Identify the correct labels of A, B and C in the following graph from the options given below:

Root mean square speed (V$$_{rms}$$); most probable speed (V$$_{mp}$$); Average speed (V$$_{av}$$)

For a gas of molecular mass $$M$$ at a temperature $$T$$, the three characteristic molecular speeds are defined as follows:

1. Most Probable Speed ($$V_{\text{mp}}$$): The speed possessed by the maximum fraction of total molecules, representing the absolute peak of the curve.

   $$V_{\text{mp}} = \sqrt{\frac{2RT}{M}} \approx 1.414 \sqrt{\frac{RT}{M}}$$



2. Average Speed ($$V_{\text{av}}$$): The mathematical mean of the speeds of all individual molecules.

   $$V_{\text{av}} = \sqrt{\frac{8RT}{\pi M}} \approx 1.596 \sqrt{\frac{RT}{M}}$$



3. Root Mean Square Speed ($$V_{\text{rms}}$$): The square root of the mean of the squares of individual molecular speeds.

   $$V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \approx 1.732 \sqrt{\frac{RT}{M}}$$

Comparing the Speeds:

By comparing the numerical values under identical conditions of temperature and molar mass, we get a definitive increasing order of magnitude:

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

The precise ratio between these speeds is:

$$V_{\text{mp}} : V_{\text{av}} : V_{\text{rms}} = 1 : 1.128 : 1.224$$

Mapping to the Graph Axis:

On the horizontal "speed" axis of the Maxwell-Boltzmann distribution graph, values systematically increase from left to right:

• Label A sits at the leftmost position (lowest speed value), directly identifying the peak of the curve: $$A = V_{\text{mp}}$$
• Label B occupies the intermediate position on the curve: $$B = V_{\text{av}}$$
• Label C sits at the rightmost position (highest speed value), stretching down toward the high-speed tail: $$C = V_{\text{rms}}$$

Conclusion:

The values increase moving rightwards along the x-axis, matching the order of most probable, average, and root mean square speed values.

Answer: Option D — A = V$$_{mp}$$; B = V$$_{av}$$; C = V$$_{rms}$$