Name: Properties and Examples of Gases
Uploaded: 2019-07-19
Duration: 3 min 58 s
Description: The invisible and yet crucial state of matter is the gaseous state. It plays a vital role in sustaining life on Earth. Learn about its unique properties in this

Question

State the postulates of the kinetic theory of an ideal gas and show that the pressure

P

exerted by the gas on the walls of the container is

\frac{2}{3}

times the translational kinetic energy per unit volume.

Accepted Answer

Following are the postulates of the kinetic theory of gases:

$1 .$ All the molecules of a gas are identical, rigid and very small. The volume of the molecules themselves is negligible compared to the volume occupied by the gas.

$2 .$ Normally no force acts between the molecules. Hence, they move in straight lines with constant speeds. But when two molecules happen to come very dose to each other, a repulsive force begins to act between them due to which the speeds and the directions of motion of the molecules change and new straight-line motions start.

$3 .$ The collision between two molecules is perfectly elastic, that is, the kinetic energy remains conserved in the collision.

$4 .$ The duration of a collision is negligible compared with the time spent by a molecule between collisions.

$5 .$ Because the molecules inside the vessel keep on moving continuously in all possible directions, the distribution of molecules in the whole vessel remains uniform.

$6 .$ Because the mass of a molecule is negligibly small and the speed is very large, there is no effect of gravity on the motion of the molecules. If this effect were there, the density of the gas would have been greater at the bottom of the vessel.

Let us consider an ideal gas contained in a cubical vessel whose walls are perfectly elastic. Let $l$ be the length of each side of the cube. Let $m$ be the mass of each molecule and $n$ the total number of molecules in the gas. These molecules are moving in all directions with different speeds. Let $v$ be the velocity of a molecule. If $v_{x}$ , $v_{y}$ and $v_{z}$ be the components of this velocity along $X$ , $Y$ and $Z$ axes respectively, then $v^{2} = {v_{x}}^{2} + {v_{y}}^{2} + {v_{z}}^{2}$ .

Suppose this molecule is moving towards the wall $A$ which is perpendicular to the $X$ axis. On colliding with the wall, its $x$ component of velocity changes from $v_{x}$ to $- v_{x}$ (other components remain unchanged). Thus, its momentum along the $X$ axis changes from $m v_{x}$ to $- m v_{x}$ , that is, in one collision the change in momentum of the molecule is $- 2 m v_{x}$ . Hence, according to the law of conservation of momentum, in one collision the momentum transferred to the wall $P$ by the molecule is $\frac{2}{3}$ .

The molecule returned from $A$ collides with the wall opposite to $A$ , rebounds and again collides with $A$ . Thus, the molecule covers a distance $2 l$ between two successive collisions with $A$ . Since, the velocity of the molecule is $v_{x}$ , it takes $\frac{2 l}{v_{x}}$ second in covering the distance $2 l$ . Thus, the molecule collides with the wall $A$ after every $\frac{2 l}{v_{x}}$ second. Hence, the momentum transferred
to the wall $A$ in $1$ second is,

$\frac{m}{l} {v_{x}}^{2}$ .

According to Newton’s law of motion, the rate of change of momentum of a body is equal to the force exerted on it. Hence, the force exerted by the molecule on the wall $A$ will be $\frac{m {v_{x}}^{2}}{l}$ . Since, there are $n$ molecules, the total force exerted on the wall $A$ is,

$\frac{m}{l} ({v_{x 1}}^{2} + {v_{x 2}}^{2} + - - - + {v_{x n}}^{2})$ .

Now, pressure $P$ will be,

$P = \frac{m n}{v} (\frac{{v_{x 1}}^{2} + {v_{x 2}}^{2} + - - - + {v_{x n}}^{2}}{n})$ or

$P = \frac{1}{3} \frac{m n}{V} v^{2}$ .

This is the expression for the pressure of the gas. $v^{2}$ is called the mean square velocity of the molecules. Since, $m n$ is the mass of the whole gas and $V$ is the volume, so $\frac{m n}{V}$ is the density $ρ$ of the gas. Therefore, the above expression can be written as,

$P = \frac{1}{3} ρ v^{2} = \frac{2}{3} (\frac{1}{3} ρ v^{2}) = \frac{2}{3} E$ .

State the postulates of the kinetic theory of an ideal gas and show that the pressure $P$ exerted by the gas on the walls of the container is $\frac{2}{3}$ times the translational kinetic energy per unit volume.

Important Questions on Kinetic Theory of Gases

Learn and Prepare for any exam you want !

State the postulates of the kinetic theory of an ideal gas and show that the pressure P exerted by the gas on the walls of the container is 23 times the translational kinetic energy per unit volume.

Important Questions on Kinetic Theory of Gases

Learn and Prepare for any exam you want !

State the postulates of the kinetic theory of an ideal gas and show that the pressure $P$ exerted by the gas on the walls of the container is $\frac{2}{3}$ times the translational kinetic energy per unit volume.