Specific isochoric heat capacity

Values for calculation

Inverse reduced temperature

$τ$

Second partial derivative of $ γ $ with respect to $ τ $

$γ_{ττ}$

Derivative of $ γ $ with respect to the dimensionless pressure $ π $

$γ_π$

Cross derivative of $ γ $ with respect to $ π $ and temperature $ τ $

$γ_{πτ}$

Second partial derivative of $ γ $ with respect to $ π $

$γ_{ππ}$

Specific gas constant of ordinary water

$R$ $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$

Second partial derivative of $ γ^o $ with respect to $ τ $

$γ^o_{ττ}$

Second partial derivative of $ γ^r $ with respect to $ τ $

$γ^r_{ττ}$

Reduced pressure

$π$

Derivative of $ γ^r $ with respect to the dimensionless pressure $ π $

$γ^r_π$

Cross derivative of $ γ^r $ with respect to $ π $ and temperature $ τ $

$γ^r_{πτ}$

Second partial derivative of $ γ^r $ with respect to $ π $

$γ^r_{ππ}$

Second partial derivative of $ φ $ with respect to $ τ $

$φ_{ττ}$

Region

$\text{Region}$

$\text{if }\ \text{Region}= 1$

$$c_ν=\left(-τ^2\cdot γ_{ττ}+\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{γ_{ππ}}\right)\cdot R$$

$\text{else if }\ \text{Region}= 2$

$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$

$\text{else if }\ \text{Region}= 3$

$$c_ν=\left(-τ^2\cdot φ_{ττ}\right)\cdot R$$

$\text{else if }\ \text{Region}= 5$

$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$

$\text{else}$

$$c_ν=0$$

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