Question

The \[{{\nu }_{rms}},{{v}_{av}}\]and ${{\nu }_{mp}}$ are root mean square, average and most probable speeds of molecules of a gas obeying Maxwell velocity distribution. Which of the following statements is correct?
A.${{\nu }_{rms}}<{{\nu }_{av}}<{{\nu }_{mp}}$
B.${{\nu }_{rms}}>{{\nu }_{av}}>{{\nu }_{mp}}$
C.${{\nu }_{mp}}<{{\nu }_{rms}}<{{\nu }_{av}}$
D.${{\nu }_{mp}}>{{\nu }_{rms}}<{{\nu }_{av}}$

Answer 1

Hint: in kinetic theory of gases, the gas molecules are in random rapid motions. During their motion, every molecule has different velocities and therefore the molecules keep on colliding with each other. Hence, we can describe the movement of molecules with velocity.

Complete step-by-step answer:Root mean square velocity is defined as the square root of mean of squares of the velocity of individual gas molecules. The formula can be written as-
${{\nu }_{rms}}=\sqrt{\dfrac{3RT}{M}}$
In this equation, ${{v}_{rms}}$ is root mean square velocity, $R$ is the universal gas constant, $T$ is the temperature, $M$ is the molar mass.
Average velocity is defined as the arithmetic mean of velocities of different molecules at a given temperature. The formula can be written as-
${{\nu }_{av}}=\sqrt{\dfrac{8RT}{\pi M}}$
In this equation, ${{v}_{av}}$ is average velocity, $R$ is the universal gas constant, $T$ is the temperature, $M$ is the molar mass.
Most probable velocity is defined as the velocity of maximum number of molecules at same temperature. The formula can be written as-
${{\nu }_{mp}}=\sqrt{\dfrac{2RT}{M}}$
In this equation, ${{v}_{mp}}$ is most probable velocity, $R$ is the universal gas constant, $T$ is the temperature, $M$ is the molar mass.
If we compare these velocities, the ratio can be written as:
${{\nu }_{rms}}:{{\nu }_{av}}:{{\nu }_{mp}}$
$\sqrt{\dfrac{3RT}{M}}:\sqrt{\dfrac{8RT}{\pi M}}:\sqrt{\dfrac{2RT}{M}}$
As $R,T$ and $M$ is present in all the three velocities hence we can cancel out.
$\sqrt{3}:\sqrt{\dfrac{8}{\pi }}:\sqrt{2}$
It can also be written as:
$1.732:1.596:1.414$
Hence, we can say that ${{\nu }_{rms}}>{{\nu }_{av}}>{{\nu }_{mp}}$

Therefore, the correct option is B.

Note:Postulates of kinetic theory of gases:
-All the molecules of particular gas are identical in shape and size.
-Volume occupied by the gas molecule is negligible when compared with total volume occupied by the gas.
-There is no force of attraction and force of repulsion between the particles.
-All the laws of motion are applicable.