熱力位能

描述與解釋

${\displaystyle H}$ ${\displaystyle U+pV}$ ${\displaystyle S,p,\{N_{i}\}}$

• 內能是做功的能力和放熱的能力。
• 吉布斯能是做非機械功的能力。
• 是做非機械功的能力和放熱的能力。
• 亥姆霍茲自由能是系统做機械功和非機械功的能力。

• 當一個封閉系統的熵與「外部參數」（如：體積）保持恒定时，內能(U)會降低並在热力平衡点達到最小值。
• 當一個封閉系統的溫度與外部參數保持恒定时，亥姆霍茲自由能(F)會降低並在热力平衡点達到最小值。
• 當一個封閉系統的壓力與外部參數保持恒定时，焓(H)會降低並在热力平衡点達到最小值。
• 當一個封閉系統的溫度、壓力與外部參數保持恒定时，吉布斯能(G)會降低並在热力平衡点達到最小值。

自然變數

 公式 自然變數 ${\displaystyle U[\mu _{j}]=U-\mu _{j}N_{j}\,}$ ${\displaystyle ~~~~~S,V,\{N_{i\neq j}\},\mu _{j}\,}$ ${\displaystyle F[\mu _{j}]=U-TS-\mu _{j}N_{j}\,}$ ${\displaystyle ~~~~~T,V,\{N_{i\neq j}\},\mu _{j}\,}$ ${\displaystyle H[\mu _{j}]=U+pV-\mu _{j}N_{j}\,}$ ${\displaystyle ~~~~~S,p,\{N_{i\neq j}\},\mu _{j}\,}$ ${\displaystyle G[\mu _{j}]=U+pV-TS-\mu _{j}N_{j}\,}$ ${\displaystyle ~~~~~T,p,\{N_{i\neq j}\},\mu _{j}\,}$

热力学基本关系

${\displaystyle \mathrm {d} U=\delta Q-\delta W+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}}$

${\displaystyle \delta Q=T\,\mathrm {d} S\,}$
${\displaystyle \delta W=p\,\mathrm {d} V\,}$

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$

${\displaystyle dU=T\,dS-\sum _{i}X_{i}\,dx_{i}+\sum _{j}\mu _{j}\,dN_{j}\,}$

 ${\displaystyle \mathrm {d} U\,}$ ${\displaystyle \!\!=\!\!}$ ${\displaystyle T\mathrm {d} S\,}$ ${\displaystyle -\,}$ ${\displaystyle p\mathrm {d} V\,}$ ${\displaystyle +\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$ ${\displaystyle \mathrm {d} F\,}$ ${\displaystyle \!\!=\!\!}$ ${\displaystyle -\,}$ ${\displaystyle S\,\mathrm {d} T\,}$ ${\displaystyle -\,}$ ${\displaystyle p\mathrm {d} V\,}$ ${\displaystyle +\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$ ${\displaystyle \mathrm {d} H\,}$ ${\displaystyle \!\!=\!\!}$ ${\displaystyle T\,\mathrm {d} S\,}$ ${\displaystyle +\,}$ ${\displaystyle V\mathrm {d} p\,}$ ${\displaystyle +\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$ ${\displaystyle \mathrm {d} G\,}$ ${\displaystyle \!\!=\!\!}$ ${\displaystyle -\,}$ ${\displaystyle S\,\mathrm {d} T\,}$ ${\displaystyle +\,}$ ${\displaystyle V\mathrm {d} p\,}$ ${\displaystyle +\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$

狀態方程式

${\displaystyle \mathrm {d} \Phi =\sum _{i}x_{i}\,\mathrm {d} y_{i}\,}$

${\displaystyle x_{j}=\left({\frac {\partial \Phi }{\partial y_{j}}}\right)_{\{y_{i\neq j}\}}}$

${\displaystyle +T=\left({\frac {\partial U}{\partial S}}\right)_{V,\{N_{i}\}}=\left({\frac {\partial H}{\partial S}}\right)_{p,\{N_{i}\}}}$
${\displaystyle -p=\left({\frac {\partial U}{\partial V}}\right)_{S,\{N_{i}\}}=\left({\frac {\partial F}{\partial V}}\right)_{T,\{N_{i}\}}}$
${\displaystyle +V=\left({\frac {\partial H}{\partial p}}\right)_{S,\{N_{i}\}}=\left({\frac {\partial G}{\partial p}}\right)_{T,\{N_{i}\}}}$
${\displaystyle -S=\left({\frac {\partial G}{\partial T}}\right)_{p,\{N_{i}\}}=\left({\frac {\partial F}{\partial T}}\right)_{V,\{N_{i}\}}}$
${\displaystyle ~\mu _{j}=\left({\frac {\partial \phi }{\partial N_{j}}}\right)_{X,Y,\{N_{i\neq j}\}}}$

