}) {\ frac {P _ {\ mathrm {r}} ^ {2}} {(1 + T _ {\ mathrm {r}}) ^ {4} + P _ {\ mathrm {r}} ^ {4}}} \ , -}

-\,[0{,}11386032-\omega (12{,}55135462-12{,}5583112T_{\mathrm {r} })]{\frac {P_{\mathrm {r} }^{3}}{(1+T_{\mathrm {r} })^{4}+P_{\mathrm {r} }^{4}}},

{\ displaystyle - \, [0 {,} 11386032- \ omega (12 {,} 55135462-12 {,} 5583112T _ {\ mathrm {r}}]] {\ frac {P _ {\ mathrm {r}} ^ { 3}} {(1 + T _ {\ mathrm {r}}) ^ {4} + P _ {\ mathrm {r}} ^ {4}}},}

Where $T_{\mathrm {r} }={\frac {T}{T_{\mathrm {k} }}}$ ${\ displaystyle T _ {\ mathrm {r}} = {\ frac {T} {T _ {\ mathrm {k}}}}}$ - reduced temperature $P_{\mathrm {r} }={\frac {P}{P_{\mathrm {k} }}}$ ${\ displaystyle P _ {\ mathrm {r}} = {\ frac {P} {P _ {\ mathrm {k}}}}}$ - reduced pressure $\omega$ ${\ displaystyle \ omega}$ - the factor of acentricity .

Gray and others modification obtained for $T_{\mathrm {r} }<1{,}1$ ${\ displaystyle T _ {\ mathrm {r}} <1 {,} 1}$ and $P_{\mathrm {r} }\leqslant 2{,}0$ ${\ displaystyle P _ {\ mathrm {r}} \ leqslant 2 {,} 0}$ .

Other Modifications

Another way to obtain modifications of the original Redlich - Kwong equation of state is to write it in the form:

Z={\frac {V}{V-b}}-{\frac {1}{3({\sqrt[{3}]{2}}-1)^{2}}}{\frac {b}{V+b}}F(\omega ,\;T_{\mathrm {r} }),

{\ displaystyle Z = {\ frac {V} {Vb}} - {\ frac {1} {3 ({\ sqrt [{3}] {2}} - 1) ^ {2}}} {\ frac { b} {V + b}} F (\ omega, \; T _ {\ mathrm {r}}),}

Where $F(\omega ,\;T_{\mathrm {r} })$ ${\ displaystyle F (\ omega, \; T _ {\ mathrm {r}})}$ - modifying function.

For the Redlich - Kwong equation itself $F(\omega ,\;T_{\mathrm {r} })=F(T_{\mathrm {r} })=T_{\mathrm {r} }^{-1{,}5}$ ${\ displaystyle F (\ omega, \; T _ {\ mathrm {r}}) = F (T _ {\ mathrm {r}}) = T _ {\ mathrm {r}} ^ {- 1 {,} 5}}$ .

Wilson Modification

G. Wilson ^[5] ^[6] ( GM Wilson ) modifying function has the form:

F(\omega ,\;T_{\mathrm {r} })=1+(1{,}57+1{,}62\omega )\left({\frac {1}{T_{\mathrm {r} }}}-1\right).

{\ displaystyle F (\ omega, \; T _ {\ mathrm {r}}) = 1+ (1 {,} 57 + 1 {,} 62 \ omega) \ left ({\ frac {1} {T _ {\ mathrm {r}}}} - 1 \ right).}

Wilson showed that his form of the equation gives good results on corrections to the enthalpy of pressure not only for polar (including ammonia ), but also for non-polar substances .

Barnet King Modification

Barnet ^[7] ( FJ Barnès ), and later King ^[8] ( CJ King ) proposed the following modification in 1973–74:

F(\omega ,\;T_{\mathrm {r} })=1+(0{,}9+1{,}21\omega )(T_{\mathrm {r} }^{-1{,}5}-1).

{\ displaystyle F (\ omega, \; T _ {\ mathrm {r}}) = 1+ (0 {,} 9 + 1 {,} 21 \ omega) (T _ {\ mathrm {r}} ^ {- 1 {,} 5} -1).}

Barnet and King also applied their modification to mixtures of both hydrocarbons and non-hydrocarbons.

Soave Modification

G. Soave was proposed ^{[9] the} following equation:

F(\omega ,\;T_{\mathrm {r} })={\frac {1}{T_{\mathrm {r} }}}[1+(0{,}480+1{,}574\omega -0{,}176\omega ^{2})(1-T_{\mathrm {r} }^{0{,}5})]^{2}.

