## Heat Capacities of Ideal Gases

All gases behave as ideal gases under moderate conditions not too extreme. This means that all gases obey the ideal gas equation
$pV=nRT$
and that the kinetic energy of the molecules of a gas is on average - for a monatomic gas is
$\frac{3}{2}kT$
. Neither of these equation mentions the mass of the gas atoms, implying the the specific heat capacities of all monatomic gases is the same.
The total internal energy of a mol of gas is then
$U=N_A \times \frac{3}{2}kT= \frac{3}{2}RT$
where
$k, \; N_A, \; R$
are Boltzmann's, Avagadro's and the Gas Constant respectively. Hence the molar heat capacity - required to raise the temperature by 1 Degree - of all monatomic gases are the same, and this is true for any set of ideal gases with the same physical characteristics. The same is true of
$C_P$
- as implied by the relationship
$C_P=C_V+R$
.
Not the the specific heat capacity is not the same as the molar heat capacity. In fact is
$m_R$
is the mass of one mol in kg then the number of mols in a kg is
$\frac{1}{m_R}$
and the specific heat capacity at constant volume will be
$\frac{C_V}{m_R}$
.