At constant pressure, δQ = dU + PdV (isobaric process)

At constant pressure, heat supplied to the system contributes to both the work done and the change in internal energy, according to the first law of thermodynamics. The heat capacity is called ${\displaystyle C_{p}}$ and defined as:

${\displaystyle C_{p}={\frac {\delta Q}{dT}}{\Bigr |}_{p=const}}$

From the first law of thermodynamics follows ${\displaystyle \delta Q=dU+pdV}$ and the inner energy as a function of ${\displaystyle p}$ and ${\displaystyle T}$ is:

${\displaystyle \delta Q=\left({\frac {\partial U}{\partial T}}\right)_{p}dT+\left({\frac {\partial U}{\partial p}}\right)_{T}dp+p\left[\left({\frac {\partial V}{\partial T}}\right)_{p}dT+\left({\frac {\partial V}{\partial p}}\right)_{T}dp\right]}$

For constant pressure ${\displaystyle (dp=0)}$ the equation simplifies to:

${\displaystyle C_{p}={\frac {\delta Q}{dT}}{\Bigr |}_{p=const}=\left({\frac {\partial U}{\partial T}}\right)_{p}+p\left({\frac {\partial V}{\partial T}}\right)_{p}}$

At constant volume, dV = 0, δQ = dU (isochoric process)

A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted ${\displaystyle C_{V}.}$ The value of ${\displaystyle C_{V}}$ is always less than the value of ${\displaystyle C_{p}.}$ (${\displaystyle C_{V}}$ < ${\displaystyle C_{p}.}$)

Expressing the inner energy as a function of the variables ${\displaystyle T}$ and ${\displaystyle V}$ gives:

${\displaystyle \delta Q=\left({\frac {\partial U}{\partial T}}\right)_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV+pdV}$

For a constant volume (${\displaystyle dV=0}$) the heat capacity reads:

${\displaystyle C_{V}={\frac {\delta Q}{dT}}{\Bigr |}_{V={\text{const}}}=\left({\frac {\partial U}{\partial T}}\right)_{V}}$

The relation between ${\displaystyle C_{V}}$ and ${\displaystyle C_{p}}$ is then:

${\displaystyle C_{p}=C_{V}+\left(\left({\frac {\partial U}{\partial V}}\right)_{T}+p\right)\left({\frac {\partial V}{\partial T}}\right)_{p}}$

Calculating C_p and C_V for an ideal gas

Mayer's relation:

${\displaystyle C_{p}-C_{V}=nR.}$

${\displaystyle C_{p}/C_{V}=\gamma ,}$

where

${\displaystyle n}$ is the number of moles of the gas,

${\displaystyle R}$ is the universal gas constant,

${\displaystyle \gamma }$ is the heat capacity ratio (which can be calculated by knowing the number of degrees of freedom of the gas molecule).

Using the above two relations, the specific heats can be deduced as follows:

${\displaystyle C_{V}={\frac {nR}{\gamma -1}},}$

${\displaystyle C_{p}=\gamma {\frac {nR}{\gamma -1}}.}$

At constant temperature (Isothermal process)

No change in internal energy (as the temperature of the system is constant throughout the process) leads to only work done by the total supplied heat, and thus an infinite amount of heat is required to increase the temperature of the system by a unit temperature, leading to infinite or undefined heat capacity of the system.

At the time of phase change (Phase transition)

Heat capacity of a system undergoing phase transition is infinite, because the heat is utilized in changing the state of the material rather than raising the overall temperature.