Vnitřní energie jako funkce objemu

$$\begin{array}{rlrl}& dU={\left(\frac{\partial U}{\partial V}\right)}_{T}dV+{\left(\frac{\partial U}{\partial T}\right)}_{V}dT& & C_V={\left(\frac{\partial U}{\partial T}\right)}_V\end{array}$$

Ideální plyn $\Rightarrow U=U(T)\Rightarrow$

$$\begin{array}{r}dU=C_{V}dT\end{array}$$

1. TD zákon pro objemovou práci:

$$\begin{array}{r}\delta Q=C_{V}dT+PdV\end{array}$$ $$\begin{array}{r}{\left(\frac{\partial U}{\partial V}\right)}_T(=0)=\frac{C_P-C_V}{V\beta }-P\Rightarrow C_P-C_V=\beta PV\end{array}$$ $$\begin{array}{r}V=\frac{nRT}{P}\Rightarrow \beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_P=\frac{nR}{PV}\end{array}$$

Mayerův vztah:

$$\begin{array}{rlrl}{C}_P-C_V& =nR& C_{mP}-C_{mV}& =R\end{array}$$