Ideal gas: Heat capacity

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The heat capacity at constant volume is given by

C_V = \left. \frac{\partial U}{\partial T} \right\vert_V

where U is the internal energy. Given that an ideal gas has no interatomic potential energy, the only term that is important is the kinetic energy of an ideal gas, which is equal to (3/2)RT. Thus

C_V =  \frac{\partial ~ }{\partial T}  \left( \frac{3}{2}RT \right) = \frac{3}{2} R

At constant pressure one has

C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p

we can see that, just as before, one has

\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R

and from the equation of state of an ideal gas

p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R


C_p =  C_v + R  =  \frac{5}{2} R

where R is the molar gas constant.


  1. Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
  2. Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11