User:Inconspicuum/Physics (A Level)/Boltzmann Factor
Particles in a gas lose and gain energy at random due to collisions with each other. On average, over a large number of particles, the proportion of particles which have at least a certain amount of energy ε is constant. This is known as the Boltzmann factor. It is a value between 0 and 1. The Boltzmann factor is given by the formula:
,
where n is the number of particles with kinetic energy above an energy level ε, n0 is the total number of particles in the gas, T is the temperature of the gas (in kelvin) and k is the Boltzmann constant (1.38 x 10-23 JK-1).
This energy could be any sort of energy that a particle can have - it could be gravitational potential energy, or kinetic energy, for example.
Derivation[edit | edit source]
In the atmosphere, particles are pulled downwards by gravity. They gain and lose gravitational potential energy (mgh) due to collisions with each other. First, let's consider a small chunk of the atmosphere. It has horizontal cross-sectional area A, height dh, molecular density (the number of molecules per. unit volume) n and all the molecules have mass m. Let the number of particles in the chunk be N.
Therefore:
(which makes sense, if you think about it)
By definition:
The total mass Σ m is the mass of one molecule (m) multiplied by the number of molecules (N):
Then work out the weight of the chunk:
The downwards pressure P is force per. unit area, so:
We know that, as we go up in the atmosphere, the pressure decreases. So, across our little chunk there is a difference in pressure dP given by:
(1) In other words, the pressure is decreasing (-) and it is the result of the weight of this little chunk of atmosphere.
We also know that:
So:
But:
So, by substitution:
So, for our little chunk:
(2)
If we equate (1) and (2):
Rearrange to get:
Integrate between the limits n0 and n:
Since we are dealing with gravitational potential energy, ε = mgh, so:
A Graph of this Function[edit | edit source]
This topic comes up in Q10 494 June 2010. The Values used for various things in that question are