和達三樹《物理用數學》習題1.4.2，Maxwell熱力學關係式

==問題==

令溫度為T，壓力為p，體積為V，熵為S，根據熱力學公式
\begin{align*}
dU&=TdS-pdV &\quad& (U\text{為內能})\\
dH&=TdS+Vdp &\quad& (H\text{為焓})\\
dF&=-SdT-pdV &\quad& (F\text{為亥姆霍茲(Helmholtz)自由能})\\
dG&=-SdT+Vdp &\quad& (G\text{為吉布斯(Gibbs)自由能})
\end{align*}
證明Maxwell關係式
\begin{align*}
\left( \frac{\partial p}{\partial S} \right)_V = -\left( \frac{\partial T}{\partial V} \right)_S, & \left( \frac{\partial V}{\partial S} \right)_p = \left( \frac{\partial T}{\partial p} \right)_S \\
\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{align*}

==解答==

由全微分(total differential)
$$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy.$$
對比
\begin{align*}
dU&=TdS-pdV,\\
dH&=TdS+Vdp,\\
dF&=-SdT-pdV,\\
dG&=-SdT+Vdp.
\end{align*}

可得

\begin{align*} &\frac{\partial U}{\partial S} = T, \frac{\partial U}{\partial V} = -p.\\ &\frac{\partial H}{\partial S} = T, \frac{\partial H}{\partial p} = V.\\ &\frac{\partial F}{\partial T} = -S, \frac{\partial F}{\partial V} = -p.\\ &\frac{\partial G}{\partial T} = -S, \frac{\partial G}{\partial p} = V. \end{align*}

於是
\begin{align*} &\frac{\partial p}{\partial S} = \frac{\partial}{\partial S}p = \frac{\partial}{\partial S} \left( -\frac{\partial U}{\partial V} \right) = - \frac{\partial^2 U}{\partial S \partial V} = - \frac{\partial^2 U}{\partial V \partial S} = - \frac{\partial}{\partial V} \left( \frac{\partial U}{\partial S} \right) = - \frac{\partial}{\partial V} T = - \frac{\partial T}{\partial V}, \\ &\frac{\partial V}{\partial S} = \frac{\partial}{\partial S} V = \frac{\partial}{\partial S} \left( \frac{\partial H}{\partial p} \right) = \frac{\partial^2 H}{\partial S \partial p} = \frac{\partial^2 H}{\partial p \partial S} = \frac{\partial}{\partial p} \left( \frac{\partial H}{\partial S} \right) = \frac{\partial}{\partial p} T = \frac{\partial T}{\partial p}, \\ &\frac{\partial S}{\partial V} = \frac{\partial}{\partial V} S = \frac{\partial}{\partial V} \left( -\frac{\partial F}{\partial T} \right) = - \frac{\partial^2 F}{\partial V \partial T} = - \frac{\partial^2 F}{\partial T \partial V} = - \frac{\partial}{\partial T} \left( \frac{\partial F}{\partial V} \right) = - \frac{\partial}{\partial T} (-p) = \frac{\partial p}{\partial T}, \\ &\frac{\partial S}{\partial p} = \frac{\partial}{\partial p} S = \frac{\partial}{\partial p} \left( -\frac{\partial G}{\partial T} \right) = - \frac{\partial^2 G}{\partial p \partial T} = - \frac{\partial^2 G}{\partial T \partial p} = - \frac{\partial}{\partial T} \left( \frac{\partial G}{\partial p} \right) = - \frac{\partial}{\partial T} V = - \frac{\partial V}{\partial T}. \end{align*}

(解答結束)