# Relation between heat capacities

$C_p$ here means specific heat at constant pressure and $C_V$ at constant volume. My book says that $C_p$ is "generally" greater that $C_V$ because at constant pressure a part of heat given maybe used for expansion whereas at constant volume all the added heat produces a rise in temperature. The term "generally" has been used because substances generally expand with increase of temperature at constant pressure but in a few exceptional cases there may be contraction. After a few pages the relation $C_p-C_V=R$ is derived I know $R=8.3$ so this means $C_p = C_V+R$ or $C_p>C_V$. so according to this relation $C_p$ is always greater than $C_V$ but the book claims that this is not always true!! How is it possible?? Why are the two statements contradicting each other??

• Perhaps the "edge cases" here are not ideal gases? Commented Oct 15, 2015 at 15:26
• Thnx!! @kyle and , I didn't realised that!!got the answer!! Strange how a word (IDEAL) can change everything!! Commented Oct 15, 2015 at 16:52

The equation $C_p - C_v = R$ will have been derived for an ideal gas. For any other substance the relationship between $C_p$ and $C_v$ will be more complicated. However in the vast majority of cases materials expand as they get hotter so if the pressure is kept constant the material will do work as it expands. That means $C_p$ must be greater than $C_v$ even though the difference will no longer simply be $R$.

However materials do exist that have a negative thermal expansion coefficient i.e. the material contracts as it gets hotter. In this case if the pressure is kept constant the material will have work done on it and $C_p$ will be less than $C_v$. These materials are special cases and they are few and far between. Nevertheless such materials exit.

• In light of Erik Pillon's answer, which (correctly) states that $C_P$ is never less than $C_V$ (as derived here, for example), can you cite any experimental data that support your claim? Commented Apr 8, 2018 at 4:56
• Ge et al.'s "Can $C_P$ Be Less Than $C_V$?" discusses the misconception—seen even in some textbooks, as well as this answer—that $C_P$ can ever be less than $C_V$, even for materials that contract upon heating. $C_P$ exceeds $C_V$ by a factor that scales with the thermal expansion coefficient squared, which is always a positive number. Commented Jun 28, 2022 at 4:04

That's true because Mayer relation $c_p-c_v$ relates the specific gas constant to the specific heats for a calorically perfect gas and a thermally perfect gas.

By the way, that's not still valid for general substances...

## 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:

• 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.

• 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. $$C_{{P,m}} \geq C_{V,m}$$ Anyway, some very strange materials can undergo a negative thermal expansion (NTE), which is a physicochemical process in which some materials contract upon heating rather than expand as most materials do. Materials which undergo this unusual process have a range of potential engineering, photonic, electronic, and structural applications.(see Wikipedia source)