ギブズ-ヘルムホルツの式

${\displaystyle U=F-T\left({\frac {\partial F}{\partial T}}\right)_{V}}$

${\displaystyle H=G-T\left({\frac {\partial G}{\partial T}}\right)_{p}}$

${\displaystyle {\begin{aligned}U&=-T^{2}\left[{\frac {\partial }{\partial T}}\left({\frac {F}{T}}\right)\right]_{V}\\H&=-T^{2}\left[{\frac {\partial }{\partial T}}\left({\frac {G}{T}}\right)\right]_{p}\end{aligned}}}$

${\displaystyle {\begin{aligned}\left[{\frac {\partial }{\partial T}}\left({\frac {F}{T}}\right)\right]_{V}&=-{\frac {U}{T^{2}}}\\\left[{\frac {\partial }{\partial T}}\left({\frac {G}{T}}\right)\right]_{p}&=-{\frac {H}{T^{2}}}\end{aligned}}}$

${\displaystyle {\begin{aligned}\left[{\frac {\partial (F/T)}{\partial (1/T)}}\right]_{V}&=U\\\left[{\frac {\partial (G/T)}{\partial (1/T)}}\right]_{p}&=H\end{aligned}}}$

${\displaystyle H({\text{products}};T,p)=G({\text{products}};T,p)-T\left[{\frac {\partial G({\text{products}};T,p)}{\partial T}}\right]_{p}}$

${\displaystyle H({\text{reactants}};T,p)=G({\text{reactants}};T,p)-T\left[{\frac {\partial G({\text{reactants}};T,p)}{\partial T}}\right]_{p}}$

${\displaystyle \Delta _{\text{r}}H(T,p)=H({\text{products}};T,p)-H({\text{reactants}};T,p)}$

${\displaystyle \Delta _{\text{r}}G(T,p)=G({\text{products}};T,p)-G({\text{reactants}};T,p)}$

${\displaystyle \Delta _{\text{r}}H(T,p)=\Delta _{\text{r}}G(T,p)-T\left[{\frac {\partial \Delta _{\text{r}}G(T,p)}{\partial T}}\right]_{p}}$

${\displaystyle \Delta _{\text{r}}U(T,V)=\Delta _{\text{r}}F(T,V)-T\left[{\frac {\partial \Delta _{\text{r}}F(T,V)}{\partial T}}\right]_{V}}$

${\displaystyle \Delta _{\text{r}}G=-z{\boldsymbol {F}}E_{\text{cell}}}$

${\displaystyle \Delta _{\text{r}}H=-z{\boldsymbol {F}}E_{\text{cell}}+z{\boldsymbol {F}}T\left({\frac {\partial E_{\text{cell}}}{\partial T}}\right)_{p}}$

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V}$

${\displaystyle F=U-TS}$

${\displaystyle \mathrm {d} F=\mathrm {d} U-T\mathrm {d} S-S\mathrm {d} T=-S\mathrm {d} T-p\mathrm {d} V}$

${\displaystyle \mathrm {d} F=\left({\frac {\partial F}{\partial T}}\right)_{V}\mathrm {d} T+\left({\frac {\partial F}{\partial V}}\right)_{T}\mathrm {d} V}$

${\displaystyle U=F-T\left({\frac {\partial F}{\partial T}}\right)_{V}}$

${\displaystyle \left[{\frac {\partial (F/T)}{\partial (1/T)}}\right]_{V}=-T^{2}\left[{\frac {\partial }{\partial T}}\left({\frac {F}{T}}\right)\right]_{V}=F-T\left({\frac {\partial F}{\partial T}}\right)_{V}}$

${\displaystyle {\begin{aligned}S(T,V)&=-\left({\frac {\partial F}{\partial T}}\right)_{V}\\p(T,V)&=-\left({\frac {\partial F}{\partial V}}\right)_{T}\end{aligned}}}$

