Heat capacity

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From the first law of thermodynamics one has

$\left.\delta Q\right. = dU + pdV$

where $Q$ is the heat, $U$ is the internal energy, $p$ is the pressure and $V$ is the volume. The heat capacity is given by the differential of the heat with respect to the temperature,

$C := \frac{\delta Q}{\partial T}$

[edit] At constant volume

At constant volume (denoted by the subscript $V$ ),

$C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V$

[edit] At constant pressure

At constant pressure (denoted by the subscript $p$ ),

$C_p := \left.\frac{\delta Q}{\partial T} \right\vert_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p$