Question

For the molecules of an ideal gas, which of the following velocity averages cannot be zero?
A) $\left\langle V \right\rangle $
B) $\left\langle {{V^4}} \right\rangle $
C) $\left\langle {{V^3}} \right\rangle $
D) $\left\langle {{V^5}} \right\rangle $

Answer 1

Hint
The velocity of the gas molecules can be positive or negative. The even power of the velocity will make the negatives also positive. Only in odd powers negative velocity can exist to cancel out to make velocity zero. So, even the power of velocity cannot be zero.

Complete step-by-step answer
Gas is made up of many molecules and every molecule moves in a random direction. The velocity of each molecule depends on its direction of motion. It can be positive, negative or zero.
The average velocity of all the molecules in the gas is given by,
$\left\langle {{{\bar v}^k}} \right\rangle = {\left( {\dfrac{{\sum v }}{n}} \right)^{\dfrac{1}{k}}}$
Where, n is the number of molecules.
The average velocities according to the options are:
Case A: $\left\langle {\bar v} \right\rangle = \left( {\dfrac{{\sum v }}{n}} \right)$
Case B: $\left\langle {{{\bar v}^4}} \right\rangle = {\left( {\dfrac{{\sum v }}{n}} \right)^{\dfrac{1}{4}}}$
Case C:$\left\langle {{{\bar v}^3}} \right\rangle = {\left( {\dfrac{{\sum v }}{n}} \right)^{\dfrac{1}{3}}}$
Case D: $\left\langle {{{\bar v}^5}} \right\rangle = {\left( {\dfrac{{\sum v }}{n}} \right)^{\dfrac{1}{5}}}$
Since the molecules of the gas move randomly in all directions the average velocity is zero. So A is zero.
The only possibility for the average velocity to be equal to zero is when for each positive velocity there exists a negative velocity. But negative velocity cannot exist in even powers like 2,4,6,8. Since, the even power of velocity will be positive the average cannot be equal to zero.
Hence, average velocity cannot be zero in $\left\langle {{V^4}} \right\rangle $.
The correct option is (B).

Note
The speed possessed by the maximum fraction of the total number of molecules of the gas is called the most probable speed. The rms speed of the molecules depends on temperature, molecular weight and does not depend on pressure.