Ncert Exercises & Solutions

In a diatomic molecule, the rotational energy at a given temperature:

(a)	obeys Maxwell’s distribution.
(b)	have the same value for all molecules.
(c)	equals the translational kinetic energy for each molecule.
(d)	is \(\frac{2}{3}\text{rd}\) the translational kinetic energy for each molecule.

Choose the correct alternatives:

1.	(a), (b)	2.	(a), (d)
3.	(c), (d)	4.	(a), (c)

(2) Hint: Use the law of equipartition of energy.

Step 1: Find the total energy of the molecule.

Consider a diatomic molecule as shown in the diagram.
The total energy associated with the molecule is,

E = \frac{1}{2} m v_{x}^{2} + \frac{1}{2} m v_{y}^{2} + \frac{1}{2} m v_{z}^{2} + \frac{1}{2} I_{x} ω_{x}^{2} + \frac{1}{2} I_{y} ω_{y}^{2}

The above expression contains translational kinetic energy

(\frac{1}{2} m v^{2})

corresponding to velocities in each x, y and z-directions as well as rotational K.E.

(\frac{1}{2} I ω^{2})

associated with the axis of rotations x and y.

Step 2: Find the amount of translational and rotational kinetic energy.

The number of independent terms in the above expression is 5. As we can predict velocities of molecules by Maxwell's distribution, hence the above expression also obeys Maxwell's distribution.

∵

2 rotational and 3 translational energies are associated with each molecule.

∴

Rotational energy at a given temperature is

(\frac{2}{3})

of translational K.E. of each molecule.

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