Notes on deriving kinetic theory equation

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Preasure is defined as force per unit area so to find an expression for preasure we must 1st find an expression for the force of the gas on the walls of its container.

Consider a gas particle on a container with velocity ux, once it rebounds of the opposite sids of the box its velocity will be -ux, with mass m the momentum change will be -2mux

From newtons 3rd law we know that a force is needed to cause a momentum change. From Newtons 2nd law we know that Fdt = dmv so F = dmv/dt

dt can be writen as an expression of ux, as v = s/t where s = 2L and t = s/v, so dt = 2l/ux

by subsituting this back into F=dmv/dt we find that F = mux^2/L

thus as preasure is force/area p = mux^2/L^3

to find the overall preasure of the gas, we need to find the force exerted by all the gas particles in the container. All of the molecules will have random speeds and the preasure is the sum of the partial preasures in the container. so p= m/L^3 x sum(ux1^2 + ux2^2 + ux3^2 + .....) which statisticaly be writen as p = mN<ux^2>/L^3

This will be the total preasure exerted on the container, however as we have only considered the velocity in the x-axis the preasure will only be 1/3 that as above p = mN<ux^2>/3L^3

and as L^3 is the volume of the box the equation can be writen as p = mN<ux^2>/3V

thus pV = mN<ux^2>/3 QED