\[ \newcommand{dn}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d} #2^{#3}}}
\newcommand{\d}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}}
\newcommand{\dd}[2]{\frac{\mathrm{d}^2 #1}{\mathrm{d} {#2}^2}}
\newcommand{\ddd}[2]{\frac{\mathrm{d}^3 #1}{\mathrm{d} {#2}^3}}
\newcommand{\pdn}[3]{\frac{\partial^{#3} #1}{\partial {#2}^{#3}}}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pdd}[2]{\frac{\partial^2 #1}{\partial {#2}^2}}
\newcommand{\pddd}[2]{\frac{\partial^3 #1}{\partial {#2}^3}}
\newcommand{\p}{\partial}
\newcommand{\D}[2]{\frac{\mathrm{D} #1}{\mathrm{D} #2}}
\newcommand{\Re}{\mathrm{Re}}
\newcommand{\Im}{\mathrm{Im}}
\newcommand{\bra}[1]{\left\langle #1 \right|}
\newcommand{\ket}[1]{\left|#1 \right\rangle}
\newcommand{\braket}[2]{\left\langle #1 \middle|#2 \right\rangle}
\newcommand{\inner}[2]{\left\langle #1 ,#2 \right\rangle}
\newcommand{\l}{\left} \newcommand{\m}{\middle} \newcommand{\r}{\right}
\newcommand{\f}[2]{\frac{#1}{#2}} \newcommand{\eps}{\varepsilon}
\newcommand{\ra}{\rightarrow} \newcommand{\F}{\mathcal{F}}
\newcommand{\L}{\mathcal{L}} \newcommand{\t}{\quad}
\newcommand{\intinf}{\int_{-\infty}^{+\infty}}
\newcommand{\R}{\mathcal{R}} \newcommand{\C}{\mathcal{C}}
\newcommand{\Z}{\mathcal{Z}} \newcommand{\bm}[1]{\boldsymbol{#1}} \]
基本法則
第一法則 \[ dU = δQ + δW \] 第二法則 \[ \oint δQ/T \leq 0 \]
- 準静的 \[ δW = -pdV \]
- 可逆 \[ dS = δQ/T = 0 \]
- 準静的・可逆 \[ dU = TdS - pdV \]
- 物質の増減 \[ dU = TdS - pdV + \mu dN \]
マクスウェルの関係式
エネルギー \(U(S,V)\) \[ dU = TdS - pdV \]
\[
dU = \left(\frac{\partial U}{\partial V}\right)dV + \left(\frac{\partial U}{\partial S}\right)dS
\]
を比較すると，
\[ -p = \left(\frac{\partial U}{\partial V}\right) \] \[ T = \left(\frac{\partial U}{\partial S}\right) \]
U が C^2 以上なら二階微分は可換で
\[
\left(\frac{\partial p}{\partial S}\right)_V = -\left(\frac{\partial T}{\partial V}\right)_S
\]
ヘルムホルツエネルギー \(F(T,V) = U - TS\)
ギブスエネルギー \(G = U - TS + pV\)
一般の熱力学的状態方程式
エネルギー保存
\[
dU=Tds-PdV
\]
\[
\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial S}{\partial V}\right)_T-P
\]
マクスウェルの関係式
\[
\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V
\]
より，
\[
\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial P}{\partial T}\right)_V-P
\]
※ \(U,p,V,T\) の関係式
