Specific Heat Relationships

Typically, for a given constant volume specific heat and constant pressure specific heat, the following relationships hold:

$$\begin{array}{rl}{q}_v& <q_p\text{for same }dT\\ \Delta u& <\Delta h\text{where }dh=du+Pdv\\ c_v& <c_p\end{array}$$

Specific Heat Ratio

The specific heat ratio of a substance is:

$$k=\frac{c_p}{c_v}$$

Ideal Gas

For an ideal gas, we have:

$$h=u+Pv=u+RT$$

thus, enthalpy of an ideal gas is a function of temperature such that $h=h(T)$.

At constant pressure, we have:

$$c_p=\frac{dh}{dT}=\frac{du}{dT}+R$$

which leads to

$$\boxed{c_p=c_v+R}$$

Solids and Liquids

For incompressible solids and liquids, we have

$$\begin{array}{rl}h& =u+Pv\\ dh& =du+Pdv\\ dh& =du\end{array}$$

which means that

$$\boxed{c_p=c_v=c}$$