Tepelná kapacita při konstantním objemu

Jednoduchá TS - její stav je určen třemi stavovými veličinami: $P$, $V$, $T$.

$$\begin{array}{r}U=U(P,V,T)\Rightarrow P=g(V,T)\Rightarrow U=U(g(V,T),V,T)\Rightarrow U=U(V,T)\end{array}$$

$V$, $T$ - nezávislé stavové proměnné

$$\begin{array}{r}dU={\left(\frac{\partial U}{\partial V}\right)}_T+{\left(\frac{\partial U}{\partial T}\right)}_{V}dT\end{array}$$

1. TD zákon: $\delta Q=dU+PdV\Rightarrow$

$$\begin{array}{r}\delta Q=\left[{\left(\frac{\partial U}{\partial V}\right)}_T+P\right]dV+{\left(\frac{\partial U}{\partial T}\right)}_{V}dT\end{array}$$

$V=konst.\Rightarrow dV=0\Rightarrow$

$$\begin{array}{r}{C}_V={\left(\frac{\delta Q}{dT}\right)}_V={\left(\frac{\partial U}{\partial T}\right)}_V\end{array}$$ $$\begin{array}{rlrl}& \delta Q=\left[{\left(\frac{\partial U}{\partial V}\right)}_T+P\right]dV+{\left(\frac{\partial U}{\partial T}\right)}_{V}dT& & C_V={\left(\frac{\partial U}{\partial T}\right)}_V\end{array}$$ $$\begin{array}{r}\delta Q=\left[{\left(\frac{\partial U}{\partial V}\right)}_T+P\right]dV+C_{V}dT\end{array}$$