Editing Ideal gas: Heat capacity

The [[heat capacity]] at constant volume is given by

:<math>C_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>

where <math>U</math> is the [[internal energy]]. Given that an [[ideal gas]] has no interatomic potential energy, the only term that is important is the [[Ideal gas: Energy | kinetic energy of an ideal gas]], which is equal to <math>(3/2)RT</math>. Thus 

:<math>C_V =  \frac{\partial ~ }{\partial T}  \left( \frac{3}{2}RT \right) = \frac{3}{2} R </math>.

At constant [[pressure]] one has

:<math>C_p = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math>

we can see that, just as before, one has 

:<math>\left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R </math>

and from the [[Equation of State: Ideal Gas | equation of state of an ideal gas]] 

:<math>p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R</math>

thus 

{{resultbox|<math>C_p =  C_v + R  =  \frac{5}{2} R</math>}}

where <math>R</math> is the [[molar gas constant]].
==References==
#Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
#Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11
[[Category: Ideal gas]]