Question

At which temperature the mean kinetic energy of a gas molecule will be $\dfrac{1}{3}$ of its kinetic energy at $27{\text{ }}^\circ C$ ?

Answer 1

In order to answer this question, to know about the temperature at which the mean kinetic energy of the gas molecule will be $\dfrac{1}{3}rd$ of its K.E, we will use the kinetic energy of molecule and find that kinetic energy will be proportional to temperature and hence from this relation we will get the solution accordingly.

Complete step by step solution:
The average kinetic energy of the particles in a substance is measured by temperature. It's a typical particle's kinetic energy. Thermal equilibrium occurs when two things are at the same temperature. When a material's temperature is raised, it expands due to thermal expansion.
Average Kinetic Energy of a molecule is given by $\dfrac{3}{2}kT$ Where, $k$ is Boltzmann’s constant and $T$ is the temperature in kelvin.
We can observe that the average kinetic energy is exactly proportional to temperature. As a result, we may state that-
For kinetic energy to be $\dfrac{1}{3}rd$ , the temperature in Kelvin should also be $\dfrac{1}{3}rd$
As a result, in this situation, the needed temperature will be :-
$\dfrac{{27 + 273}}{3}Kelvin$
$= \dfrac{{300}}{3}Kelvin \\\ = 100\,Kelvin\left( {approx} \right) \\\ = {\left( {100 - 273} \right)^ \circ }C \\\ = - {173^ \circ }C \\\$
At $- {173^ \circ }C$ the mean kinetic energy of a gas molecule will be $\dfrac{1}{3}rd$ of its kinetic energy at $27{\text{ }}^\circ C$

Additional Information:
The Boltzmann constant is a proportionality factor that connects the average relative kinetic energy of particles in a gas to the gas's thermodynamic temperature. It appears in the kelvin and gas constant definitions, as well as Planck's law of black-body radiation and Boltzmann's entropy formula.

Note:
The average kinetic energy of a sample of matter falls as it is continuously cooled. One would anticipate the particles to eventually stop moving totally. Absolute zero is the temperature at which particle motion hypothetically comes to a halt. In the laboratory, absolute zero has never been achieved, though temperatures on the order of $1{\text{ }} \times {10^{ - 10}}K$ have been achieved. Because the Kelvin temperature scale is based on molecular motion, absolute zero is also known as $0{\text{ }}K$ . The average kinetic energy of the particles of a substance is exactly proportional to the Kelvin temperature of the substance.