Editing Heat capacity (section)

==== At constant pressure, ''δQ'' = ''dU'' + ''pdV'' ([[isobaric process]]) ====
At constant pressure, heat supplied to the system contributes to both the [[Work (thermodynamics)|work]] done and the change in [[internal energy]], according to the [[first law of thermodynamics]]. The heat capacity is called <math>C_p</math> and defined as:

<math display="block">C_p = \left.\frac{\delta Q}{dT}\right|_{p = \text{const}}</math>

From the [[first law of thermodynamics]] follows <math>\delta Q = dU + p\,dV </math> and the inner energy as a function of <math>p</math> and <math>T</math> is:

<math display="block">\delta Q = \left(\frac{\partial U}{\partial T}\right)_p dT + \left(\frac{\partial U}{\partial p}\right)_T dp + p\left[ \left(\frac{\partial V}{\partial T}\right)_p dT + \left(\frac{\partial V}{\partial p}\right)_T dp  \right]</math>

For constant pressure <math>(dp = 0)</math> the equation simplifies to:

<math display="block">C_p = \left.\frac{\delta Q}{dT}\right|_{p = \text{const}}
= \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p
= \left(\frac{\partial H}{\partial T}\right)_p</math>

where the final equality follows from the appropriate [[Maxwell relations]], and is commonly used as the definition of the isobaric heat capacity.