\[dU = T\,dS - p\,dV,\quad dH = T\,dS + V\,dp,\quad dA = -S\,dT - p\,dV,\quad dG = -S\,dT + V\,dp\]
开放体系（含化学势）open systems with chemical potentials
\[dG = -S\,dT + V\,dp + \sum_i\mu_i\,dn_i\]含义Meaning
把第一、第二定律合并（可逆过程 δq = TdS、δw = −pdV），给出四个特性函数的全微分。由它们可直接读出 \(T = (\partial U/\partial S)_V\)、\(V = (\partial G/\partial p)_T\) 等关系。
Combining the first and second laws (δq = TdS, δw = −pdV) gives the differentials of the four characteristic functions, from which relations like T = (∂U/∂S)_V and V = (∂G/∂p)_T follow.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\mu_i\) | 化学势chemical potential | J·mol⁻¹ |
推导Derivation
可逆：\(dU = \delta q + \delta w = TdS - pdV\)。虽由可逆路径导出，U 是状态函数，所以对任意过程成立。
Reversibly dU = TdS − pdV; as U is a state function it holds generally.
勒让德变换：H = U + pV，A = U − TS，G = H − TS，分别求全微分。
Legendre transforms give dH, dA, dG.
组成可变时添加 \(\sum\mu_idn_i\)。
Add Σμ dn for variable composition.
适用条件与局限Assumptions & limitations
只含体积功；有表面功、电功时添加相应项（γ dA、φ dq）。
Add terms for surface or electrical work.