Q.Given the values of the two principal specific heats, S_P = 3400 cal kg⁻¹ K⁻¹ and S_V = 2400 cal kg⁻¹ K⁻¹ for the hydrogen gas, find the value of J if the universal gas constant R = 8300 J kg⁻¹ K⁻¹.

Treating a water molecule (3 atoms) by the equipartition/Dulong-Petit-type argument, each atom contributes $3R$ to the molar specific heat.

Using the law of equipartition of energy extended to a solid-like/associated liquid model, each atom is treated as a three-dimensional oscillator with 6 degrees of freedom (3 kinetic + 3 potential), each contributing $\tfrac12 R T$ to the molar internal energy, i.e. $3R$ per mole per atom (a Dulong–Petit-type result).

A water molecule (H$_2$O) has 3 atoms, so its molar specific heat is

$C = 3\times(3R) = 9R$

Count total degrees of freedom (translational + rotational + $2\times$vibrational) and apply $C_V=\frac{n}{2}R$, $C_P=C_V+R$.

A general polyatomic (non-linear) molecule has:

• 3 translational degrees of freedom,
• 3 rotational degrees of freedom,
• and, for each vibrational mode, 2 degrees of freedom (one for kinetic energy of vibration, one for the associated potential energy of the vibrating bond) — so $f$ vibrational modes contribute $2f$.

Total degrees of freedom:

$n=3+3+2f=6+2f$

By the law of equipartition of energy, each degree of freedom contributes $\frac12 RT$ of internal energy per mole, so

$C_V=\frac{n}{2}R=\frac{6+2f}{2}R=(3+f)R$

Using Mayer's relation $C_P-C_V=R$:

$C_P=C_V+R=(3+f)R+R=(4+f)R$

Ratio:

$\gamma=\frac{C_P}{C_V}=\frac{(4+f)R}{(3+f)R}=\frac{4+f}{3+f}$