20.26 Boltzmann and other distribution laws

Before you read this, I suggest you read post 20.25


Normally, we think that objects spontaneously minimise their potential energy so that, for example, an object falls to the ground when it is released (post 16.21) and a stretched spring recoils when released (post 16.49). But this can’t be true for the molecules in a gas because they move to occupy the available space. When they move upwards, they increase their potential energy. The moving molecules have kinetic energy because their temperature is greater than zero; the heat associated with this non-zero temperature (post 16.34) is kinetic energy of molecular motion (post 16.35). So the heat energy does work to increase the potential energy of the molecules, in exactly the same way that we could use the (mechanical) potential energy of a compressed spring to project an object upwards.

In post 20.25, we saw that the number of molecules with potential energy E, in a gas, is given by

μ = μ0e-E/(kT)                     (1)

where μ0 is the number of molecules with the lowest energy, k is Boltzmann’s constant (post 18.25) and T is the temperature measure on the Kelvin scale (post 16.34). The number e is defined in post 18.15. This result can be applied to other particles in other situations and is called Boltzmann’s distribution law. Equation 1 is often written in the form

μ = λe-E/(kT)                   (2)

where λ has the same value for every energy state and is called the activity or absolute activity. Here “activity” has a different meaning to that in post 19.3 – be careful of confusion (post 17.14)! Note that λ is not a constant because its value depends on the system being considered.

Now let’s think about numbers of electrons with different energies. According to quantum theory, small particles, like electrons, cannot have any energy (post 16.29). They can only occupy allowed energy levels (posts 16.29 and 16.31); an analogy is that objects on stairs can’t have any potential energy – they are confined to the potential energy levels defined by the heights of each step (post 16.21). There is a further complication – a maximum of two electrons can occupy an energy level and they must have opposite spins (post 16.29). Equation 2 does not apply to this situation and must be replaced by

μi = λexp(-Ei/kT)/[1 + λexp(-Ei/kT)]                    (3)

where μi is the number of electrons in the ith level that has energy Ei; exp(x) is another way of writing ex (post 18.15). Equation 3 is a Fermi-Dirac distribution and spinning particles that follow this distribution when they occupy quantum states are called fermions.

Now let’s think about photons (post 19.23) occupying different quantum states. There is no restriction on the number of photons that can occupy a single quantised energy level – so they don’t obey the Fermi-Dirac distribution. Instead, they are distributed according to

μi = λexp(-Ei/kT)/[1 – λexp(-Ei/kT)].                   (4)

Equation 4 is a Bose-Einstein distribution; particles that follow this distribution are called bosons. Equation 3 can be used to explain how the intensity of radiated energy depends on wavelength (post 19.19).

When λ is very small, both [1 + λexp(-Ei/kT)] and [1 – λexp(-Ei/kT)] are then approximately equal to 1. Equations 3 and 4 are then identical to the Boltzmann distribution (equation 2). From its definition, a small value of λ corresponds to few particles in the lowest energy state. This implies that the separation between allowed energy levels is small; as this separation tends to zero the allowed energy levels become a continuum – so quantum effects are negligible. The Boltzmann distribution is often used as an approximate way of determining the number of particles in different quantised energy levels.

It can be shown that, in a gas of bosons, at very low temperatures, the particles tend to crowd into the lowest energy level. This phenomenon is called Bose-Einstein condensation and the resulting phase is called a Bose-Einstein condensate. Bose-Einstein condensates are the subject of considerable research activity because of their relevance to lasers and quantum computers.

I have stated a lot of results without proof in this post. That is because the ideas required to develop equations 3 and 4 (and Bose-Einstein condensation) have not been developed in previous posts. But I have written about Bose-Einstein and Fermi-Dirac statistics because they tell us more about the nature of the Boltzmann distribution.

Related posts

20.25 Pressure distribution in fluids

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