${\displaystyle U(T,V)=F-T\left({\frac {\partial F}{\partial T}}\right)_{V}}$

${\displaystyle {\begin{aligned}G(T,V)&=F+pV=F-V\left({\frac {\partial F}{\partial V}}\right)_{T}\\H(T,V)&=U+pV=F-T\left({\frac {\partial F}{\partial T}}\right)_{V}-V\left({\frac {\partial F}{\partial V}}\right)_{T}\end{aligned}}}$

${\displaystyle {\begin{aligned}S(T,p)&=-\left({\frac {\partial G}{\partial T}}\right)_{p}\\V(T,p)&=\left({\frac {\partial G}{\partial p}}\right)_{T}\end{aligned}}}$

${\displaystyle H(T,p)=G-T\left({\frac {\partial G}{\partial T}}\right)_{p}}$

${\displaystyle {\begin{aligned}F(T,p)&=G-pV=G-p\left({\frac {\partial G}{\partial p}}\right)_{T}\\U(T,p)&=H-pV=G-T\left({\frac {\partial G}{\partial T}}\right)_{p}-p\left({\frac {\partial G}{\partial p}}\right)_{T}\end{aligned}}}$

${\displaystyle G(S,p)=H-TS=H-S\left({\frac {\partial H}{\partial S}}\right)_{p}}$

${\displaystyle G-T\left({\frac {\partial G}{\partial T}}\right)_{p}=H}$

${\displaystyle {\begin{aligned}{\frac {G(T_{2},p)}{T_{2}}}&={\frac {G(T_{1},p)}{T_{1}}}-\int _{T_{1}}^{T_{2}}{\frac {H(T,p)}{T^{2}}}\,\mathrm {d} T\\&={\frac {H(T_{1},p)}{T_{1}}}-S(T_{1},p)-\int _{T_{1}}^{T_{2}}{\frac {H(T,p)}{T^{2}}}\,\mathrm {d} T\end{aligned}}}$

${\displaystyle H_{\text{m}}(T,p)=\mu (T,p)-T{\frac {\partial }{\partial T}}\mu (T,p)}$

${\displaystyle \mu (T,p)=\mu ^{\circ }(T)+RT\ln(p/p^{\circ })}$

${\displaystyle H_{\text{m}}(T)=\mu ^{\circ }(T)-T{\frac {\partial }{\partial T}}\mu ^{\circ }(T)}$

${\displaystyle H(T,p,{\boldsymbol {n}})=G(T,p,{\boldsymbol {n}})-T{\frac {\partial }{\partial T}}G(T,p,{\boldsymbol {n}})}$

${\displaystyle {\overline {H}}_{i}(T,p,{\boldsymbol {n}})=\mu _{i}(T,p,{\boldsymbol {n}})-T{\frac {\partial }{\partial T}}\mu _{i}(T,p,{\boldsymbol {n}})}$

${\displaystyle {\overline {H}}_{i}(T,p,{\boldsymbol {n}})={\frac {\partial }{\partial n_{i}}}H(T,p,{\boldsymbol {n}})}$

${\displaystyle \mu _{i}(T,p,{\boldsymbol {n}})={\frac {\partial }{\partial n_{i}}}G(T,p,{\boldsymbol {n}})}$

${\displaystyle \mu _{i}(T,p^{\circ },m_{i})=\mu _{i}^{\circ }(T)+RT\ln {\frac {m_{i}}{\mathrm {mol\cdot kg^{-1}} }}}$

${\displaystyle {\overline {H}}_{i}(T,p,{\boldsymbol {n}})=\mu _{i}(T,p,{\boldsymbol {n}})-T{\frac {\partial }{\partial T}}\mu _{i}(T,p,{\boldsymbol {n}})}$

${\displaystyle {\overline {H}}_{i}(T,p^{\circ })=\mu _{i}^{\circ }(T)-T{\frac {\partial }{\partial T}}\mu _{i}^{\circ }(T)}$

${\displaystyle H_{\text{m}}^{\circ }({\text{B}};T)=\mu ^{\circ }({\text{B}};T)-T{\frac {\partial }{\partial T}}\mu ^{\circ }({\text{B}};T)}$

${\displaystyle \Delta _{\text{r}}H^{\circ }(T)=\Delta _{\text{r}}G^{\circ }(T)-T{\frac {\partial }{\partial T}}\Delta _{\text{r}}G^{\circ }(T)}$

