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Specific isochoric heat capacity

Values for calculation

$ \{τ\in\mathbb{R}\mid τ>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ τ $
$ \{γ_{ττ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ_{ττ} $
$ \{γ_π\in\mathbb{R}\} $
This set includes all real numbers.
$ γ_π $
$ \{γ_{πτ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ_{πτ} $
$ \{γ_{ππ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ_{ππ} $
$ R $ $ \mathrm{J\cdot\ kg^{-1}\cdot\ K^{-1}} $
$ \{γ^o_{ττ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ^o_{ττ} $
$ \{γ^r_{ττ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ^r_{ττ} $
$ \{π\in\mathbb{R}\mid π>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ π $
$ \{γ^r_π\in\mathbb{R}\} $
This set includes all real numbers.
$ γ^r_π $
$ \{γ^r_{πτ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ^r_{πτ} $
$ \{γ^r_{ππ}\in\mathbb{R}\} $
This set includes all real numbers.
$ γ^r_{ππ} $
$ \{φ_{ττ}\in\mathbb{R}\} $
This set includes all real numbers.
$ φ_{ττ} $
$ \text{Region} $

Calculation

Specific isochoric heat capacity

$\text{if }\ \text{Region}= 1$
$\text{if }\ \text{Region}= 1$
Equations in LaTeX code
\text{if }\ \text{Region}= 1
$$c_ν=\left(-τ^2\cdot γ_{ττ}+\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{γ_{ππ}}\right)\cdot R$$
$c_ν=\left(-τ^2\cdot γ_{ττ}+\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{γ_{ππ}}\right)\cdot R$
Equations in LaTeX code
c_ν=\left(-τ^2\cdot γ_{ττ}+\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{γ_{ππ}}\right)\cdot R
$\text{else if }\ \text{Region}= 2$
$\text{else if }\ \text{Region}= 2$
Equations in LaTeX code
\text{else if }\ \text{Region}= 2
$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$
$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$
Equations in LaTeX code
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
$\text{else if }\ \text{Region}= 3$
$\text{else if }\ \text{Region}= 3$
Equations in LaTeX code
\text{else if }\ \text{Region}= 3
$$c_ν=\left(-τ^2\cdot φ_{ττ}\right)\cdot R$$
$c_ν=\left(-τ^2\cdot φ_{ττ}\right)\cdot R$
Equations in LaTeX code
c_ν=\left(-τ^2\cdot φ_{ττ}\right)\cdot R
$\text{else if }\ \text{Region}= 5$
$\text{else if }\ \text{Region}= 5$
Equations in LaTeX code
\text{else if }\ \text{Region}= 5
$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$
$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$
Equations in LaTeX code
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
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$c_ν=0$$
$c_ν=0$
Equations in LaTeX code
c_ν=0
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