Question

The kinetic energy of a molecule of a gas is directly proportional to the absolute temperature of the gas.
(A) True
(B) False

Answer 1

The physical behaviour of a gas can be explained by Kinetic molecular theory of gases. Average kinetic energy is related both to the molecular speed, and the absolute temperature. We could find the relation between them by deriving the equation from average pressure of gas molecules in terms of root mean square speed.

Complete step by step answer:
- According to the kinetic theory, all gaseous particles are in constant random motions at temperatures above absolute zero
- We can write the average pressure of gas molecules in terms of root mean square speed as follows
$p=\dfrac{mn{{\nu }^{2}}}{3}$ → equation (1)
Where m is the mass of gas molecules
n is the number of molecules per unit volume V
$\nu$ is the rms speed
We can write, $n=\dfrac { N }{ V }$ → equation (2)
Where N is the number of molecules

Thus, by substituting (2) in (1) we can write
$pV=\dfrac{mN{{\nu }^{2}}}{3}$ → equation (3)
We know kinetic energy E = $\dfrac {m{{\nu }^{2}}}{2}$ →equation (4)

On substituting (4) in (3) we get
$pV=\dfrac{2NE}{3}$ →equation (5)

From the ideal gas equation, we know $pV=nRT$ (R is the molar gas constant and T is the absolute temperature). Thus, we can rewrite the equation (5) as follows
$nRT=\dfrac{2NE}{3}$ → equation (6)

Where $n=\dfrac{N}{{{N}_{A}}}$. Here ${{N}_{A}}$ is the Avogadro number. Thus equation (6) can be rewritten as
$\dfrac{N}{{{N}_{A}}}RT=\dfrac{2NE}{3}$ →equation(7)

On rearranging the above equation, we get kinetic energy of a molecule as
$E=\dfrac{3}{2}{{K}_{_{B}}}T$ → equation (8)
Where ${{K}_{_{B}}}=\dfrac{R}{{{N}_{A}}}$ .This value is the Boltzmann constant.

Thus, from equation (8) it's clear that the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas. This indicates that all gases at a given temperature have the same average kinetic energy.
So, the correct answer is “Option A”.

Note: We can explain the answer in a simpler way. From the equation of rms speed (${{\nu }_{rms}}=\sqrt{\dfrac{3RT}{M}}$), we know that molecular velocity is directly proportional to absolute temperature. Since kinetic energy is directly proportional to velocity, we can assume that the kinetic energy of a gas is directly proportional to absolute temperature.