# 内能

## 维基百科，自由的百科全书

U

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

## 描述和定义

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

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

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

### 内能变化量

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

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

## 理想气体的内能

${\displaystyle U=cNT,}$

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

## 封闭热力学系统的内能

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

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

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

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

### 随温度与容量而变的变化量

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

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

${\displaystyle SO_{2))$ 1.272

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

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

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

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

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

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

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

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

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

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

${\displaystyle T\left({\frac {\partial S}{\partial T))\right)_{V))$ 为固定容量下的热容量 ${\displaystyle C_{V}.}$

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

A 相对于 T 与 V 之二阶导数的对称性，可给出麦克斯韦关系式

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

### 随温度与压力而变的变化量

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

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

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

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

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

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

### 在固定温度下，随容量而变的变化量

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

## 多成分系统的内能

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

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

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

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

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

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

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

## 弹性介质里的内能

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

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

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

## 注记

1. 在本条目里，机械功的正负值与在化学里所定义的一样，但不同于在物理里所使用的习惯。在化学里，系统对环境所做的功（如系统膨涨）为负值，而在物理里则为正值。

## 参考资料

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