Parts one and two of modeling the atmosphere derived a way to describe atmospheric pressure as a function of altitude, assuming a constant temperature and acceleration due to gravity throughout the ideal gas. In this installment of Modeling the Atmosphere, the same function will be derived, this time using the Boltzmann distribution.
The Boltzmann distribution states that the fraction of particles in an energy state is given by the equation
where represents the degeneracy of the energy level and the partition function
This distribution works for discrete energy levels, so a continuum approximation must be made for classical systems. By using a density of states , which supposes that there are states in the energy interval to , the probability distribution for the energy
The first goal in when using the Boltzmann distribution to model an atmosphere is to find an expression for the energy .
The equipartition theorem states that each degree of freedom for a particle receives of energy. So, in an isothermal atmosphere of ideal gas, each particle receives the same amount of kinetic energy . The potential energy of a particle depends on its altitude and is given by the equation , where is the mass of a particle, is the acceleration due to gravity (once again assumed to be constant), and represents altitude. Thus, the total energy of an ideal gas particle in an isothermal atmosphere is given by
Assuming that the density of states is a constant , then the continuous distribution
Calculating the integral:
Evaluate at zero:
And at infinity:
For a large system, this equation describes the fraction of particles at a height between and :
Notice that the coefficient has units , which cancels out to be unitless.
How can this expression for the factional number of particles at a height be transformed to describe pressure as a function of altitude? The ideal gas law allows pressure to be expressed in terms of the number of particles :
The number of particles between the height and is given by the fraction of particles between and times the total number of particles in the system:
So the pressure as a function of altitude is given by:
Notice that the coefficient now has units of pressure:
Using the boundary condition at , this coefficient is equal to the initial pressure :
makes sense conceptually. To see why, note that , or the mass of all of the particles in the atmosphere, and , where is the surface area of the ground. Then
which describes the force of the entire atmosphere across the surface.
Thus, the exact same expression for pressure as a function of altitude (previously derived using a different method) has been found, this time starting from the Boltzmann distribution!