Tepelná kapacita při konstantním tlaku

$P$, $T$ - nezávislé stavové proměnné $\Rightarrow U=U(P,T)$, $V=V(P,T)$

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

1. TD zákon: $\delta Q=dU+PdV$:

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

$P=konst.\Rightarrow dP=0\Rightarrow$

$$\begin{array}{r}{C}_P={\left(\frac{\delta Q}{dT}\right)}_P={\left(\frac{\partial U}{\partial T}\right)}_P+P{\left(\frac{\partial V}{\partial T}\right)}_P\end{array}$$

Stavová veličina $H$- entalpie (tepelný obsah):

$$\begin{array}{r}H=U+PV\end{array}$$

($PV$ je „zásoba energie“, která by byla potřeba k tomu, aby si soustava vytvořila svůj objem $V$ proti okolnímu tlaku $P$)

$$\begin{array}{r}dH=dU+PdV+VdP=\delta Q+VdP\Rightarrow dP=0\Rightarrow dH=\delta Q\end{array}$$ $$\begin{array}{r}H=H(T,P)\Rightarrow dH={\left(\frac{\partial H}{\partial T}\right)}_{P}dT+{\left(\frac{\partial H}{\partial P}\right)}_{T}dP\Rightarrow dP=0\Rightarrow dH={\left(\frac{\partial H}{\partial T}\right)}_{P}dT\end{array}$$ $$\begin{array}{r}dH={\left(\frac{\partial H}{\partial T}\right)}_{P}dT=\delta Q\Rightarrow \end{array}$$ $$\begin{array}{r}{C}_P={\left(\frac{\delta Q}{dT}\right)}_P={\left(\frac{\partial H}{\partial T}\right)}_P\end{array}$$

1. TD zákon: $\delta Q=dU+PdV\Rightarrow$ pro plyny $C_P>C_V$ , pro kapaliny a pevné látky je $dV\approx 0\Rightarrow C_P\approx C_V$.