## Specific Heat Ratio of Real Gas

Equation of state is used to derive variety of thermodynamic properties. This article illustrate calculation of specific heat ratio from Peng Robinson Equation of state.

Example

**Calculate specific heat ratio ( γ = Cp/Cv ) for methane gas
at 11 Bar & 300 °K. Critical constants for Methane are as
following**

- Critical temperature, Tc : 190.6°K
- Critical Pressure, Pc : 46.002 bar
- Accentric Factor, ω : 0.008

Ideal gas specific heat constants Cp^{IG} = A + B.T +
C.T² + D.T³ are as following

- A = 4.5980
- B = 0.0125
- C = 2.86 x 10
^{-6} - D = -2.7 x 10
^{-9}

where Cp is in cal/mol-K

Peng Robinson equation of state is defined as

` P = RT / (V - b) - a / [V(V + b) + b(V - b)]`

where

`a`

_{c}= 0.45723553 R²Tc²/Pc`b = 0.077796074 RTc/Pc`

`m = 0.37464 + 1.54226ω - 0.26992ω²`

`a = a`

_{c}[1 + m(1 - (T/Tc)^{0.5})]²

Above equation is translated into polynomial form and solved for values of Z using Newton-Raphson method.

`Z³ - (1 - B)Z² + Z (A - 2B - 3B²) - (AB - B² - B³) = 0`

`Z = PV/RT`

`A = aP/ (RT)²`

`B = bP/ RT`

Following partial derivatives are required for calculating thermodynamic properties. First derivative is obtained by differentiation of P with respect to V at constant T.

`(δP/ δV)`

_{T}= -RT/(v - b)² + 2a(v + b)/[v(v + b) + b(v - b)]²`(δP/ δV)`

_{T}= -0.00485 bar/(cm^{3}/mol)

Second derivative is obtained by differentiation of P with respect to T at constant V.

`(δP/ δT)`

_{V}= R/(v - b) - a'/[v(v + b) + b(v - b)]`(δa/ δT)`

_{V}= -ma_{c}/[(TTc)^{0.5}(1 + m( 1 - (T/Tc)^{0.5}))]`(δP/ δT)`

_{V}= 0.039 bar/K`(δT/ δP)`

_{V}= 25.814 K/bar

Third derivative is obtained by differentiation of V with respect to T at constant P.

`(δV/ δT)`

_{P}= (R/P)[ T(δZ/δT)_{P}+ Z]`(δZ/ δT)`

_{P}= Num / Denom`Num = (δA/δT)`

_{P}(B-Z) + (δB/δT)_{P}(6BZ+2Z-3B²-2B+A-Z²)`Denom = 3Z² + 2(B-1)Z + (A-2B-3B²)`

where,

`(δA/δT)`

_{P}= (P/(RT)²)(a' - 2a/T)`(δB/δT)`

_{P}= -bP/(RT²)

Calculation of Heat Capacities

Ideal gas heat capacity Cp^{IG} is calculated at 300 °K
from polynomial equation provided above. Specific heat at
constant volume for ideal gas, Cv^{IG} is calculated
using following relation.

` Cv`^{IG} = Cp^{IG} - R

Residual heat capacity at constant volume Cv ^{R} is
calculated from internal energy U ^{R} as following.

`Cv`

^{R}= (δU^{R}/δT)_{V}`U`

^{R}= [(Ta'-a)/b(8)^{0.5}] ln[(Z+B(1+2^{0.5}))/(Z+B(1-2^{0.5}))]`Cv`

^{R}= [Ta"/b(8)^{0.5}] ln[(Z+B(1+2^{0.5}))/(Z+B(1-2^{0.5}))]

where,

` a" = a`_{c} m (1 + m)(Tc/T)^{0.5}/ (2TTc)

Specific heat capacity at constant pressure and volume is calculated using following equation.

`Cp`

^{R}= Cv^{R}+ T(δP/δT)_{V}(δV/δT)_{P}- R`Cp = Cp`

^{IG}+ Cp^{R}`Cv = Cv`

^{IG}+ Cv^{R}

Specific heat ratio is obtained as :

`γ = Cp / Cv`

`γ = 1.338`

Resources

- Spreadsheet for Specific heat ratio calculaton from Peng Robinson EOS