# 吉布斯能

${\displaystyle G\ {\stackrel {def}{=}}\ U-TS+pV=H-TS}$

## 概述

${\displaystyle \Delta G\leq W'}$

${\displaystyle -\Delta G=-W'}$

## 自發過程與平衡過程

${\displaystyle \Delta S_{\mathrm {tot} }\geq 0}$

${\displaystyle \Delta S_{\mathrm {tot} }=\Delta S+\Delta S_{\mathrm {surr} }\geq 0}$

${\displaystyle \Delta S_{\mathrm {surr} }=-Q/T}$

${\displaystyle T\Delta S-Q\geq 0}$

${\displaystyle \Delta G=\Delta H-T\Delta S}$

${\displaystyle H\ {\stackrel {def}{=}}\ U+pV}$

${\displaystyle \Delta H=\Delta U+p\Delta V}$

### 非體積功是零的狀況

${\displaystyle \Delta U=Q-p\Delta V}$

${\displaystyle \Delta G=Q-T\Delta S}$

${\displaystyle \Delta G\leq 0}$

• ${\displaystyle \Delta G<0}$：過程具有自发性。
• ${\displaystyle \Delta G=0}$：過程處於平衡状态。
• ${\displaystyle \Delta G>0}$：過程被嚴格禁止。

${\displaystyle \Delta G=-T\Delta S_{\mathrm {tot} }}$

### 非體積功不是零的狀況

${\displaystyle \Delta U=Q-p\Delta V+W'}$

${\displaystyle \Delta G=\Delta U-T\Delta S+p\Delta V}$

${\displaystyle \Delta G=Q-T\Delta S+W'\leq W'}$

${\displaystyle -\Delta G\geq -W'}$

## 化學反應

### 標準生成吉布斯能

(kJ/mol)
${\displaystyle \Delta _{\mathrm {f} }G_{\mathrm {m} }^{\ominus }}$
(kcal/mol)
NO g 87.6 20.9
NO2 g 51.3 12.3
N2O g 103.7 24.78
H2O g -228.6 -54.64
H2O l -237.1 -56.67
CO2 g -394.4 -94.26
CO g -137.2 -32.79
CH4 g -50.5 -12.1
C2H6 g -32.0 -7.65
C3H8 g -23.4 -5.59
C6H6 g 129.7 29.76
C6H6 l 124.5 31.00

### 标准反应吉布斯能

${\displaystyle a\;\mathrm {A} +b\;\mathrm {B} \rightarrow c\;\mathrm {C} +d\;\mathrm {D} }$

${\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=[c\Delta _{\mathrm {f} }G_{\mathrm {C} }^{\ominus }+d\Delta _{\mathrm {f} }G_{\mathrm {D} }^{\ominus }]-[a\Delta _{\mathrm {f} }G_{\mathrm {A} }^{\ominus }+b\Delta _{\mathrm {f} }G_{\mathrm {B} }^{\ominus }]}$

## 開放系統

${\displaystyle U=U(S,V,n_{1},n_{2},n_{3},\dots )}$

${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i=1}^{N}\mu _{i}\mathrm {d} n_{i}}$

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

${\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial n_{i}}}\right)_{T,p,n_{j\neq i}}}$

${\displaystyle dG=\sum _{i=1}^{N}\mu _{i}\mathrm {d} n_{i}}$

${\displaystyle dG=\sum _{i=1}^{N}\mu _{i}n_{i}\mathrm {d} x}$

${\displaystyle G=\sum _{i=1}^{N}\mu _{i}n_{i}}$

### 理想氣體混合物

${\displaystyle \left({\frac {\partial G_{m}}{\partial p}}\right)_{T}=V_{m}={\frac {RT}{p}}}$

${\displaystyle G_{m}(T,p)=G_{m0}(T)+RT\ln(p/p_{0})}$

${\displaystyle G_{m}(T,p)=G_{m0}(T)+RT\ln(p)}$

${\displaystyle p=\sum _{i=1}^{N}p_{i}}$

${\displaystyle G_{m}(T,p_{i})=G_{m0i}(T)+RT\ln(p_{i})}$

${\displaystyle \mu _{i}=\Delta _{\mathrm {f,i} }G_{\mathrm {m} }^{\ominus }+RT\ln(p_{i})}$

${\displaystyle a\;\mathrm {A} +b\;\mathrm {B} \rightarrow c\;\mathrm {C} +d\;\mathrm {D} }$

{\displaystyle {\begin{aligned}\Delta G&=[[c\Delta _{\mathrm {f} }G_{\mathrm {C} }^{\ominus }+d\Delta _{\mathrm {f} }G_{\mathrm {D} }^{\ominus }]-[a\Delta _{\mathrm {f} }G_{\mathrm {A} }^{\ominus }+b\Delta _{\mathrm {f} }G_{\mathrm {B} }^{\ominus }]+RT\ln(Q)\\&=\Delta G_{0}+RT\ln(Q)\\\end{aligned}}}

