Question

At what temperature will the mean kinetic energy of ${O_2}$ be the same for ${H_2}$ molecules at $ - {73^0}C$?
A. $ - {173^0}C$
B. $ - {73^0}C$
C. ${2927^0}C$
D. ${527^0}C$

Answer 1

Hint:This problem is solved by the help of law of equipartition of energy and the degree of freedom. So, students should have a clear-cut idea about the topic. Degree of freedom is the number of independent variables required to fully understand the motion of a body.

Complete step by step answer:
The number of degrees of freedom of a gas is given by $f = 3N - K$ where, $N$ = number of atoms in the molecule and, $K$ = number of independent relations between atoms of a molecule. Law of equipartition of energy states that for a system in thermodynamic equilibrium, the energy per particle per degree of freedom is equal to $\dfrac{1}{2}{\text{ kT}}$ .

In the question it is asked what temperature will mean kinetic energy of oxygen is the same for the hydrogen molecules at $ - {73^0}C$.From the law of equipartition of energy we have,kinetic energy per molecule per degree of freedom is equal to $\dfrac{1}{2}{\text{ kT}}$. Therefore, kinetic energy per molecule per ‘f’ degree of freedom is equal to $\dfrac{f}{2}{\text{ kT}}$.

As the ${H_2}$ is a diatomic gas it has 5 degrees of freedom similarly the ${O_2}$ is also a diatomic gas it also has 5 degrees of freedom.Therefore, Kinetic energy per molecule per degree of freedom for ${H_2}$ is equal to $\dfrac{5}{2}{\text{ kT}}$. Kinetic energy per molecule per degree of freedom for ${O_2}$ is equal to $\dfrac{5}{2}{\text{ kT}}$. And let the temperature of hydrogen be ${T_1}$ and oxygen be ${T_2}$.
$K{E_{{H_2}}} = K{E_{{O_2}}}$
Therefore,
$\dfrac{5}{2}{\text{ k}}{{\text{T}}_1} = \dfrac{5}{2}{\text{ k}}{{\text{T}}_2}$
Further simplifying, we get
${T_1} = {T_2}$
As the temperature $\left( {{T_2}} \right)$ of ${O_2}$ is equal to $ - {73^0}C$ then the temperature $\left( {{T_1}} \right)$ of the ${H_2}$ is also equal to $ - {73^0}C$ as ${T_1} = {T_2}$ .

Hence, the correct answer is option B.

Note:From the above relation we conclude that the kinetic energy of oxygen and hydrogen molecules only depend on temperature so if the temperature of the molecules is equal then the kinetic energy is also equal or vice versa. Students should know the degrees of freedom of mono, di and tri atomic particles are 3, 5 and 6 respectively. In the Law of equipartition of energy These points require attention that is Thermodynamic system, Degree of freedom and k (Boltzmann constant). The law depends on these factors.