# 内能

U

${\displaystyle U=\sum _{i}E_{i}\!}$

## 描述和定義

${\displaystyle \Delta U=\sum _{i}E_{i}\,}$

${\displaystyle U=U_{\mathrm {micro,pot} }+U_{\mathrm {micro,kin} }}$

${\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}\ .}$

### 內能變化量

${\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }}$ [note 1]

${\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }+\Delta U_{\mathrm {matter} }}$

## 理想氣體的內能

${\displaystyle U=cNT,}$

${\displaystyle U(S,V,N)=const\cdot e^{\frac {S}{cN}}V^{\frac {-R}{c}}N^{\frac {R+c}{c}},}$

## 封閉熱力學系統的內能

${\displaystyle dU=\delta Q+\delta W\,}$

${\displaystyle \delta W=-p\mathrm {d} V\,}$ .

${\displaystyle \delta Q=T\mathrm {d} S\,}$ .

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

### 隨溫度與容量而變的變化量

${\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV\,\,{\text{ (1)}}.\,}$

${\displaystyle H_{2}}$  1.410
${\displaystyle O_{2}}$  1.397
${\displaystyle N_{2}}$  1.402

${\displaystyle SO_{2}}$  1.272

${\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV.\,}$

${\displaystyle pV=nRT.\,}$

${\displaystyle p={\frac {nRT}{V}}.}$

${\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-{\frac {nRT}{V}}\right]dV.\,}$

${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}}.}$

${\displaystyle dU=C_{V}dT+\left[{\frac {nRT}{V}}-{\frac {nRT}{V}}\right]dV.}$

${\displaystyle dU=C_{V}dT.\,}$

${\displaystyle dS=\left({\frac {\partial S}{\partial T}}\right)_{V}dT+\left({\frac {\partial S}{\partial V}}\right)_{T}dV\,}$

${\displaystyle dU=TdS-pdV.\,}$

${\displaystyle dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-p\right]dV.\,}$

${\displaystyle T\left({\frac {\partial S}{\partial T}}\right)_{V}}$  為固定容量下的熱容量 ${\displaystyle C_{V}.}$

${\displaystyle dA=-SdT-pdV.\,}$

A 相對於 T 與 V 之二階導數的對稱性，可給出麥克斯韋關係式

${\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}.\,}$

### 隨溫度與壓力而變的變化量

${\displaystyle dU=\left(C_{p}-\alpha pV\right)dT+\left(\beta _{T}p-\alpha T\right)Vdp\,}$

${\displaystyle C_{p}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}\,}$

${\displaystyle \alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}\,}$

${\displaystyle \beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T}\,}$

${\displaystyle dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dp+\left({\frac {\partial V}{\partial T}}\right)_{p}dT=V\left(\alpha dT-\beta _{T}dp\right)\,\,{\text{ (2)}}\,}$

${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{p}}{\left({\frac {\partial V}{\partial p}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}\,\,{\text{ (3)}}\,}$

### 在固定溫度下，隨容量而變的變化量

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

## 多成分系統的內能

${\displaystyle U=U(S,V,N_{1},\ldots ,N_{n})\,}$

${\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots )\,}$

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

${\displaystyle T={\frac {\partial U}{\partial S}},}$
${\displaystyle p=-{\frac {\partial U}{\partial V}},}$

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

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

${\displaystyle G=\sum _{i}\mu _{i}N_{i}\,}$

## 彈性介質裡的內能

${\displaystyle \mathrm {d} U=T\mathrm {d} S+V\sigma _{ij}\mathrm {d} \varepsilon _{ij}}$

${\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}$

${\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}$

## 註記

1. 在本條目裡，機械功的正負值與在化學裡所定義的一樣，但不同於在物理裡所使用的習慣。在化學裡，環境對系統所作的功（如系統收縮）為負值，而在物理裡則為正值。

## 參考資料

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