Question

Consider the following statements for air molecules in an airtight container.
I. The average speed of molecules is larger than root mean square speed.
II. Mean free path of molecules is larger than the mean distance between molecules.
III. Mean free path of molecules increases with temperature.
IV. The rms speed of nitrogen molecules is smaller than oxygen molecules.
The true statements are,
A. Only I
B. II & III
C. II & IV
D. I, II & IV

Answer 1

Hint: We will look at the different statements separately, and try to find out which ones are wrong. We will have to know the expressions for average velocity and the rms velocity. Also, we have to know about the expression for the mean free path.

Formula used:
$c_{rms}=\sqrt{\dfrac{3k_BT}{m}}$
$c_{avrg}=\sqrt{\dfrac{8k_BT}{\pi m}}$
$\lambda=\dfrac{1}{\sqrt{2} n\pi \sigma^2}$

Complete step-by-step solution:
If we add all the individual velocity of each molecule and divide the sum by the total number of molecules, what we obtain is called the average velocity of the gas in the container. On the other hand, the rms velocity of the gas is defined as the root of the mean of the squared individual velocities of the molecules. So, these two terms are never the same. The expression for rms velocity is,
$c_{rms}=\sqrt{\dfrac{3k_BT}{m}}$
And the expression for average velocity is,
$c_{avrg}=\sqrt{\dfrac{8k_BT}{\pi m}}$
Here $k_B$ is Boltzmann's constant. T is the absolute temperature. And m is the mass of a single molecule of the gas. So, plainly, rms velocity is larger than average velocity.
Now, the expression for mean free path is given by,
$\lambda=\dfrac{1}{\sqrt{2} n\pi \sigma^2}$
In this expression, $\sigma$ is the diameter of a molecule and n is the number density of molecules which is the number of molecules per unit volume. Since the enclosure is air tight, n is constant. So, a free path does not depend on temperature. So, III is wrong.
Now, the mass of a nitrogen molecule is 28 amu. And that of oxygen molecules is 32 amu. So, from the expression of rms velocity, it is clear that rms velocity of oxygen is less than that of nitrogen. So, IV is also wrong.
Hence from the given options, option A is the correct answer.

Note: At first sight, it may seem that the mean free path will depend on temperature. But you have to check the mathematical expression and answer accordingly. If temperature is increased in an airtight container, the pressure will increase but the number density of molecules will remain the same.