|
| | \begin{equation*} \begin{split}
dU = &\;\; T dS - p dV = \left(\frac{\partial U}{\partial S} \right)_V dS + \left(\frac{\partial U}{\partial V} \right)_S
dV \quad \Rightarrow \quad \left(\frac{\partial U}{\partial S} \right)_V = T \qquad \left(\frac{\partial U}{\partial V}
\right)_S = - p\\ dF = &\;\; - S dT - p dV = \left(\frac{\partial F}{\partial T} \right)_V dT + \left(\frac{\partial F}{\partial
V} \right)_T dV \quad \Rightarrow \quad \left(\frac{\partial F}{\partial T} \right)_V = -S \qquad \left(\frac{\partial F}{\partial
V} \right)_T = - p\\ dH = &\;\; T dS + V dp = \left(\frac{\partial H}{\partial S} \right)_p dS + \left(\frac{\partial H}{\partial
p} \right)_S dp \quad \Rightarrow \quad \left(\frac{\partial H}{\partial S} \right)_p = T \qquad \left(\frac{\partial H}{\partial
p} \right)_S = V\\ dG = &\;\; - S dT + V dp = \left(\frac{\partial G}{\partial T} \right)_p dT + \left(\frac{\partial G}{\partial
p} \right)_T dp \quad \Rightarrow \quad \left(\frac{\partial G}{\partial T} \right)_p = - S \qquad \left(\frac{\partial
G}{\partial p} \right)_T = V\\ \end{split} \label{eq:Guggenheim_exact_differentials} \end{equation*} | (3.61) |
