マクスウェルの関係式

概要[編集]

温度 $T$ ,圧力 $P$ ,エントロピー $S$ ,体積 $V$ に対して

{(\frac{\partial T}{\partial V})}_{S} = - {(\frac{\partial P}{\partial S})}_{V}

{(\frac{\partial T}{\partial P})}_{S} = {(\frac{\partial V}{\partial S})}_{P}

{(\frac{\partial S}{\partial V})}_{T} = {(\frac{\partial P}{\partial T})}_{V}

{(\frac{\partial S}{\partial P})}_{T} = - {(\frac{\partial V}{\partial T})}_{P}

が成り立つ。ヤコビアンを使うと1つの式にまとめられて

\frac{\partial (T, S)}{\partial (P, V)} = 1

である。

各式は4つの状態量(内部エネルギー,エンタルピー,ヘルムホルツの自由エネルギー,ギブズの自由エネルギー)の偏微分を計算することで求められる。内部エネルギー $U$ ,エンタルピー $H$ ,ヘルムホルツの自由エネルギー $F$ ,ギブズの自由エネルギー $G$ に対して

d U = T d S - P d V = {(\frac{\partial U}{\partial S})}_{V} d S + {(\frac{\partial U}{\partial V})}_{S} d V

d H = T d S + V d P = {(\frac{\partial H}{\partial S})}_{P} d S + {(\frac{\partial H}{\partial P})}_{S} d P

d F = - S d T - P d V = {(\frac{\partial F}{\partial T})}_{V} d T + {(\frac{\partial F}{\partial V})}_{T} d V

d G = - S d T + V d P = {(\frac{\partial G}{\partial T})}_{P} d T + {(\frac{\partial G}{\partial P})}_{T} d P

なので

T = {(\frac{\partial U}{\partial S})}_{V}, - P = {(\frac{\partial U}{\partial V})}_{S}

T = {(\frac{\partial H}{\partial S})}_{P}, V = {(\frac{\partial H}{\partial P})}_{S}

- S = {(\frac{\partial F}{\partial T})}_{V}, - P = {(\frac{\partial F}{\partial V})}_{T}

- S = {(\frac{\partial G}{\partial T})}_{P}, V = {(\frac{\partial G}{\partial P})}_{T}

これを

\frac{\partial}{\partial V} (\frac{\partial U}{\partial S}) = \frac{\partial}{\partial S} (\frac{\partial U}{\partial V})

\frac{\partial}{\partial P} (\frac{\partial H}{\partial S}) = \frac{\partial}{\partial S} (\frac{\partial H}{\partial P})

\frac{\partial}{\partial V} (\frac{\partial F}{\partial T}) = \frac{\partial}{\partial T} (\frac{\partial F}{\partial V})

\frac{\partial}{\partial P} (\frac{\partial G}{\partial T}) = \frac{\partial}{\partial T} (\frac{\partial G}{\partial P})

に代入して

{(\frac{\partial T}{\partial V})}_{S} = - {(\frac{\partial P}{\partial S})}_{V}

{(\frac{\partial T}{\partial P})}_{S} = {(\frac{\partial V}{\partial S})}_{P}

{(\frac{\partial S}{\partial V})}_{T} = {(\frac{\partial P}{\partial T})}_{V}

{(\frac{\partial S}{\partial P})}_{T} = - {(\frac{\partial V}{\partial T})}_{P}

となる。