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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}$


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