Animated Solution for Chemistry - States of Matter: Identify the correct labels of $A$, $B$ and $C$ in the following graph from the options given below : Root mean square speed ($v_{\text{rms}}$); most probable speed ($v_{\text{mp}}$); Average speed ($v_{\text{av}}$)

Imagine a container filled with gas molecules zipping around in random directions. Not all molecules travel at the same speed; some are sluggish, some are incredibly fast, but most travel at some intermediate speed. This behavior is beautifully captured by the Maxwell-Boltzmann distribution curve.

When we look at the graph provided in the question, the x-axis represents the speed of the molecules, and the y-axis represents the number of molecules (or fraction of molecules) possessing that particular speed. The curve starts at zero, rises to a peak, and then gradually tails off towards higher speeds.

The peak of this curve is of special significance. It represents the speed possessed by the maximum number of molecules in the gas sample. We call this the most probable speed ($v_{\text{mp}}$).

To fully describe the kinetic behavior of a gas, physicists define three characteristic speeds. Let's look at their mathematical definitions derived from the kinetic theory of gases:

1. Most Probable Speed ($v_{\text{mp}}$): As discussed, this is the speed at the peak of the distribution.

2. Average Speed ($v_{\text{av}}$): This is the simple arithmetic mean of the speeds of all molecules.

3. Root Mean Square Speed ($v_{\text{rms}}$): This is the square root of the average of the squares of the speeds. It is directly related to the average kinetic energy of the gas.

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

To identify the labels $A$, $B$, and $C$ on the graph, we need to determine the relative magnitudes of these three speeds. Notice that the term $\sqrt{\frac{RT}{M}}$ is common to all three formulas. Therefore, we only need to compare their numerical coefficients:

For $v_{\text{mp}}$, the coefficient is $\sqrt{2}\approx 1.414$ For $v_{\text{av}}$, the coefficient is $\sqrt{\frac{8}{\pi }}\approx \sqrt{2.546}\approx 1.595$ * For $v_{\text{rms}}$, the coefficient is $\sqrt{3}\approx 1.732$

Comparing these values, we arrive at a universal inequality for any ideal gas:

$$v_{\text{mp}}<v_{\text{av}}<v_{\text{rms}}$$

Now, let's return to our Maxwell-Boltzmann curve. As we move from left to right along the x-axis, the value of speed increases.

Point A is exactly at the peak of the curve. By definition, this is the most probable speed ($v_{\text{mp}}$). Point B lies slightly to the right of the peak, meaning it represents a higher speed. Based on our inequality, this must be the average speed ($v_{\text{av}}$). Point C is the furthest to the right among the three, representing the highest speed. This corresponds to the root mean square speed* ($v_{\text{rms}}$).

Therefore, the correct labeling is $A\to v_{\text{mp}}$, $B\to v_{\text{av}}$, and $C\to v_{\text{rms}}$. This perfectly matches option (c).

Understanding this fundamental order not only helps in solving direct questions like this but also builds a strong intuition for more complex thermodynamics problems!