${\displaystyle {\frac {\partial }{\partial T}}{\frac {\Delta _{\text{r}}G^{\circ }(T)}{T}}=-{\frac {\Delta _{\text{r}}H^{\circ }(T)}{T^{2}}}}$

${\displaystyle \Delta _{\text{r}}H^{\circ }(T)=\sum _{\text{B}}\nu _{\text{B}}H_{\text{m}}^{\circ }(\mathrm {B} ;T)}$

${\displaystyle \Delta _{\text{r}}H^{\circ }(T)=cH_{\text{m}}^{\circ }({\text{C}};T)+dH_{\text{m}}^{\circ }({\text{D}};T)-aH_{\text{m}}^{\circ }({\text{A}};T)-bH_{\text{m}}^{\circ }({\text{B}};T)}$

${\displaystyle \Delta _{\text{r}}G^{\circ }(T)=\sum _{\text{B}}\nu _{\text{B}}\mu ^{\circ }(\mathrm {B} ;T)}$

${\displaystyle \Delta _{\text{r}}G^{\circ }(T)=c\mu ^{\circ }({\text{C}};T)+d\mu ^{\circ }({\text{D}};T)-a\mu ^{\circ }({\text{A}};T)-b\mu ^{\circ }({\text{B}};T)}$

${\displaystyle H_{\text{m}}^{\circ }({\text{B}};T)=\mu ^{\circ }({\text{B}};T)-T{\frac {\partial }{\partial T}}\mu ^{\circ }({\text{B}};T)}$

${\displaystyle \Delta _{\text{r}}H^{\circ }(T)=\Delta _{\text{r}}G^{\circ }(T)-T{\frac {\partial }{\partial T}}\Delta _{\text{r}}G^{\circ }(T)}$

${\displaystyle \Delta _{\text{r}}H(T,p)=\Delta _{\text{r}}G(T,p)-T\left[{\frac {\partial \Delta _{\text{r}}G(T,p)}{\partial T}}\right]_{p}}$

${\displaystyle \Delta _{\text{r}}H^{\circ }(T)=\Delta _{\text{r}}G^{\circ }(T)-T\left[{\frac {\partial \Delta _{\text{r}}G^{\circ }(T)}{\partial T}}\right]_{p=p^{\circ }}}$

${\displaystyle {\frac {\partial }{\partial T}}\ln K(T)={\frac {\Delta _{\text{r}}H^{\circ }(T)}{RT^{2}}}}$

${\displaystyle {\frac {\partial }{\partial T}}{\frac {\Delta _{\text{r}}G^{\circ }(T)}{T}}=-{\frac {\Delta _{\text{r}}H^{\circ }(T)}{T^{2}}}}$

${\displaystyle \mathrm {d} G=-S\mathrm {d} T+V\mathrm {d} p+\sum _{i={\text{A,B,C,D}}}\mu _{i}\mathrm {d} n_{i}}$

${\displaystyle \mathrm {d} G=\mu _{\text{A}}\mathrm {d} n_{\text{A}}+\mu _{\text{B}}\mathrm {d} n_{\text{B}}+\mu _{\text{C}}\mathrm {d} n_{\text{C}}+\mu _{\text{D}}\mathrm {d} n_{\text{D}}}$

${\displaystyle \mathrm {d} G=-a\mu _{\text{A}}\mathrm {d} \xi -b\mu _{\text{B}}\mathrm {d} \xi +c\mu _{\text{C}}\mathrm {d} \xi +d\mu _{\text{D}}\mathrm {d} \xi }$

${\displaystyle a\mu _{\text{A}}+b\mu _{\text{B}}=c\mu _{\text{C}}+d\mu _{\text{D}}}$

${\displaystyle a\,\mu _{\text{A}}(T,p,{\boldsymbol {n}})+b\,\mu _{\text{B}}(T,p,{\boldsymbol {n}})=c\,\mu _{\text{C}}(T,p,{\boldsymbol {n}})+d\,\mu _{\text{D}}(T,p,{\boldsymbol {n}})}$

