Animated Solution for Physics - Thermodynamics: A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds $\left(\frac{v_{\text{rms}}(\text{helium})}{v_{\text{rms}}(\text{argon})}\right)$ is

The kinetic theory of gases provides us with a beautiful lens to peer into the microscopic chaos of a gas. When we look at a container of gas, we see a static, calm volume. But zoom in, and it's a bustling metropolis of atoms zipping around at breakneck speeds!

In this problem, we are asked to compare the speeds of two different types of atoms—Helium and Argon—sharing the exact same container.

Imagine you are standing inside this container. The temperature is a constant $300\text{ K}$. Because both the Helium and Argon gases are mixed together in this single space, they are in perfect thermal equilibrium. This means they share the exact same temperature.

But does sharing the same temperature mean they move at the same speed? Not quite! To understand why, we need to look at the master equation for the speed of gas molecules.

The root mean square (rms) speed of a gas molecule is given by the elegant formula:

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

Notice something crucial here: for our mixture, both $R$ and $T$ are identical for Helium and Argon. The only thing that differentiates them is their mass, $M$. Therefore, we can see that the rms speed is inversely proportional to the square root of the molar mass:

This makes perfect physical sense. At a given temperature, all molecules have the same average kinetic energy. Since kinetic energy is $\frac{1}{2}m{v}^2$, a lighter molecule must travel much faster to pack the same energetic punch as a heavier, slower molecule.

Now, let's set up the ratio of their speeds. By dividing the rms speed of Helium by that of Argon, the $3RT$ terms completely cancel out, leaving us with:

We are given the atomic masses: $M_{\text{Ar}}=40\text{ amu}$ and $M_{\text{He}}=4\text{ amu}$. Let's plug these in:

This simplifies beautifully to:

We know that $3^2=9$, so $\sqrt{10}$ must be slightly larger than $3$. Calculating it gives us approximately $3.16$.

This means the nimble Helium atoms are zipping around more than three times faster than the heavier Argon atoms! The correct option is (d).