Calculate where F is the force on a particle of the ideal g

Calculate , where F, is the force on a particle of the ideal gas and in given exclusively by the momentum reversal of the container walls, and can therefore be expressed by the pressure of the gas. Use this results to obtain the equation of an ideal gas.

Solution

Without losing any generality, we need to make certain assumptions for the derivation:

1.) We will consider the container to be a rectangular section of length L and A1 and A2 as the area of the end walls.

2.) The number of particles travelling in any of the three directions is same. That is, if there are a total of N particles, the number of particles travelling in the X direction would be N/3

We will make use of the above to derive the required expression:

The time between collisions with the wall is the distance of travel between wall collisions divided by the speed, that is, t = 2L / V

Now force can be defined as the rate of change of the momentum, that is, F = (Pf - Pi) / t

For before and after the collision, Pf - Pi = 2mV

That is, F = 2mV / t = 2mV / (2L/V) = mV²/L

Also, the pressure, P, exerted by a single molecule is the average force per unit area, A. Also V=AL which is the volume of the rectangular box.

So we get: P = mV²/AL = mV²/ Volume

Now, for all the particles travelling along the horizontal direction, we can take the root mean squared velocities of the N/3 particles. That is, P = NmV²/3 x Volume

Also, we know that the root mean squared speed is given as: V = Sqrt[3RT/m]

Using this in the expression above, we get:

P = Nm(3RT/m) / 3 x Volume = nRT/Volume [N x mass of each particle / molar mass = Number of moles]

or, PV = nRT which is the required equation for ideal gase