Animated Solution for Chemistry - States of Matter: Points I, II and III in the following plot respectively correspond to ($v_{\text{mp}}$ : most probable velocity)

When we look at a gas trapped in a container, not all molecules are moving at the exact same speed. Some are sluggish, some are zipping around like bullets, but most are moving at a speed somewhere in between. This beautiful chaos is perfectly captured by the Maxwell-Boltzmann distribution curve.

In the given plot, the y-axis represents the fraction of molecules $f(v)$, and the x-axis represents their speed $v$. The peak of each curve is a very special point: it corresponds to the most probable velocity ($v_{\text{mp}}$). This is the speed possessed by the maximum number of molecules in the sample.

By simply observing the graph, we can establish a clear relationship between the peaks. Curve I peaks at the lowest speed, Curve II peaks at a medium speed, and Curve III peaks at the highest speed. Mathematically, we can write this as:

Most Probable Velocity

To figure out which gas belongs to which curve, we need to bring in the heavy machinery from the kinetic theory of gases. The formula for the most probable velocity is given by:

Here, $R$ is the universal gas constant, $T$ is the absolute temperature in Kelvin, and $M$ is the molar mass of the gas. Since $2$ and $R$ are constants, we can strip away the noise and focus on the core proportionality:

This elegant relationship tells us everything we need to know. A gas will have a higher most probable velocity if it is at a higher temperature or if it is physically lighter (lower molar mass).

Now, let's evaluate the $\sqrt{T/M}$ ratio for the three gas samples provided in the options. We don't need to calculate the exact velocity in meters per second; we just need the relative values to rank them.

1. Hydrogen Gas (${\text{H}}_2$) at $300\text{ K}$: Hydrogen is the lightest element. Its molar mass $M$ is $2\text{ g/mol}$.

2. Nitrogen Gas (${\text{N}}_2$) at $300\text{ K}$: Nitrogen is much heavier, with a molar mass $M$ of $28\text{ g/mol}$.

3. Oxygen Gas (${\text{O}}_2$) at $400\text{ K}$: Oxygen is the heaviest of the bunch with $M=32\text{ g/mol}$, but it is also at a higher temperature.

With our numbers ready, the comparison is straightforward. We can clearly see that:

Translating this back to our velocities, we get:

Now, we just match this ranking with our initial visual observation of the graph. Curve I has the lowest velocity, so it must be ${\text{N}}_2$ at $300\text{ K}$. Curve II is in the middle, corresponding to ${\text{O}}_2$ at $400\text{ K}$. Finally, Curve III has the highest velocity, which perfectly matches the incredibly light ${\text{H}}_2$ at $300\text{ K}$.

This makes option (c) the undeniably correct answer. As a final thought, notice how the hydrogen curve is much broader and flatter than the nitrogen curve. Lighter gases at higher temperatures have a much wider spread of molecular speeds, even though the total area under the curve (representing the total number of molecules) remains constant!