A magnetic field of  2 T is applied to a paramagnetic gas. The atoms of the gas have magnetic dipole moment of 4.5 × 10-23 J/T. At what temperature, will the mean translation kinetic energy kinetic energy of an atom of the gas be equal to the energy required to change the alignment of atom’s magnetic dipole form antiparallel to parallel (to the magnetic field) (Boltzmann constant = 1.38 × 10-23 J/K)

CONCEPT:

• The average translational energy of a molecule is given by the equipartition theorem as,

\(E=\frac{3kT}{2}\) where k is the Boltzmann constant and T is the absolute temperature.

• The equipartition theorem relates the temperature of a system to its average energies.
• The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T
• The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms: for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that in rotational motion.

EXPLANATIONS:

​The mean kinetic energy of translation of the atoms is given by:

\(K=\frac{f}{2}kT\)

Here k = 1.38 × 10^-23 J/K is the Boltzmann constant, and f = 3 is the degree of freedom number, so:

\(K=\frac{3}{2}kT\) -----(1)

the magnitude energy required to reverse such a dipole end for the end is given by:

U= |μ̅ ⋅ B̅ - (- μ̅ ⋅ B̅)|

U = 2μB  ----(2)

To find the temperature, we set (1) equal to (2) and then solve for T, so:

K = U

\(\frac{3}{2}kT=2\mu B\)

\(T=\frac{4\mu B}{3k}\)

Substituting the given values 

\(T=\frac{4\times 4.5 \times 10^{-23}\times 2T}{3 \times 1.38 \times 10^{-23}}\)

T = 8.69 K

Option 1 is the correct option.