# Specific Heat Capacity of Gases

## Specific Heat Capacity

(i) Mono atomic gases :
• Mono atomic gas molecules has three translational degrees of freedom.
• From law of equipartition of energy average energy of an molecule at temperature T is (3/2)KBT
• Total internal energy of one mole of such gas is
$U= \frac {3}{2}K_B T N$
$U = \frac {3}{2} RT$ ---(12)
• If $C_V$ is molar specific heat at constant volume then
$C_V = \frac {dU}{dT}$
$= \frac {3}{2}R$ ---(13)
Now for an ideal gas
$C_P - C_V = R$
$C_P$ - molar specific heat capacity at constant pressure
$C_P = \frac {5}{2} R$--- (14)
Thus for a mono atomic gas ratio of specific heats is
$\gamma _{mono} = \frac {C_P}{C_V}= \frac {5}{3}$ ---(15)

(ii) Diatomic gases :
• A diatomic gas molecule is treated as a rigid rotator like dumb-bell and has 5 degrees of freedom out of which three degrees of freedom are translational and two degrees of freedom are rotational.
• Using law of equipartition of energy the total internal energy of one mole of diatomic gas is
$U= \frac {5}{2}K_BN T$
$= \frac {5}{2} RT$ ---(16)
• Specific heats are thus
$C_V =\frac {dU}{dT}= \frac {5}{2}R$
Now for an ideal gas
$C_P - C_V = R$
$C_P$ - molar specific heat capacity at constant pressure
$C_P = \frac {7}{2} R$--- (14)
$\gamma _{dia}= \frac {7}{2}$ (rigid rotator)
• If diatomic molecule is not only rigid but also has an vibrational mode in addition, then
$U = \frac {7}{2} RT$
and $C_V=\frac {7}{2} R$
Now for an ideal gas
$C_P - C_V = R$
$C_P$ - molar specific heat capacity at constant pressure
$C_P=\frac {9}{2} R$
and $\gamma _{dia}=\frac {C_P}{C_V}= \frac {9}{7}$