## 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 CpIG = 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

```` ac = 0.45723553 R²Tc²/Pc`
` b = 0.077796074 RTc/Pc`
` m = 0.37464 + 1.54226ω - 0.26992ω² `
` a = ac[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````

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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/(cm3/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 = -mac/[(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 = 3Z&sup2 + 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 CpIG is calculated at 300 °K from polynomial equation provided above. Specific heat at constant volume for ideal gas, CvIG is calculated using following relation.

`` CvIG = CpIG - R``

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

```` CvR = (δUR/δT)V`
` UR = [(Ta'-a)/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]`
` CvR = [Ta"/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]````

where,

`` a" = ac m (1 + m)(Tc/T)0.5/ (2TTc)``

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

```` CpR = CvR + T(δP/δT)V(δV/δT)P - R`
` Cp = CpIG + CpR`
` Cv = CvIG + CvR````

Specific heat ratio is obtained as :

```` γ = Cp / Cv`
` γ = 1.338````

Spreadsheet for Specific heat ratio calculaton from Peng Robinson EOS