In an ideal gas at temperature $T$, the average force that a molecule applies on the walls of a closed container depends on $T$ as $T^q$. A good estimate for $q$ is :

Question

In a mixture of gases, the average number of degrees of freedom per molecule is 6. If the rms speed of the molecule is $c$, what is the velocity of sound in the gas?

Answer 1

To determine the relationship between the average force that a molecule applies on the walls of a closed container and the temperature $ T $, let's analyze the behavior of an ideal gas.

According to the kinetic theory of gases, the pressure $ P $ exerted by an ideal gas is given by:

$P = \frac{1}{3} N m \overline{v^2} / V$

where:

$ N $ is the number of molecules,
$ m $ is the mass of each molecule,
$ \overline{v^2} $ is the mean square velocity of the molecules,
$ V $ is the volume of the container.

The mean square velocity $ \overline{v^2} $ is related to the temperature by:

$\overline{v^2} = \frac{3kT}{m}$

where:

$ k $ is the Boltzmann constant,
$ T $ is the absolute temperature.

Substituting $ \overline{v^2} $ in the equation for pressure:

$P = \frac{1}{3} N \frac{3kT}{V} = \frac{NkT}{V}$

The average force ($ F $) exerted by a molecule on the walls is related to the pressure by:

$F = PA$

Here, $ A $ is the area of the wall. Since we consider molecular impacts, we can relate force per collision and frequency of collisions, showing how force is proportional to pressure.

From kinetic theory: $P \propto T$.

Thus, the force is $F \propto T $, and hence\)

Therefore, the correct estimate for $ q $ is 1, meaning that the average force exerted by a gas molecule depends linearly on the temperature $ T $.

In an ideal gas at temperature $T$, the average force that a molecule applies on the walls of a closed container depends on $T$ as $T^q$. A good estimate for $q$ is :

The Correct Option is B

Solution and Explanation

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