Question

From the following statements , regarding an ideal gas at any given temperature $T$, choose the correct alternative$\left(s\right)$

• A. The coefficient of volume expansion at constant pressure is the same for all ideal gases
• B. The average translational kinetic energy per molecule of oxygen gas is $3kT$, where $k$ being Boltzmann constant
• C. The mean -free path of molecules increases with decrease in the pressure
• D. In a gaseous mixture, the average translational kinetic energy of the molecules of each component is different.

Answer 1

Answer: (A), (C)

The mean -free path of molecules increases with decrease in the pressure

For one mole of an ideal gas $PV=RT$
Differentiating the above equation with respect to $T$ on both sides we get,

${\left(\frac{\Delta V}{\Delta T}\right)}_P=\frac{R}{P}$

The coefficient of volume expansion at constant pressure is given by

$\gamma =\frac{1}{V}{\left(\frac{\Delta V}{\Delta T}\right)}_P=\frac{1}{T}$

At a given temperature and pressure, coefficient of volume expansion is same for all the ideal gases.

$\therefore$ Option (a) is correct.

The average translational kinetic enrgy per molecule is $\left(\frac{3}{2}\right)kT$ and not $3kT$.

With decrease in pressure, collisions between molecules will be less frequent, hence mean free path will increase.

$\lambda =\frac{kT}{\sqrt{2}\pi d^{2}P}\Rightarrow \lambda \propto \frac{1}{P}$

$\therefore$ Option (c) is also correct.

The average translational kinetic energy of the molecules is independent of their nature and is only a function of temperature.

$\Rightarrow$ Each component of the gaseous mixture will have the same value of average translational kinetic energy, at the given temperature.

Thus, Options (a) and (c) are the correct alternatives.