The molar heat capacity , `C_(v)` of helium gas is `3//2R` and is independent of temperature. For hydrogen gas, `C_(v)` approaches `3//2R` at very low temperature, equal `5//2R` at moderate temperature and is higher than `5//2 R` at high temperatures. Give a reason for the temperature dependence of `C_(v)` in case of hydrogen, in not more than two or three sentences.

Answer

Hydrogen is diatomic so at high temperature rational and vibrational motion also counts.

Explanation

Helium `(He)` gas is monoatomic and it has three translational degree of freedom. Hence, contribution of each transiational degree of freeedom towards `C_(v)` being `R//2`, so the total contribution towards `C_(v)=3xxR//2`. Hydrogen molecule is diatomic. At low temperature, rotational and vibrational contribution for `H_(2)` are zero. So, `C_(v)` for `H_(2)` at low temperature continues to be `3R//2`. At moderate temperature, rotational contribution `(=2xxR//2)` also becomes dominant and hence total contribution towards `C_(v)=(3R)/(2)+R=(5R)/(2)`. At even high temperature, vibrational contribution `(=1xxR)` also becomes significant. Hence total contibution towards `C_(v)=(3R)/(2)+R+R=(7R)/(2)`