My question is with regard to the derivation for the kinetic theory of gases that allows us to relate temperature to the motion of the particles.(adsbygoogle = window.adsbygoogle || []).push({});

I've looked at several introductory Physics texts and the same derivation is given (derivation in italics, my question in red, regular font):

To proceed we need an expression for the force

(1) Consider an ideal gas of N identical particles in a cubical container of length L.

(2)These particles do not interact except for elastic collisions.

Let’s just concentrate on one of these particles of mass m moving towards a wall with velocity +v and momentum +mv. Since the wall essentially has infinite mass, the particle bounces off with a velocity –v and momentum –mv. Now the time between collisions with the right hand wall is the round trip distance 2L divided by the particle speed or t=2L/v.

We now have enough information to calculate the average force on the wall given by the particle's change in momentum. Using the impulse-momentum theorem,

Fave(on particle) = (Final p – Initial p) / (Time between Collisions with the wall)

Substituting our information

Fave(on particle) = (-mv – mv) / (2L/v) = -2mv/2L/v = - mv2/L

I'm okay with the rest of it (based on the above), but I'm unsure why the "time between collisions" is being used for the time in the above equation? In that equation, shouldn't the time of the collision be used?

Now this is the average force exerted on the particle by the wall. But according to Newton’s 3rd Law the opposite but equal force will be exerted by the particle on the wall,

Fave(on wall) = +mv2/L.

The total force exerted on the right wall is equal to the force of each particle striking the wall. Since there are N particles and three dimensions, on average one-third will strike the right wall during the time t with an average value of the squared speed,

Ftotal(on wall) = [N/3]*[m(v2)ave/L]

Note we are using the average value of the squared velocity. The average must be used since the particles have a Maxwell distribution. Often one uses the definition: the square root of (v2)ave equals the root-mean-square speed or rms speed, vrms = ((v2)ave)1/2. So we can re-express the total force on the wall as

Ftotal(on wall) = [N/3]*[mv2rms/L]

So let’s convert to the pressure on the wall by dividing by the wall’s area L2,

P = Ftotal(on wall)/L2 = [N/3]* [mv2rms/L3]

But not, and this is nice, that the length cubed is just the volume of the box,so

PV = [N/3]* [mv2rms].

This is starting to look a lot like the ideal gas law! Note how we’ve related quantitatively macroscopic properties of the gas on the left to microscopic properties on the right. A little more manipulation and

PV = 2/3 * N *[mv2rms /2]

Since [mv2rms /2] is the average translational kinetic energy of an individual particle we can also state,

PV = 2/3 * N * KEave

Let’s compare this last result with the second version of the ideal gas law

PV = NkT

This shows that

2/3 *KEave = kT

or

KEave = ½*mv2rms = 3/2*kT

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# Derivation for Kinetic Theory of Gases

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