Points I, II and III in the following plot respectively correspond to (V$$_{mp}$$: most probable velocity)

The formula for the most probable velocity (Vmp) of a gas molecule based on the Maxwell-Boltzmann distribution is:

Vmp = square root of (2 * R * T / M)

Where:

• R is the universal gas constant.
• T is the absolute temperature in Kelvin (K).
• M is the molar mass of the gas.

From this formula, we can see that:

• Vmp is directly proportional to the square root of Temperature (T).
• Vmp is inversely proportional to the square root of Molar Mass (M).

Therefore, Vmp increases when Temperature (T) increases or Molar Mass (M) decreases.

Given Data

We need to compare the values of Vmp for three specific cases:

1. N2 at 300 K (Molar mass of N2 = 28 g/mol)
2. O2 at 400 K (Molar mass of O2 = 32 g/mol)
3. H2 at 300 K (Molar mass of H2 = 2 g/mol)

Let's look at the ratio of (T / M) for each option to determine the relative order of their speeds:

• For N2 at 300 K: Ratio = 300 / 28 = 10.71
• For O2 at 400 K: Ratio = 400 / 32 = 12.50
• For H2 at 300 K: Ratio = 300 / 2 = 150.00

Comparing the Values

Comparing the ratios calculated above: 10.71 < 12.50 < 150.00

This gives us the order of increasing most probable velocity: Vmp(N2 at 300 K) < Vmp(O2 at 400 K) < Vmp(H2 at 300 K)

Matching with the Graph

Looking at the provided Maxwell-Boltzmann distribution curves along the x-axis (Speed, v):

• Point I has the lowest speed value.
• Point II has an intermediate speed value.
• Point III has the highest speed value.

Therefore:

• Point I corresponds to Vmp of N2 at 300 K
• Point II corresponds to Vmp of O2 at 400 K
• Point III corresponds to Vmp of H2 at 300 K