Question

Let V_a V_mp, V_rms represent average speed, most probable speed and root mean square speed of a gas molecule at a given temperature.Which of the following is correct
1. V_rms > V_a = V_mp
2. V_rms > V_mp > V_a
3. V_rms < V_mp < V_a
4. V_rms > V_a > V_mp

Answer 1

Correct Answer - Option 4 : V_rms > V_a > V_mp

Correct option - 4

Concept:

I.RMS Velocity

• The root-mean-square (RMS) velocity of a gas is the value of the square root of the sum of the squares of all the individual velocity values divided by the number of values. 
• It is the speed of a wave through sub-surface layers of different interval velocities along a specifically directed path. 
• RMS velocity is always higher than the average velocity of the gas.

The expression for RMS speed is given by Maxwell as-

\({{v}_{rms}}=\sqrt{\frac{3RT}{M}}\)

where,

\({{v}_{rms}}=\) RMS speed

II.Mean Speed or Average Speed

• The particles or molecules of a gas have a range of speeds.
• The average speed of all the molecules at a given temperature is found by taking the average of the speeds of all the particles at a given instant.
• Remember that the speed is a positive scalar since it is the magnitude of the velocity.

For example, let N number of molecules in a container possess speeds v1, v2, v3, .... vN. then the average or mean speed of the molecules is written as-

\({{v}_{a}}=\frac{{{v}_{1}}+{{v}_{2}}+..+{{v}_{N}}}{N}\)

From Maxwell's speed distribution law, it is given by-

\({{v}_{a}}=\sqrt{\frac{8RT}{\pi M}}\)

where,

\({{v}_{a}}=\) Average speed

III.Most Probable speed

• It is defined as the speed which is possessed by a maximum fraction of a total number of molecules of the gas.

For example, if the speeds of 15 molecules of gas are, 1, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 2, 6, 6, 1 km/s, then the most probable speed is 6 km/s, as the maximum fraction of total molecules possess this speed. 

From Maxwell's speed distribution law,

The Most probable speed is given by-

\({{v}_{mp}}=\sqrt{\frac{2RT}{M}}\)

where,

\({{v}_{mp}}=\) Most probable speed

R = Molar gas constant

T = Temperature and

M = Molar mass of a given gas molecule.

Calculation:

From the above expressions of V_rms, V_mp, and V_a

We have 

\({{v}_{rms}}=\sqrt{\frac{3RT}{M}}\)

\({{v}_{a}}=\sqrt{\frac{8RT}{\pi M}}\) and

\({{v}_{mp}}=\sqrt{\frac{2RT}{M}}\)

Now,

\({{v}_{rms}}:{{v}_{a}}:{{v}_{mp}}=\sqrt{3}:\sqrt{\frac{8}{\pi }}:\sqrt{2}\)

\(\because \frac{8}{\pi }\approx 2.5\)

\(\Rightarrow {{v}_{rms}}:{{v}_{a}}:{{v}_{mp}}=\sqrt{3}:\sqrt{2.5}:\sqrt{2}\)

\(\therefore {{v}_{rms}}>{{v}_{a}}>{{v}_{mp}}\)

Hence, option - 4 is correct