${\displaystyle -N_{j}=\left({\frac {\partial U[\mu _{j}]}{\partial \mu _{j}}}\right)_{S,V,\{N_{i\neq j}\}}}$

麦克斯韦关系式

${\displaystyle \left({\frac {\partial }{\partial y_{j}}}\left({\frac {\partial \Phi }{\partial y_{k}}}\right)_{\{y_{i\neq k}\}}\right)_{\{y_{i\neq j}\}}=\left({\frac {\partial }{\partial y_{k}}}\left({\frac {\partial \Phi }{\partial y_{j}}}\right)_{\{y_{i\neq j}\}}\right)_{\{y_{i\neq k}\}}}$

${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S,\{N_{i}\}}=-\left({\frac {\partial p}{\partial S}}\right)_{V,\{N_{i}\}}}$
${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{S,\{N_{i}\}}=+\left({\frac {\partial V}{\partial S}}\right)_{p,\{N_{i}\}}}$
${\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T,\{N_{i}\}}=+\left({\frac {\partial p}{\partial T}}\right)_{V,\{N_{i}\}}}$
${\displaystyle \left({\frac {\partial S}{\partial p}}\right)_{T,\{N_{i}\}}=-\left({\frac {\partial V}{\partial T}}\right)_{p,\{N_{i}\}}}$

${\displaystyle \left({\frac {\partial T}{\partial N_{j}}}\right)_{V,S,\{N_{i\neq j}\}}=\left({\frac {\partial \mu _{j}}{\partial S}}\right)_{V,\{N_{i}\}}}$

${\displaystyle \left({\frac {\partial N_{j}}{\partial V}}\right)_{S,\mu _{j},\{N_{i\neq j}\}}=-\left({\frac {\partial p}{\partial \mu _{j}}}\right)_{S,V\{N_{i\neq j}\}}}$
${\displaystyle \left({\frac {\partial N_{j}}{\partial N_{k}}}\right)_{S,V,\mu _{j},\{N_{i\neq j,k}\}}=-\left({\frac {\partial \mu _{k}}{\partial \mu _{j}}}\right)_{S,V\{N_{i\neq j}\}}}$

歐拉積分

${\displaystyle U(\{\alpha y_{i}\})=\alpha U(\{y_{i}\})\,}$

${\displaystyle U(\{y_{i}\})=\sum _{j}y_{j}\left({\frac {\partial U}{\partial y_{j}}}\right)_{\{y_{i\neq j}\}}}$

${\displaystyle U=TS-pV+\sum _{i}\mu _{i}N_{i}\,}$

${\displaystyle F=-pV+\sum _{i}\mu _{i}N_{i}\,}$
${\displaystyle H=TS+\sum _{i}\mu _{i}N_{i}\,}$
${\displaystyle G=\sum _{i}\mu _{i}N_{i}\,}$

吉布斯－迪昂關係

${\displaystyle \mathrm {d} G=\left.{\frac {\partial G}{\partial p}}\right|_{T,N}\mathrm {d} p+\left.{\frac {\partial G}{\partial T}}\right|_{p,N}\mathrm {d} T+\sum _{i=1}^{I}\left.{\frac {\partial G}{\partial N_{i}}}\right|_{p,N_{j\neq i}}\mathrm {d} N_{i}\,}$

${\displaystyle \mathrm {d} G=V\mathrm {d} p-S\mathrm {d} T+\sum _{i=1}^{I}\mu _{i}\mathrm {d} N_{i}\,}$

${\displaystyle G=\sum _{i=1}^{I}\mu _{i}N_{i}\,}$
${\displaystyle \mathrm {d} G=\sum _{i=1}^{I}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{I}N_{i}\,\mathrm {d} \mu _{i}\,}$

${\displaystyle \sum _{i=1}^{I}N_{i}\,\mathrm {d} \mu _{i}=-S\,\mathrm {d} T+V\,\mathrm {d} p\,}$