# Why would we need to know $c_p - c_v$

I understand that it's easier to calculate $c_p$ experimentally since we pretty much experience constant pressure conditions all our lives. It's also more difficult to calculate $c_p$ mathematically since $$c_p = \Bigg{(}\dfrac{\partial H}{\partial T}\Bigg{)}_p$$ Mathematically $$c_v= \Bigg{(}\dfrac{\partial E}{\partial T}\Bigg{)}_V$$Is easier to calculate, But I've just proved $c_p - c_v$ and it took me a while so I was wondering why would we need to know $$c_p-c_v = \Bigg{[}p + \Bigg{(}\dfrac{\partial E}{\partial V}\Bigg{)}_T\Bigg{]}\Bigg{(}\dfrac{\partial V}{\partial T}\Bigg{)}_p$$

Basically why would we need to know $c_p - c_v$ at all.

• As is not uncommon there's an answer from @John-Rennie that gives some physical insight to the significance of $c_p - c_v$. Oct 26, 2017 at 21:38

The difference between the heat capacities is the definition of the specific gas constant:

$$R_{\text{specific}} = c_p - c_v$$

And so depending on your use-case, that could be handy. Usually we use it the other way around to get either $c_p$ or $c_v$ from the universal gas constant and molecular weight and whichever heat capacity we know.

The importance of the quantity $c_p-c_v$, where $c_p$ is the specific heat for a constant pressure and $c_v$ is the specific heat for a constant volume, is given by the Mayer Relation that relates the specific gas constant to the specific heats for a calorically perfect gas and a thermally perfect gas.

## Generalizations

For more general homogeneous substances, not just ideal gases, the difference takes the form,

$$C_{P}-C_{V}=VT\frac{\alpha^{2}}{\beta_{T}},$$
where $C_{P}$ is the heat capacity of a body at constant pressure, $C_{V}$ is the heat capacity at constant volume, $V$ is the volume, $T$ is the temperature, $\alpha _{T}$ is the thermal expansion coefficient and $\beta$ is the isothermal compressibility.

From this relation, several inferences can be made:

• Since isothermal compressibility $\beta _{T}$ is positive for all phases and the square of thermal expansion coefficient $\alpha$ is a positive quantity or zero, the specific heat at constant-pressure is always greater than or equal to specific heat at constant-volume. $$C_{{P,m}} \geq C_{V,m}$$

• As the absolute temperature of the system approaches zero, the difference between $C_{P,m}$ and $C_{V,m}$ also approaches zero.

• For incompressible substances, $C_{P,m}$ and $C_{V,m}$ are identical. Also for substances that are nearly incompressible, such as solids and liquids, the difference between the two specific heats is negligible.