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How does the concept of pressure follow from the kinetic molecular theory?

How does the concept of pressure follow from the kinetic molecular theory?

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How does the concept of pressure follow from the kinetic molecular theory?

Expert Solution
Step 1

This is a problem from the kinetic theory of gases.

According to the kinetic theory, the pressure due to the ideal gas in a cubical container can be written as,

P=13NVmv2Where,N-Number of moleculesm-Mass of a each moleculev2-Mean of the square of velocities of all the molecules

We will prove this in the next step.

Step 2

Let us consider a cubic box having the sides of length L containing an ideal gas inside it.

Consider a typical molecule, moving in positive x-direction with a speed v and which going to collide with the wall parallel to the yz-plane and bounce back  (Consider this to be an elastic collision).

The components of velocities are vx , vy and vz . 

The change in momnetum of a molecule is given by,

∆px=-mvx-mvx=-2mvx                                                             ...[mv before colliding and -mv after colliding]     ...(1)

Time taken by the molecule between the two successive collisions can be written as,

∆t=2Lvx                                          ...[distance in 2L and velocity is vx]                                                                           ...(2) 

Now, The force exerted by a single molecule on the wall of a container is given by the,

F=∆p∆t                                                                                                                         ....(3)

Substituting the equation 1 and 2 in equation 3, we get,

F=2mvx12Lvx1=mvx12L                                                                                                                     ...(4)

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