Heat Capacity

Topics

• Heat Capacity Ratio

$C_y={\left(\frac{\partial Q}{\partial T}\right)}_y$

• $y=V$ or $P$

$C_V={\left(\frac{\mathrm{d}Q}{\mathrm{d}\tau }\right)}_P={\left(\frac{\partial U}{\partial \tau }\right)}_V=\tau {\left(\frac{\partial \sigma }{\partial \tau }\right)}_V$

$C_V=\frac{12{\pi }^{4}N{k}_B}{5}{\left(\frac{T}{\theta }\right)}^3$

• The Debye low temperature limit of the heat capacity of a dielectric solid
• $\theta$ is the Debye temperature

$C_P={\left(\frac{\mathrm{d}Q}{\mathrm{d}\tau }\right)}_P={\left(\frac{\partial U}{\partial \tau }\right)}_P=\tau {\left(\frac{\partial \sigma }{\partial \tau }\right)}_P=\frac{\mathrm{d}U+P\mathrm{d}V}{\mathrm{d}\tau }$

$C_P=C_V+N$

$C_P=\frac{5}{2}N$

• Heat capacity at constant pressure for an ideal monatomic gas

$C_{\text{electron gas}}=\frac{{\pi }^{2}N\tau }{2{\tau }_F}=$