目录
Thermodynamic Potentials: Natural Variables & Exact Differential
| Thermodynamic Potential | Natural Variables | Exact Differential | Key Partial Derivatives | Maxwell Relation |
|---|---|---|---|---|
| Internal Energy $U$ | $S, V$ | \(dU = TdS – pdV\) | \(\left( \frac{\partial U}{\partial S} \right)_V = T\)；\(\left( \frac{\partial U}{\partial V} \right)_S = -p\) | \(\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial p}{\partial S} \right)_V\) |
| Enthalpy $H$ | $S, p$ | \(dH = TdS + Vdp\) | \(\left( \frac{\partial H}{\partial S} \right)_p = T\)；\(\left( \frac{\partial H}{\partial p} \right)_S = V\) | \(\left( \frac{\partial T}{\partial p} \right)_S = \left( \frac{\partial V}{\partial S} \right)_p\) |
| Gibbs Energy $G$ | $p, T$ | \(dG = Vdp – SdT\)（基本方程） | \(\left( \frac{\partial G}{\partial p} \right)_T = V\)；\(\left( \frac{\partial G}{\partial T} \right)_p = -S\) | \(\left( \frac{\partial V}{\partial T} \right)_p = -\left( \frac{\partial S}{\partial p} \right)_T\) |
| Helmholtz Energy $A$ | $V, T$ | \(dA = -pdV – SdT\) | \(\left( \frac{\partial A}{\partial V} \right)_T = -p\)；\(\left( \frac{\partial A}{\partial T} \right)_V = -S\) | \(\left( \frac{\partial p}{\partial T} \right)_V = \left( \frac{\partial S}{\partial V} \right)_T\) |
| – | – | Δ（状态函数有限变化） d（精确微分） δ（非精确微分） ∂（偏导数） | – | – |
Tips
Natural Variable 指的是与特定 Thermodynamics potentials（$U$, $A$, $H$, $G$）直接关联的一组状态变量；当一个热力学势被表示为其 “自然变量” 的函数时，它的微分形式会变得简洁且仅包含状态函数（$T$, $P$, $V$, $S$），该变量组合是热力学势最 “原生”、最能直接反映其本质的描述方式.
Partial Derivatives & Maxwell Relations
U→H→G→A，这是推导的顺序.
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