Question

If the molecule of a gas moves in a plane and oscilates in one dimension, then the total energy of the molecule is equal to:
1. \(\frac{3}{2}k_BT\)
2. k_BT
3. 2kBT
4. None of these

Answer 1

Correct Answer - Option 3 : 2kBT

CONCEPT:

Law of equipartition of energy:

• According to the law of equipartition of energy, for any dynamic system in thermal equilibrium, the total energy for the system is equally divided among the degree of freedom with each degree of freedom having average energy equal to \(\frac{1}{2}k_BT\).

Where kB = Boltzmann constant and T = absolute temperature

• If a molecule is constrained to move along a line then it has one degree of freedom.
• If a molecule is constrained to move in a plane then it has two degrees of freedom.
• If a molecule is free to move in space then it has three degrees of freedom.
  • Molecules of a monatomic gas like argon have only translational degrees of freedom.
  • Molecule of diatomic gas such as O2 and N2 has three translational degrees of freedom. But in addition, it has also two independent axes of rotation. The molecule thus has two rotational degrees of freedom, each of which contributes a term to the total energy consisting of translational energy and rotational energy
• Molecules like CO even at moderate temperatures have a mode of vibration, i.e. its atoms oscillate along the interatomic axis like a one-dimensional oscillator, and contribute a vibrational energy term to the total energy.
• While each translational and rotational degree of freedom has contributed only one ‘squared term’ but one vibrational mode contributes two ‘squared terms’: kinetic and potential energies.

EXPLANATION:

• According to the law of equipartition of energy, for any dynamic system in thermal equilibrium, the total energy for the system is equally divided among the degree of freedom with each degree of freedom having average energy equal to \(\frac{1}{2}k_BT\).

Where kB = Boltzmann constant and T = absolute temperature

• We know that if a molecule is constrained to move in a plane then it has two degrees of freedom.
• Therefore if the molecule of a gas can move in a plane then the energy of the molecule due to translational motion is given as,

\(⇒ E_1=2\times\frac{1}{2}k_BT\)

⇒ E1 = kBT      ----(1)

• When a molecule oscillates along the interatomic axis like a one-dimensional oscillator, then its energy is given as,

⇒ E2 = kBT      ----(2)

• So the total energy of the molecule is given as,

⇒ E = E₁ + E₂

⇒ E = kBT + kBT

⇒ E = 2kBT

• Hence, option 3 is correct.