${\displaystyle \mu _{i}(T,p,{\boldsymbol {n}})=\mu _{i}^{\circ }(T)+RT\ln {\frac {p_{i}}{p^{\circ }}}}$

${\displaystyle a\mu _{\text{A}}^{\circ }(T)+b\mu _{\text{B}}^{\circ }(T)-c\mu _{\text{C}}^{\circ }(T)-d\mu _{\text{D}}^{\circ }(T)=RT\ln \left({\frac {p_{\text{A}}}{p^{\circ }}}\right)^{-a}\left({\frac {p_{\text{B}}}{p^{\circ }}}\right)^{-b}\left({\frac {p_{\text{C}}}{p^{\circ }}}\right)^{c}\left({\frac {p_{\text{D}}}{p^{\circ }}}\right)^{d}}$

${\displaystyle -\Delta _{\text{r}}G^{\circ }(T)=RT\ln {\frac {K_{p}(T,p)}{(p^{\circ })^{c+d-a-b}}}}$

${\displaystyle K_{p}(T,p)={\frac {{p_{\text{C}}}^{c}\,{p_{\text{D}}}^{d}}{{p_{\text{A}}}^{a}\,{p_{\text{B}}}^{b}}}}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} T}}\ln K_{p}(T)&={\frac {\Delta _{\text{r}}H^{\circ }(T)}{RT^{2}}}\\{\frac {\mathrm {d} \ln K_{p}}{\mathrm {d} (1/T)}}&=-{\frac {\Delta _{\text{r}}H^{\circ }(T)}{R}}\end{aligned}}}$

${\displaystyle \mu _{i}(T,p,{\boldsymbol {n}})=\mu _{i}^{\circ }(T)+RT\ln {\frac {f_{i}(T,p,{\boldsymbol {n}})}{p^{\circ }}}}$

${\displaystyle K_{f}(T,p)={\frac {{f_{\text{C}}}^{c}{f_{\text{D}}}^{d}}{{f_{\text{A}}}^{a}{f_{\text{B}}}^{b}}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} T}}\ln K_{f}(T)={\frac {\Delta _{\text{r}}H^{\circ }(T)}{RT^{2}}}}$

${\displaystyle {\frac {\partial }{\partial T}}\ln K_{p}(T,p)={\frac {\Delta _{\text{r}}H(T,p,{\boldsymbol {n}}_{\text{eq}})}{RT^{2}}}}$^{[注 7]}

${\displaystyle \Delta _{\text{r}}H(T,p,{\boldsymbol {n}})=c{\overline {H}}_{\text{C}}(T,p,{\boldsymbol {n}})+d{\overline {H}}_{\text{D}}(T,p,{\boldsymbol {n}})-a{\overline {H}}_{\text{A}}(T,p,{\boldsymbol {n}})-b{\overline {H}}_{\text{B}}(T,p,{\boldsymbol {n}})}$

${\displaystyle {\overline {H}}_{i}(T,p,{\boldsymbol {n}})=H^{\circ }(T)-RT^{2}{\frac {\partial \ln f_{i}(T,p,{\boldsymbol {n}})}{\partial T}}}$^{[注 8]}

${\displaystyle \mu _{i}(T,p^{\circ },m_{i})=\mu _{i}^{\circ }(T)+RT\ln {\frac {m_{i}}{\mathrm {mol\cdot kg^{-1}} }}}$

${\displaystyle K_{m}(T,p^{\circ })={\frac {{m_{\text{C}}}^{c}{m_{\text{D}}}^{d}}{{m_{\text{A}}}^{a}{m_{\text{B}}}^{b}}}}$

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial T}}\ln K_{m}(T,p^{\circ })&={\frac {\Delta _{\text{r}}H^{\circ }(T)}{RT^{2}}}\\\left[{\frac {\partial \ln K_{m}}{\partial (1/T)}}\right]_{p=p^{\circ }}&=-{\frac {\Delta _{\text{r}}H^{\circ }(T)}{R}}\end{aligned}}}$

