Chapter 12 Thermodynamic Property Relations

Summary

Gibbs Equation

$$\begin{aligned}& d u=T d s-P d v \\& d h=T d s+v d P \\& d a=-s d T-p d v \\& d g=-s d T+v d p\end{aligned}$$

Maxwell Relation

$$\begin{aligned}& \left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial P}{\partial S}\right)_{V} ;\left(\frac{\partial T}{\partial P}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{P} \\& \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V} ;\left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{P}\end{aligned}$$

Clapeyron equation

$$\left(\frac{d P}{d T}\right)_{\text {sat }}=\frac{h f_{g}}{T V_{f g}}$$

For liquide-vaper and selide-vapor at low pressures

$$\ln \left(\frac{P_{2}}{P_{1}}\right)_{\text {sat }} \simeq \frac{h_{g}}{R} \left(\frac{1}{T_1}-\frac{1}{T_{2}} \right)_{\operatorname{sat}}$$

du, dh, ds

$$\begin{aligned} & d u=C_{v} \cdot d T+\left[T\left(\frac{d P}{d T}\right){v}-p\right] d v \\ & d h=C{p} \cdot d T+\left[v-T\left(\frac{\partial V}{\partial T}\right)_{p}\right] d p \end{aligned}$$ $$\begin{aligned}\quad d s & =\frac{C_v}{T} d T+\left(\frac{\partial P}{\partial T}\right)_{V} \cdot d V \\d s & =\frac{C_P}{T} d T -\left(\frac{\partial V}{\partial T}\right)_{P} \cdot d p \\U_{2}-U_{1} & =\int_{T_{1}}^{T_{2}} C_{V} d \tau+\int_{V_1}^{V_2}\left[T \cdot \left(\frac{\partial P}{\partial T}\right)_{V}-P\right] d V\end{aligned}$$

specific heats

$$\begin{aligned}& \left(\frac{\partial C_{V}}{\partial V}\right)_{T}=T\left(\frac{\partial^{2} P}{\partial T^{2}}\right)_{V} ; \quad\left(\frac{\partial C_{P}}{\partial p}\right)_{T}=-T\left(\frac{\partial^{2} V}{\partial T^{2}}\right)_{P} \\& \text { } C_{p, T}-C_{p_ 0, T}=-T \int_{0}^{p_{1}}\left(\frac{\partial^{2} V}{\partial T^{2}}\right)_{p} \cdot d p \\& \text { } C_{P}-C_{V}=-T\left(\frac{\partial V}{\partial T}\right)_{P}^{2}\left(\frac{\partial P}{\partial V}\right)_{T} \\& =\frac{V T \beta^{2}}{\alpha}, \beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}, \alpha=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{\phi T}\end{aligned}$$

Joule - Thomson coefficient

$$\mu_{J T}=\left(\frac{\partial T}{\partial p}\right)_{n}=-\frac{1}{c_ p}\left[V-T\left(\frac{\partial V}{\partial T}\right)_{p}\right]$$

Généralized enthalpy

$$\begin{aligned}& \bar{h}_{2}-\bar{h}_{1}=\left(\bar{h}_{2}-\bar{h}_{1}\right) _{i deal }-R_{u} T_{a T}\left(Z_{n_{2}}-Z_{n_{1}}\right) \\& \bar{u}_{2}-\bar{u}_{1}=\left(\bar{h}_{2}-\bar{h}_{1}\right)-R_{u}\left(Z_{2} T_{2}-Z_{2} T_{1}\right) \\& \bar{s}_{2}-\bar{s}_1=\left(\bar{s}_{2}-\bar{s}_{1}\right) _ { ideal }-R_{u}\left(Z _{s_{2}}-Z_{s_{1}}\right)\end{aligned}$$