Question

Prove that root mean square velocity of gas molecules is directly proportional to the square root of its absolute temperature.

Answer 1

A gas molecule randomly moves inside a container and exerts pressure on the walls of the container when they collide. Calculate this pressure in terms of root mean square velocity and substitute in the ideal gas equation PV=nRT. Rearrange the terms to get the expression of root mean square velocity.

Complete answer:
Assume a cubical box having side ‘a’ is filled with an ideal gas. The gas molecules having mass m collide with the walls of the box and bounce back with a velocity ${{v}_{x}}$along x-direction.
The momentum of the gas molecule along this direction is given as
$M=m{{v}_{x}}$
Let us assume a molecule will take time dt to travel between the walls.
Therefore,
$dt=\dfrac{a}{{{v}_{x}}}$
When the gas molecule collides with the wall, it transfers its momentum to the wall. Thus, the force imparted by the molecule on the wall is
$\begin{aligned} & F=\dfrac{M}{dt} \\\ & F=\dfrac{m{{v}_{x}}}{\dfrac{a}{{{v}_{x}}}} \\\ & F=\dfrac{m{{v}_{x}}^{2}}{a} \\\ \end{aligned}$
We have,
$Pressure=\dfrac{Force}{Area}$

& Pressure=\dfrac{\dfrac{m{{v}_{x}}^{2}}{a}}{{{a}^{2}}} \\\ & \therefore Pressure=\dfrac{m{{v}_{x}}^{2}}{{{a}^{3}}}=\dfrac{m{{v}_{x}}^{2}}{V} \\\ \end{aligned}$$ Where V is the volume of the box. In the cuboidal box, molecules will travel in all direction, thus, the average velocity of a molecule is given as $$\sum{{{v}^{2}}=}\sum{{{v}_{x}}^{2}}+\sum{{{v}_{y}}^{2}}+\sum{{{v}_{z}}^{2}}$$ Where v is the root mean square velocity of the molecule. Since the molecule has an equal probability of travelling in all direction, we can write, ${{v}_{x}}^{2}=\dfrac{{{v}^{2}}}{3}$ For n molecules, we can write, Pressure, $$\begin{aligned} & P=\dfrac{nm{{v}_{x}}^{2}}{V}=\dfrac{nm{{v}^{2}}}{3V} \\\ & \therefore PV=\dfrac{nm{{v}^{2}}}{3} \\\ \end{aligned}$$ The ideal gas equation is given as PV=nRT, where R represents the universal gas constant and T represents the absolute temperature. Thus, $$\begin{aligned} & \dfrac{nm{{v}^{2}}}{3}=nRT \\\ & {{v}^{2}}=\dfrac{3RT}{m} \\\ & \Rightarrow v=\sqrt{\dfrac{3RT}{m}} \\\ \end{aligned}$$ Thus, the root mean square velocity of a gas molecule is directly proportional to the square root of its absolute temperature. **Note:** There are a large number of gas molecules travelling in a different direction at different speeds. Thus, if we take average velocity then it could result in zero due to the random motion of molecules. To have some meaning full average we take root mean square velocity: the square root of the average of the square of velocities of gas molecules.