${\displaystyle \Delta G_{0}+RT\ln(Q)=0}$

${\displaystyle K\ {\stackrel {def}{=}}\ Q}$

${\displaystyle K(T)=e^{-\Delta G_{0}/RT}}$

• ${\displaystyle \Delta G_{0}>0}$ 时，${\displaystyle K<1}$
• ${\displaystyle \Delta G_{0}=0}$ 时，${\displaystyle K=1}$
• ${\displaystyle \Delta G_{0}<0}$ 时，${\displaystyle K>1}$

## 电化学

${\displaystyle -\Delta G\geq -W'}$

${\displaystyle W'=-n{\mathcal {F}}{\mathcal {E}}}$

${\displaystyle \Delta G=-n{\mathcal {F}}{\mathcal {E}}}$

${\displaystyle {\mathcal {E}}=-\Delta G/n{\mathcal {F}}}$

${\displaystyle {\mathcal {E}}^{\ominus }\ {\stackrel {def}{=}}\ -\Delta G^{\ominus }/n{\mathcal {F}}}$

${\displaystyle {\mathcal {E}}=-{\frac {\Delta G^{\ominus }+RT\ln Q}{n{\mathcal {F}}}}}$

${\displaystyle {\mathcal {E}}={\mathcal {E}}^{\ominus }-{\frac {RT}{n{\mathcal {F}}}}\ln Q}$

## 歷史

1873年，吉布斯發表論文《用曲面方法來幾何表現出物質的熱力學性質》。在這篇論文裏，他詳細論述他的新方程式的原理。這方程式可以預測或估算，當幾樣物體或系統接觸在一起之時，各種自然過程發生的趨勢。通過研究幾樣均一性物質接觸時的相互作用，例如，由一部分固體、一部分液體與一部分氣體構成的物體，又通過展示相關體積--內能三維圖，吉布斯可以判斷三種平衡狀態，即穩定平衡、中性平衡或不穩定平衡，以及是否會發生後續變化。吉布斯闡明，[4]

 假若我們想要用方程式來表達某物質呈熱力學平衡的充要條件，在等壓與等溫狀況下，這方程式可以寫為:δ(ε − Tη + pν) = 0 其中，δ指的是物體的各個部份因變化而產生的微小變分，其與物體被區分為幾個不同狀態的部分成比例。穩定平衡的判據是，在括號內的表達式，其數值必須為最小值。

## 註釋

1. 熱力系統在壓強的作用下因体积变化所做的是體積功；任何其它種類的功屬非體積功，例如，由於彈簧伸展而做的彈性功，由於電池內化學變化生成的電功，由於肌肉運動而產生的生物功等等，都是非體積功。
2. ^ 電動勢${\displaystyle {\mathcal {E}}}$以方程式定義為
${\displaystyle {\mathcal {E}}\ {\stackrel {def}{=}}\ \Phi _{cathode}-\Phi _{anode}}$
其中，${\displaystyle \Phi _{cathode}}$${\displaystyle \Phi _{anode}}$分別為電池正極與負極的電勢。

## 参考文献

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2. Perrot, Pierre. A to Z of Thermodynamics. Oxford University Press. 1998. ISBN 0-19-856552-6.
3. Peter Atkins; Loretta Jones. Chemical Principles: The Quest for Insight. W. H. Freeman. 1 August 2007. ISBN 978-1-4292-0965-6.
4. J.W. Gibbs, "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces," Transactions of the Connecticut Academy of Arts and Sciences 2, Dec. 1873, pp. 382-404 .
5. ^ Gibbs energy (function). IUPAC GOLDBOOK. 24 Feb 2014 [2016-05-18]. （原始内容存档于2016-10-12）.
6. ^ CRC Handbook of Chemistry and Physics, 2009, pp. 5-4 - 5-42, 90th ed., Lide
7. Clement John Adkins. Equilibrium Thermodynamics. Cambridge University Press. 14 July 1983. ISBN 978-0-521-27456-2.
8. ^ D.R. Crow. Principles and Applications of Electrochemistry. CRC Press. 15 September 1994. ISBN 978-0-7514-0168-4.
9. Henry Marshall Leicester. The Historical Background of Chemistry. Courier Corporation. 1971. ISBN 978-0-486-61053-5.
10. ^ Coffey, Patrick. Chemical free energies and the third law of thermodynamics (PDF). Hist Stud Phys Biol Sci. 2006, 36 (2): 365–396 [2016-06-08]. （原始内容 (PDF)存档于2017-01-07）.