Maxwell-Boltzmann Distribution: Function, Derivation & Examples

Describing what is happening with very small particles is a challenge in physics. Not only is their size difficult to work with, but in most everyday applications, you aren't dealing with a single particle, but countless many of them all interacting with each other.

Within a solid, particles do not move past each other, but instead are pretty much stuck in place. Solids can expand and contract with temperature variations, however, and sometimes even undergo interesting changes into crystalline structures in certain situations.

In liquids, particles are free to move past each other. Scientists don't tend to study fluids, however, by trying to keep track of what each individual molecule is doing. Instead they look at larger properties of the whole, such as viscosity, density and pressure.

Just as with liquids, the particles within a gas are also free to move past each other. In fact, gases can undergo dramatic changes in volume due to differences in temperature and pressure.

Again, it doesn't make sense to study a gas by keeping track of what each individual gas molecule is doing, even at thermal equilibrium. It wouldn't be feasible, especially when you consider that even in the space of an empty drinking glass there are around 10²² air molecules. There isn't even a computer powerful enough to run a simulation of that many interacting molecules. Instead scientists use macroscopic properties such as pressure, volume and temperature to study gases and make accurate predictions.

The type of gas that is easiest to analyze is an ideal gas. It is ideal because it allows for certain simplifications that make the physics much easier to understand. Many gases at standard temperatures and pressures act approximately as ideal gases, which makes the study of them useful as well.

In an ideal gas, the gas molecules themselves are assumed to collide in perfectly elastic collisions so that you don't need to worry about energy changing form as a result of such collisions. It is also assumed that the molecules are very far apart from each other, which essentially means you don't have to worry about them fighting each other for space and can treat them as point particles. Ideal gases are also not too hot and not too cold, so you don't need to worry about effects such as ionization or quantum effects.

From here the gas particles can be treated like little point particles bouncing around within their container. But even with this simplification, it still isn't feasible to understand gases by tracking what each individual particle is doing. However, it does allow scientists to develop mathematical models that describe the relationships between macroscopic quantities.

The ideal gas law relates the pressure, volume and temperature of an ideal gas. The pressure ​P​ of a gas is the force per unit area that it exerts on the walls of the container it is in. The SI unit of pressure is the pascal (Pa) where 1Pa = 1N/m². The volume ​V​ of the gas is the amount of space it takes up in SI units of m³. And the temperature ​T​ of the gas is a measure of the average kinetic energy per molecule, measured in SI units of Kelvin.

The equation describing the ideal gas law may be written as follows:

\(PV=NkT\)

Where ​N​ is number of molecules or number of particles and the Boltzmann constant ​k​ = 1.38064852×10^-23 kgm²/s²K.

An equivalent formulation of this law is:

\(PV=nRT\)

Where ​n​ is the number of moles, and the universal gas constant ​R​ = 8.3145 J/molK.

These two expressions are equivalent. Which one you choose to use simply depends on whether you are measuring your molecule count in moles or in number of molecules.

Once a gas has been approximated as ideal, you can make an additional simplification. That is, instead of considering the exact physics of each molecule – which would be impossible due to their sheer number – they are treated as though their motions are random. Because of this, statistics can be applied to understand what's going on.

In the 19th century, physicists James Clerk Maxwell and Ludwig Boltzmann developed the kinetic theory of gases based on the simplifications described.

Classically, each molecule in a gas can have a kinetic energy attributed to it of the form:

\(E_{kin} = \frac{1}{2}mv^2\)

Not every molecule in the gas, however, has the same kinetic energy because they are constantly colliding. The exact distribution of the kinetic energies of the molecules is given by the Maxwell-Boltzmann distribution.

Maxwell-Boltzmann statistics describe the distribution of ideal gas molecules over various energy states. The function that describes this distribution is as follows:

\(f(E)=\frac{1}{Ae^{\frac{E}{kT}}}\)

Where ​A​ is a normalization constant, ​E​ is energy, ​k​ is Boltzmann's constant and ​T​ is temperature.

Further assumptions made to obtain this function are that, due to their point-particle nature, there is no limit of how many particles can occupy a given state. Also, the distribution of particles among energy states necessarily takes the most probable distribution (with larger numbers of particles, the odds of the gas not being close to this distribution become increasingly small). And finally, all energy states are equally probable.

These statistics work because it is extremely unlikely that any given particle can end up with an energy significantly above the average. If it did, that would leave a lot fewer ways for the rest of the total energy to be distributed. It boils down to a numbers game – as there are far more energy states that don't have a particle far above average, the probability of the system being in such a state is vanishingly small.

However, energies lower than the average are more probable, again because of how the probabilities play out. Since all motion is considered random and there are a greater number of ways a particle can end up in a low energy state, these states are favored.

The Maxwell-Boltzmann distribution is the distribution of the speeds of ideal gas particles. This speed-distribution function can be derived from the Maxwell-Boltzmann statistics and used to derive relationships between pressure, volume and temperature.

The distribution of speed ​v​ is given by the following formula:

\(f(v)=4\pi \Big[\frac{m}{2\pi kT}\Big]^{3/2}v^2e^{[\frac{-mv^2}{2kT}]}\)

Where ​m​ is the mass of a molecule.

The associated distribution curve, with the speed distribution function on the ​y​-axis and the molecular speed on the ​x​-axis, looks roughly like an asymmetric normal curve with a longer tail on the right. It has a peak value at the most probable speed ​_v_p_​, and an average speed given by:

\(v_{avg}=\sqrt{\frac{8kT}{\pi m}}\)

Note also how it has a long narrow tail. The curve changes slightly at different temperatures, with the long tail becoming "fatter" at higher temperatures.

Use the relationship:

\(E_{int}=N\times KE_{avg}=\frac{3}{2}NkT\)

Where ​_E_int​ is the internal energy, ​KE_​_​avg​ is the average kinetic energy per molecule from the Maxwell-Boltzmann distribution. Together with the ideal gas law, it's possible to get a relationship between pressure and volume in terms of molecular motion:

\(PV = \frac{2}{3}N\times KE_{avg}\)