# 鋅銅電池

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s)

## 熱力學理論

$\mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V+ \mathcal{E}\mathrm{d}Z$

$\mathrm{d}U = T\mathrm{d}S + \mathcal{E}\mathrm{d}Z$

$\mathrm{d}S =\left(\frac{\partial S}{\partial T}\right)_Z \mathrm{d}T+ \left(\frac{\partial S}{\partial Z}\right)_T \mathrm{d}Z$

$\mathrm{d}S = \left(\frac{\partial S}{\partial Z}\right)_T \mathrm{d}Z$

$\mathrm{d}U =\left\{\mathcal{E}+T\left(\frac{\partial S}{\partial Z}\right)_T \right\}\mathrm{d}Z$

$C_T^{(Z)}\ \stackrel{def}{=}\ \frac{\mathrm{d}Q_T}{\mathrm{d}Z}=T\left(\frac{\partial S}{\partial Z}\right)_T$

$\mathrm{d}F = \mathrm{d}U - \mathrm{d}(TS)= - S\mathrm{d}T + \mathcal{E}\mathrm{d}Z$

$\left(\frac{\partial \mathcal{E}}{\partial T}\right)_Z= - \left(\frac{\partial S}{\partial Z}\right)_T$

$\mathrm{d}U =\left\{\mathcal{E} - T\left(\frac{\partial \mathcal{E}}{\partial T}\right)_Z \right\}\mathrm{d}Z$

$\mathrm{d}U =\left(\mathcal{E} - T\frac{\mathrm{d} \mathcal{E}}{\mathrm{d} T}\right) \mathrm{d}Z$

$\Delta H=\Delta(U+PV)=\Delta U$

$\Delta Z = zN_A e$

$\Delta H = zN_A e \left( \mathcal{E} - T \frac {d\mathcal{E}}{\mathrm{d}T}\right)$

## 参考文献

1. ^ 可逆电池及其电动势的测定
2. ^ Colin B P Finn. Thermal Physics. CRC Press. 1992: 163. ISBN 0748743790.