# Boltzmann distribution: isothermal atmosphere error?

• I
• strauser
In summary, there is a well-known analysis of the distribution of particles in an isothermal atmosphere which states that the probability of finding a particle at a certain height is proportional to the exponential of the negative product of the Boltzmann constant, the particle's mass, and the gravitational potential at that height. This analysis also suggests that the number of particles at a given height is proportional to this exponential. However, some argue that this is only an approximation and that the number of particles should also take into account the available volume at that height. This is not typically done due to the small variation of height compared to the Earth's radius, but some suggest that it should be considered in order to provide a more comprehensive understanding.

#### strauser

There is a well-known analysis of the distribution of particles by height in an isothermal atmosphere. It states that the probability of finding a particle at height ##h## is ##p(h) \propto e^{-\beta mgh}##, and then goes on to state that the number of particles at height ##h## is ##n(h) \propto e^{-\beta mgh}##.

Is this not strictly incorrect, and merely an approximation? Surely we need to count the number of available states at each ##h##, and this will be proportional to the available volume in a thin shell at height ##h## i.e. we should have ##n(h) \propto f(h) e^{-\beta mgh}## where ##f(h)## measures the available volume? (This is because, roughly speaking, we can fit more molecules into a thin shell as ##h## increases.)

I'm guessing that this isn't done as ##h## doesn't vary much compared to the Earth's radius?

strauser said:
I'm guessing that this isn't done as h doesn't vary much compared to the Earth's radius?
It should be obvious form the fact that the potential used is proportional to h rather than 1/h!

Shyan said:
It should be obvious form the fact that the potential used is proportional to h rather than 1/h!
Well, I take your point but I don't think that it's necessarily "obvious" - given that the way this kind of argument should be structured, it seems like poor pedagogy, to my mind, to make no mention of any kind of density-of-states derivation - this seems to abound in statistical physics treatments though.

## 1. What is the Boltzmann distribution?

The Boltzmann distribution is a statistical distribution that describes the distribution of particles in a system at thermal equilibrium. It is used to calculate the probability of a particle having a certain energy level in a system.

## 2. How is the Boltzmann distribution related to isothermal atmosphere error?

In an isothermal atmosphere, all particles have the same average kinetic energy, so the Boltzmann distribution can be used to calculate the distribution of particles in this system. However, due to errors in measuring the energy levels of particles, the actual distribution may deviate from the theoretical Boltzmann distribution.

## 3. What factors can affect the accuracy of the Boltzmann distribution in an isothermal atmosphere?

Some factors that can affect the accuracy of the Boltzmann distribution in an isothermal atmosphere include measurement errors, non-ideal conditions, and interactions between particles.

## 4. How is the Boltzmann distribution used in atmospheric science?

The Boltzmann distribution is used in atmospheric science to understand the distribution of particles in the atmosphere, such as gas molecules or pollutants. It can also be used to calculate the rates of chemical reactions and to model temperature changes in the atmosphere.

## 5. Can the Boltzmann distribution be applied to non-isothermal atmospheres?

Yes, the Boltzmann distribution can be applied to non-isothermal atmospheres, but it may require more complex mathematical models to account for the varying energy levels of particles. In these cases, deviations from the theoretical Boltzmann distribution can also occur due to factors such as temperature gradients and energy transfer between particles.