Animated Solution for Physics - Thermodynamics: Consider a mixture of gas molecule of types $A$, $B$ and $C$ having masses $m_A<m_B<m_C$. The ratio of their root mean square speeds at normal temperature and pressure is

The kinetic theory of gases provides a beautiful microscopic perspective on macroscopic properties like temperature and pressure. In this problem, we are tasked with comparing the root mean square (rms) speeds of three different gases in a mixture. Let's dive into the physics behind this!

A Mixture of Gases

Imagine a container holding a mixture of three different gases: $A$, $B$, and $C$. The problem states that their masses follow a specific order: $m_A<m_B<m_C$. This means gas $A$ consists of the lightest molecules, while gas $C$ consists of the heaviest.

Because these gases are mixed together in the same container at "normal temperature and pressure," they are all in thermal equilibrium. This is a crucial piece of information: it means the temperature $T$ is exactly the same for all three gases.

RMS Speed

To compare their speeds, we need to recall the formula for the root mean square speed of a gas molecule. According to the kinetic theory of gases, the rms speed $v_{rms}$ is given by:

Here, $R$ is the universal gas constant, $T$ is the absolute temperature, and $m$ is the molar mass of the gas.

Let's analyze this equation. Since $R$ is a constant and $T$ is the same for all three gases in our mixture, the numerator $\sqrt{3RT}$ is a constant value. Therefore, the rms speed depends solely on the molar mass $m$. Specifically, it is inversely proportional to the square root of the mass:

What does this mean physically? It means that at a given temperature, all gas molecules have the same average kinetic energy ($\frac{3}{2}kT$). Because kinetic energy is $\frac{1}{2}m{v}^2$, a lighter molecule must travel much faster to have the same kinetic energy as a heavier molecule.

Given our mass relationship $m_A<m_B<m_C$, we can directly apply our inverse proportionality. The lightest gas, $A$, will have the highest speed, and the heaviest gas, $C$, will have the lowest speed.

However, if we look at the options provided, they are expressed in terms of reciprocals. In mathematics, when you take the reciprocal of positive numbers in an inequality, the direction of the inequality sign reverses.

Therefore, taking the reciprocal of our speed relationship gives:

This perfectly matches option (d). It's a simple yet elegant demonstration of how mass dictates the microscopic frenzy of gas molecules!