Physics · Kinetic Theory · NEET
Cv counts every active degree of freedom, not just translation. A polyatomic molecule has 3 translational + 3 rotational quadratic terms, giving internal energy U = (3/2 + 3/2)RT = 3RT from those alone. Each vibrational mode adds a full RT (see next question), so U = (3 + f)RT. Then Cv = dU/dT = (3 + f)R. The 3R part is only the monatomic piece; polyatomic gases have much more.
Two. NCERT is clear: each vibrational mode has both kinetic energy and potential energy, so it contributes TWO squared (quadratic) terms. Each quadratic term gives (1/2)kBT, so one vibrational mode contributes kBT per molecule, i.e. a full R per mole. That is why in U = (3/2 + 3/2 + f)RT the vibrational part is f times R (not f times R/2).
A diatomic (dumbbell) molecule is linear. Rotation about the axis joining the two atoms carries almost no moment of inertia, so that rotation does not store energy: only 2 rotational modes count. A polyatomic molecule is generally non-linear (bent, like water, or tetrahedral), so all 3 rotation axes have real moment of inertia and all 3 rotational modes are active.
Yes. NCERT states Cp - Cv = R holds for ANY ideal gas, whether monatomic, diatomic, or polyatomic. That is why Cp = (4 + f)R follows directly from Cv = (3 + f)R: you simply add one R. This is Mayer's relation and it does not depend on the number of degrees of freedom.
Set f = 0. Then Cv = 3R, Cp = 4R, and gamma = 4/3. This is the standard NEET default for a polyatomic gas when vibration is ignored (for example CO2 treated as non-vibrating, or NH3, CH4 in the rigid approximation). Only add vibrational modes when the question mentions them or gives a Cp/Cv ratio that forces f.
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 Cp/Cv of the gas is 8/7, then the value of f is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Cv = (3 + f)R and Cp = (4 + f)R, where f is the number of vibrational modes. If vibration is ignored (f = 0), Cv = 3R and Cp = 4R.
gamma = (4 + f)/(3 + f). For a rigid polyatomic gas (no vibration, f = 0), gamma = 4/3 which is about 1.33.
At least 6 without vibration: 3 translational and 3 rotational. Each additional vibrational mode adds 2 more (kinetic + potential), so the total is 6 + 2f-type contributions in energy terms.
A vibrating pair of atoms stores both kinetic energy and elastic potential energy. Each is a quadratic (squared) term worth (1/2)kBT, so one vibration gives (1/2 + 1/2)kBT = kBT per molecule.
Yes, Cp - Cv = R (Mayer's relation) is true for every ideal gas regardless of atomicity, so it holds for monatomic, diatomic, and polyatomic gases alike.
