# Thermodynamics – Relation Between Heat Capacities

thermodynamics

$C_p$ 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??

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.