An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and f number of vibrational modes. If the ratio of heat capacities C P / C V of the ga

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An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and f number of vibrational modes. If the ratio of heat capacities C P ​ / C V ​ of the gas is 8/7 , then the value of f is :NEET 2026 (Re-NEET)

TestHub · Accepted Answer

The internal energy of one mole of the polyatomic gas is given by the equipartition theorem. Each translational and rotational degree of freedom contributes $\frac{1}{2} RT$ to the internal energy, while each vibrational mode contributes $RT$ (since it has both kinetic and potential energy components).

$U = 3 (\frac{1}{2} RT) + 3 (\frac{1}{2} RT) + f (RT) = (3 + f) RT$

The molar heat capacity at constant volume is:

$C_{V} = \frac{d U}{d T} = (3 + f) R$

The molar heat capacity at constant pressure is:

$C_{P} = C_{V} + R = (4 + f) R$

The ratio of heat capacities is given as $\frac{8}{7}$ :

$\frac{C _{P}}{C _{V}} = \frac{4 + f}{3 + f} = \frac{8}{7}$

$28 + 7 f = 24 + 8 f$

$f = 4$

Answer: $4$

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