Physics · Kinetic Theory · NEET
A single atom is treated as a point mass. It can move along the x, y and z directions, giving 3 translational degrees of freedom. It has no meaningful rotation because all its mass is at one point, so its moment of inertia is almost zero. There is no bond, so no vibration either. So f = 3 for a monatomic gas.
Picture a dumbbell (two atoms on an axis). It can spin about two axes that are perpendicular to the line joining the atoms. Spinning about the third axis (the line through both atoms) does not count, because the atoms lie on that axis and the moment of inertia about it is nearly zero. So only 2 rotations store energy, giving f = 3 translation + 2 rotation = 5.
At ordinary temperature the two atoms in a diatomic molecule are treated as rigidly joined, so we ignore vibration and use f = 5. At high temperature the bond acts like a spring and the atoms vibrate. Each vibrational mode adds 2 to f (one for kinetic energy of vibration, one for potential energy of the spring). NEET questions say 'rigid' for no vibration and mention 'vibrational mode' when you must add it.
Energy is stored in a quadratic form. A vibrating bond stores kinetic energy (from speed, proportional to velocity squared) AND potential energy (from stretch, proportional to displacement squared). Each squared term is one degree of freedom by the law of equipartition, so one vibration adds 2. That is why a vibrating diatomic gas has f = 5 + 2 = 7.
By the law of equipartition, each degree of freedom carries (1/2)kB T per molecule, so internal energy of n moles is U = (f/2) nRT. From this Cv = (f/2)R, Cp = Cv + R = (f/2 + 1)R, and gamma = Cp/Cv = 1 + 2/f. So the whole specific-heat topic depends on getting f correct first.
A gas mixture consists of 2 moles of O2 and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:
The values of Cp/Cv for hydrogen, helium and another ideal diatomic gas X (whose molecules are not rigid but have an additional vibrational mode) are respectively equal to:
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.
They are the independent ways a molecule can store energy: moving in the x, y, z directions (translation), spinning about axes (rotation), and shaking along its bonds (vibration). The symbol is f.
Monatomic gas: f = 3 (translation only). Diatomic gas: f = 5 (3 translation + 2 rotation), or 7 if vibration is active. Non-linear polyatomic gas: f = 6 (3 translation + 3 rotation), plus 2 for each active vibration.
Cv = (f/2)R, Cp = (f/2 + 1)R, and gamma = Cp/Cv = 1 + 2/f. A larger f means gamma is closer to 1.
At room temperature the vibrational energy is not excited, so bonds behave as rigid. NEET tells you to add vibration only when it says the molecule is 'non-rigid' or has a 'vibrational mode'.
A linear molecule has 3 translational + 2 rotational = 5 rotational-translational degrees of freedom (like a diatomic), not 6. The third rotation about the long axis is not counted. Vibrational modes are added when they are active.