${\displaystyle \mu _{i}(T,p^{\circ },{\boldsymbol {m}})=\mu _{i}^{\circ }(T)+RT\ln a_{i}(T,p^{\circ },{\boldsymbol {m}})}$

${\displaystyle K^{\circ }={\frac {{a_{\text{C}}}^{c}{a_{\text{D}}}^{d}}{{a_{\text{A}}}^{a}{a_{\text{B}}}^{b}}}}$

${\displaystyle K^{\circ }(T)=\exp \left(-{\frac {\Delta _{\text{r}}G^{\circ }(T)}{RT}}\right)}$

${\displaystyle {\frac {\partial }{\partial T}}\ln K^{\circ }(T)={\frac {\Delta _{\text{r}}H^{\circ }(T)}{RT^{2}}}}$

${\displaystyle {\frac {\partial }{\partial T}}\ln K_{m}(T,p^{\circ },{\boldsymbol {m}})={\frac {\Delta _{\text{r}}H(T,p^{\circ },{\boldsymbol {m}}_{\text{eq}})}{RT^{2}}}}$

${\displaystyle \Delta _{\text{r}}H(T,p^{\circ },{\boldsymbol {m}})=\Delta _{\text{r}}H^{\circ }(T)-RT^{2}\sum _{i={\text{A,B,C,D}}}\nu _{i}{\frac {\partial }{\partial T}}\ln \gamma _{i}(T,p^{\circ },{\boldsymbol {m}})}$

${\displaystyle \gamma _{i}(T,p,{\boldsymbol {m}})={\frac {a_{i}(T,p,{\boldsymbol {m}})}{m_{i}/\mathrm {mol\cdot kg^{-1}} }}}$

${\displaystyle {\begin{aligned}\mathrm {d} 'Q&=\mathrm {d} H\\&=H(T+\mathrm {d} T,p,n_{\text{A}}-a\mathrm {d} \xi ,n_{\text{B}}-b\mathrm {d} \xi ,n_{\text{C}}+c\mathrm {d} \xi ,n_{\text{D}}+d\mathrm {d} \xi )-H(T,p,n_{\text{A}},n_{\text{B}},n_{\text{C}},n_{\text{D}})\\&=\left({\frac {\partial H}{\partial T}}\right)_{p,{\boldsymbol {n}}}\mathrm {d} T+\left[-a\left({\frac {\partial H}{\partial n_{\text{A}}}}\right)_{T,p,n_{j}}-b\left({\frac {\partial H}{\partial n_{\text{B}}}}\right)_{T,p,n_{j}}+c\left({\frac {\partial H}{\partial n_{\text{C}}}}\right)_{T,p,n_{j}}+d\left({\frac {\partial H}{\partial n_{\text{D}}}}\right)_{T,p,n_{j}}\right]\mathrm {d} \xi \\&=\left({\frac {\partial H}{\partial T}}\right)_{p,{\boldsymbol {n}}}\mathrm {d} T+\Delta _{\text{r}}H({\boldsymbol {n}}_{\text{eq}})\mathrm {d} \xi \\&=\left[C_{p,{\boldsymbol {n}}}+\Delta _{\text{r}}H({\boldsymbol {n}}_{\text{eq}}){\frac {\mathrm {d} \xi }{\mathrm {d} T}}\right]\mathrm {d} T\end{aligned}}}$

${\displaystyle 0<\mathrm {d} T<{\frac {\mathrm {d} 'Q}{C_{p,{\boldsymbol {n}}}}}}$

${\displaystyle \Delta _{\text{r}}H({\boldsymbol {n}}_{\text{eq}}){\frac {\mathrm {d} \xi }{\mathrm {d} T}}>0}$

${\displaystyle {\begin{aligned}{\frac {\partial \ln K_{p}}{\partial T}}&={\frac {\Delta _{\text{r}}H(T,p,{\boldsymbol {n}}_{\text{eq}})}{RT^{2}}}\simeq {\frac {\Delta _{\text{r}}H^{\circ }}{RT^{2}}}\\{\frac {\partial \ln K_{m}}{\partial T}}&={\frac {\Delta _{\text{r}}H(T,p^{\circ },{\boldsymbol {m}}_{\text{eq}})}{RT^{2}}}\simeq {\frac {\Delta _{\text{r}}H^{\circ }}{RT^{2}}}\end{aligned}}}$

