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`

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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/(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`

Spreadsheet for Specific heat ratio calculaton from Peng Robinson EOS