An ideal gas is made of polyatomic molecules. Each molecule has three translational, three rotational and f vibrational modes. If the ratio of heat capacities C_P/C_V of the gas is 8/7, then the value of f is:

Solution & Explanation

Count quadratic degrees of freedom (equipartition). A polyatomic molecule here has 3 translational + 3 rotational modes, plus f vibrational modes. Each vibrational mode counts as 2 quadratic degrees of freedom (one kinetic + one potential), contributing energy k_B T per molecule, i.e. R T per mole. Internal energy per mole: U = (3/2)R T + (3/2)R T + f·R T = (3 + f) R T. Hence C_V = dU/dT = (3 + f) R, and C_P = C_V + R = (4 + f) R. Set the given ratio: C_P/C_V = (4 + f)/(3 + f) = 8/7. Cross-multiply: 7(4 + f) = 8(3 + f) → 28 + 7f = 24 + 8f. So 28 − 24 = 8f − 7f → f = 4. Therefore f = 4, matching option A.