# How does the kinetic-molecular theory explain the pressure exerted by gases?

Jun 7, 2017

In the kinetic theory of gases, gas molecules move around in the most random way colliding and bumping into each other and then bouncing off in the opposite direction.

The molecules exert no intermolecular forces on each other and they have no potential energy associated with them in their motion.
Thus the energy is wholly kinetic.

The concept of pressure is explained in kinetic theory as a consequence of kinetic energy of gases. Let me explain it.

Consider an ideal gas (the ones with which kinetic theory is concerned) in a closed container.
Due to the troublesome motion of the gas molecules, they will collide with each other, some of them shall collide with the container walls and then bounce back and this process continues.

Now if $m \vec{v}$ be the momentum of a single molecules before it bumps into the wall and it makes an angle $\theta$ with the wall and since the wall is assumed to be perfectly smooth it rebounds at an angle $\theta$ again (directed opposite to the direction from which the gas molecule initially came.

Thus, it transfers a momentum, $m v C o s \theta$ and undergoes a momentum change and rebounds with a final momentum $- m v C o s \theta$.

Thus the momentum change turns out to be,

$- m v C o s \theta + \left(- m v C o s \theta\right) = - 2 m v C o s \theta$

This implies that there is a momentum transfer of $2 m v C o s \theta$ to the wall from the molecule.

Thus considering a number of molecules striking the walls, they transfer momentum.
Now force is defined to be, the time rate of change of momentum.
Thus the molecules exert a force on the walls.

Further since, pressure is defined to be force per unit area, then the molecules exert a pressure on the container walls.

This pressure is not the hydrostatic pressure of a fluid in an external field since we assumed that there is no potential energy associated with them. Rather this pressure is associated with kinetic energy of motion (the more is the kinetic energy, the greater is the motion and as a result, more number of collisions occur and from arguments as before a greater momentum is transferred in unit area per unit time.
Thus, a greater pressure is obtained.

That is the kinetic interpretation of pressure.

Using some mathematical simplifications and averaging over all molecules, one can show that the pressure is given by,

$p = \frac{m n {v}^{2}}{3}$

Where $m$ is mass of each molecules, $n$ is the number of molecules in unit volume (molecular density) and $v$ is the rms speed.