At what temperature will the molar kinetic energy of $0.3\,mol$ of helium be the same as that of $0.4\,mol$ of argon at $400\,K$ ?
A. $700\,K$
B. $533\,K$
C. $800\,K$
D. $400\,K$

Hint: In order to answer this question, first we will rewrite the given facts from the question, and then we will apply the formula of Kinetic energy as according to the question. And that’s how we can find the required temperature that is needed.

Complete step by step answer:
Given that, the quantity of helium is $0.3\,mol$ and the quantity of argon is $0.4\,mol$.If the $0.4\,mol$ of argon is at $400\,K$ , then we have to find at what temperature $0.3\,mol$ of helium is situated.According to the question, the molar kinetic energy of $0.3\,mol$ of helium be the same as that of $0.4\,mol$ of argon at $400\,K$ :-
So, we will apply the formula of Kinetic Energy-
$K.E = \dfrac{3}{2}nRT$
where, $n = 1$ for molar kinetic energy, $R$ is the gas constant which is the same for both the given gases and $T$ is temperature.

Now, according to the question,
$K.{E_{Helium}} = K.{E_{Argon}} \\
\Rightarrow {[\dfrac{3}{2} \times 0.3 \times R \times T]_{Helium}} = {[\dfrac{3}{2} \times 0.4 \times R \times 400]_{Argon}} \\
\therefore T = 533K $
Therefore, at $533\,K$ ,the molar kinetic energy of $0.3\,mol$ of helium is the same as that of $0.4\,mol$ of argon at $400\,K$ .

Hence, the correct option is B.

Note: The average kinetic energy of a mole of particles, KE avg, is consequently equal to: where $M$ denotes the molar mass in kilogrammes per mol. The average KE of a mole of gas molecules is likewise proportional to the gas's temperature, as shown by the equation: where $R$ is the gas constant and $T$ is the kelvin temperature.