## Heat Capacities of Ideal Gases

\[pV=nRT\]

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 \[\frac{C_V}{m_R}\]

.