${\displaystyle \Delta _{\text{r}}H=-z{\boldsymbol {F}}E_{\text{cell}}+z{\boldsymbol {F}}T\left({\frac {\partial E_{\text{cell}}}{\partial T}}\right)_{p}}$

${\displaystyle \left({\frac {\partial \Delta _{\text{r}}H}{\partial T}}\right)_{p}=z{\boldsymbol {F}}T\left({\frac {\partial ^{2}E_{\text{cell}}}{\partial T^{2}}}\right)_{p}}$

1. ^ 「ギッブズ-ヘルムホルツの式」『化学大辞典』共立出版.
2. ^ 『ムーア物理化学』 p. 103.
3. ^ ^{a b} 「ギブズ-ヘルムホルツの式」『岩波理化学辞典』第5版 CD-ROM版, 岩波書店.
4. ^ 『ルイス・ランドル熱力学』 p. 169.
5. ^ 堀内 (1981), pp. 118–122.
6. ^ 中村 (1995), p. 114.
7. ^ ^{a b c d e f g h} グリーンブック (2009) pp. 69-74.
8. ^ 『ムーア物理化学』 p. 235.
9. ^ 玉虫 (1983), p. 22.
10. ^ ^{a b} 『ムーア物理化学』 pp. 300-301.
11. ^ ^{a b} 『化学熱力学』pp. 89-90.
12. ^ 堀内 (1981), pp. 288–289.
13. ^ 『ボール物理化学』 p. 250.

1. ^ この比を平衡定数と呼ぶ。
2. ^ ^{a b} たとえばダニエル電池 Zn + Cu²⁺ → Zn²⁺ + Cu では、z = 2 である。
3. ^ ^{a b} この項目では F をヘルムホルツエネルギーの記号に使っているので、ファラデー定数を太字の F で表す。
4. ^ ${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}={\frac {\mathrm {d} (1/x)}{\mathrm {d} x}}{\frac {\mathrm {d} f}{\mathrm {d} (1/x)}}=-{\frac {1}{x^{2}}}{\frac {\mathrm {d} f}{\mathrm {d} (1/x)}}}$ および ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {f}{x}}\right)={\frac {1}{x}}{\frac {\mathrm {d} f}{\mathrm {d} x}}-{\frac {f}{x^{2}}}}$.
5. ^ 熱力学第三法則に従う純物質では、任意の圧力で S(0, p) = 0 が成り立ち、また 0 < T < T₂ の温度範囲で H(T, p) が既知であれば、第三法則エントロピー S(T₂, p) を計算できる。よって、熱力学第三法則に従う物質に限るならば、温度 T₂、圧力 p におけるギブズエネルギー G を関数 H(T, p) から計算できる。
6. ^ フガシティー係数 ϕ_i = f_i/p_i を使うと、${\displaystyle K_{f}={\frac {{\phi _{\text{C}}}^{c}{\phi _{\text{D}}}^{d}}{{\phi _{\text{A}}}^{a}{\phi _{\text{B}}}^{b}}}K_{p}}$ と表せる。これを K_f とΔ_rH° の間のファントホッフの式に代入して、フガシティー係数の項を右辺に移項し、個々の成分について成り立つギブズ-ヘルムホルツの式と反応エンタルピーの定義式を使うと導出できる。
7. ^ 理想気体であれば K_p は p に依らないが、実在気体の K_p は p に依るので、d 記号ではなく ∂ 記号を用いた。
8. ^ フガシティー係数 ϕ_i = f_i/p_i を使うと、${\displaystyle {\overline {H}}_{i}(T,p,{\boldsymbol {n}})=H^{\circ }(T)-RT^{2}{\frac {\partial \ln \phi _{i}(T,p,{\boldsymbol {n}})}{\partial T}}}$ と表せる。
9. ^ 理想溶液とはみなせない、実在する溶液のこと。