{\ displaystyle F (\ omega, \; T _ {\ mathrm {r}}) = {\ frac {1} {T _ {\ mathrm {r}}} [1+ (0 {,} 480 + 1 {, } 574 \ omega -0 {,} 176 \ omega ^ {2}) (1-T _ {\ mathrm {r}} ^ {0 {,} 5})] ^ {2}.}

For hydrogen , a simpler equation was obtained:

F(\omega ,\;T_{\mathrm {r} })=F(T_{\mathrm {r} })=1{,}202\exp(-0{,}30288T_{\mathrm {r} }).

{\ displaystyle F (\ omega, \; T _ {\ mathrm {r}}) = F (T _ {\ mathrm {r}}) = 1 {,} 202 \ exp (-0 {,} 30288T _ {\ mathrm { r}}).}

West ( EW West ) and Erbar ( JH Erbar ), using the Soave equation for light hydrocarbon systems, came to the conclusion ^[10] that it is very accurate in determining the parameters of the vapor – liquid phase equilibrium and corrections to the pressure enthalpy.

Literature

Reed R., Prausnits J., Sherwood T. Properties of gases and liquids: a Reference manual / Per. from English under the editorship of B.I.Sokolova. - 3rd ed. - L .: Chemistry, 1982. - 592 p.
Wales S. Phase equilibrium in chemical technology: In 2 hours. Part 1. - M .: Mir, 1989. - 304 p. - ISBN 5-03-001106-4 . .

Notes

↑ Redlich O., Kwong JNS On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions // Chemical Reviews. - 1949. - T. 44 , No. 1 . - S. 233–244 . (inaccessible link)
↑ Redlich O. On the Three-Parameter Representation of the Equation of State // Industrial and Engineering Chemistry Fundamentals. - 1975.- T. 14 , no. 3 . - S. 257-260 .
↑ Bjerre A., Bak TA Two-Parameter Equations of State // Acta Chemica Scandinavica. - 1969 .-- T. 23 . - S. 1733-1744 .
↑ Gray RD, Jr., Rent NH and Zudkevitch D. A modified Redlich - Kwong equation of state // The American Institute of Chemical Engineers Journal. - 1970. - T. 16 , no. 6 . - S. 991-998 . (inaccessible link)
↑ Wilson GM // Advances in Cryogenic Engineering. - 1964 .-- T. 9 . - S. 168 .
↑ Wilson GM // Advances in Cryogenic Engineering. - 1966. - T. 11 . - S. 392 .
↑ Barnès FJ Ph. D. thesis. Department of Chemical Engineering, University of California, Berkeley, 1973.
↑ King CJ Personal communication, 1974.
↑ Soave G. Equilibrium constants from a modified Redlich - Kwong equation of state // Chemical Engineering Science. - 1972. - T. 27 , no. 6 . - S. 1197-1203 .
↑ West EW, Erbar JH An Evaluation of Four Methods of Predicting the Thermodynamic Properties of Light Hydrocarbon Systems // Paper presented at 52d Annual Meeting NGPA, Dallas, Tex., March 26-28. - 1972.

[1] Redlich O., Kwong JNS On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions // Chemical Reviews. - 1949. - T. 44 , No. 1 . - S. 233–244 . (inaccessible link)

[2] Redlich O. On the Three-Parameter Representation of the Equation of State // Industrial and Engineering Chemistry Fundamentals. - 1975.- T. 14 , no. 3 . - S. 257-260 .

[3] Bjerre A., Bak TA Two-Parameter Equations of State // Acta Chemica Scandinavica. - 1969 .-- T. 23 . - S. 1733-1744 .

[4] Gray RD, Jr., Rent NH and Zudkevitch D. A modified Redlich - Kwong equation of state // The American Institute of Chemical Engineers Journal. - 1970. - T. 16 , no. 6 . - S. 991-998 . (inaccessible link)

[5] Wilson GM // Advances in Cryogenic Engineering. - 1964 .-- T. 9 . - S. 168 .

[6] Wilson GM // Advances in Cryogenic Engineering. - 1966. - T. 11 . - S. 392 .

[7] Barnès FJ Ph. D. thesis. Department of Chemical Engineering, University of California, Berkeley, 1973.

[8] King CJ Personal communication, 1974.

[9] Soave G. Equilibrium constants from a modified Redlich - Kwong equation of state // Chemical Engineering Science. - 1972. - T. 27 , no. 6 . - S. 1197-1203 .

[10] West EW, Erbar JH An Evaluation of Four Methods of Predicting the Thermodynamic Properties of Light Hydrocarbon Systems // Paper presented at 52d Annual Meeting NGPA, Dallas, Tex., March 26-28. - 1972.