Physical Applications Of The Maxwell

Physical applications of the Maxwell–Boltzmann distribution The Maxwell–Boltzmann distribution applies to ideal gases close to thermodynamic equilibrium, negligible quantum effects, and non-relativistic speeds. It forms the basis of the kinetic theory of gases, which explains many fundamental gas properties, including pressure and diffusion. The Maxwell–Boltzmann distribution is usually thought of as the distribution for molecular speeds, but it can also refer to the distribution for velocities, momenta, and magnitude of the momenta of the molecules, each of which will have a different probability distribution function, all of which are related. In the classical picture of an ideal gas, molecules bounce around at a variety of different velocities, never interacting with each other. Though this qualitative picture is obviously flawed (since molecules always do interact), it is a useful model for situations where the particle density is very low; in a more quantitative sense, this means that the particles themselves are very small when compared to the volume between them. Accordingly, we will want to know exactly how many of these molecules are moving around at a given speed. The Maxwell speed distribution (MSD), named after James Clerk Maxwell, is a probability distribution describing the "spread" of these molecular speeds; it is derived, and therefore only valid, assuming that we're dealing with an ideal gas. Again, no gas is truly ideal, but our own atmosphere at STP behaves enough like the ideal situation that the MSD can be used. Note that speed is a scalar quantity, describing how fast the particles are moving, regardless of direction; velocity also describes the direction that the particles are moving. It is elementary using statistical mechanics to find that the MSD must be proportional to the probability that a particle is moving at a given speed. Another important element is the fact that space is three dimensional, which implies that for any given speed, there are many possible velocity vectors. The probability of a molecule having a given speed can be found by using Boltzmann factor; considering the energy to be dependent only on the kinetic energy, we find that:

Here, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.

In 3-dimensional velocity space, the velocity vectors corresponding to a given speed v live on the surface of a sphere with radius v. The larger v is, the bigger the sphere, and the more possible velocity vectors there are. So the number of possible velocity vectors for a given speed goes like the surface area of a sphere of radius v.

Multiplying these two functions together gives us the distribution, and normalising this gives us the MSD in its entirety.

(Again, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.) Since this formula is a normalised probability distribution, it gives the probability of a molecule having a speed between v and v + dv. The probability that the velocity of a particle is between two different values v0 and v1 can be found by integrating this function with v0 and v1 as the bounds.

Derivation The original derivation by Maxwell assumed all three directions would behave in the same fashion, but a later derivation by Boltzmann dropped this assumption using kinetic theory. The Maxwell–Boltzmann distribution can now most readily be derived from the Boltzmann distribution for energies (see also the Maxwell–Boltzmann statistics of statistical mechanics):

where Ni is the number of molecules at equilibrium temperature T, in a state i which has energy Ei and degeneracy gi, N is the total number of molecules in the system and k is the Boltzmann constant. (Note that sometimes the above equation is written without the degeneracy factor gi. In this case the index i will specify an individual state, rather than a set of gi states having the same energy Ei.) Because velocity and speed are related to energy, Equation 1 can be used to derive relationships between temperature and the speeds of molecules in a gas. The denominator in this equation is known as the canonical partition function.

Distribution for the momentum vector What follows is a derivation wildly different from the derivation described by James Clerk Maxwell and later described with fewer assumptions by Ludwig Boltzmann. Instead it is close to Boltzmann's later approach of 1877. For the case of an "ideal gas" consisting of non-interacting atoms in the ground state, all energy is in the form of kinetic energy. The relationship between kinetic energy and momentum for massive particles is

where p2 is the square of the momentum vector p = [px, py, pz]. We may therefore rewrite Equation 1 as:

where Z is the partition function, corresponding to the denominator in Equation 1. Here m is the molecular mass of the gas, T is the thermodynamic temperature and k is the Boltzmann constant. This distribution of Ni/N is proportional to the probability density function fp for finding a molecule with these values of momentum components, so:

The normalizing constant c, can be determined by recognizing that the probability of a molecule having any momentum must be 1. Therefore the integral of equation 4 over all px, py, and pz must be 1. It can be shown that:

Substituting Equation 5 into Equation 4 gives:

The distribution is seen to be the product of three independent normally distributed variables px, py, and pz, with variance mkT. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with .

Distribution for the energy Using p² = 2mE, and the distribution function for the magnitude of the momentum (see below), we get the energy distribution:

Since the energy is proportional to the sum of the squares of the three normally distributed momentum components, this distribution is a chi-square distribution with three degrees of freedom:

where

The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a Distribution for the velocity vector Recognizing that the velocity probability density fv is proportional to the momentum probability density function by

and using p = mv we get

which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element [dvx, dvy, dvz] about velocity v = [vx, vy, vz] is

Like the momentum, this distribution is seen to be the product of three independent normally distributed variables vx, vy, and vz, but with variance . It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity [vx, vy, vz] is the product of the distributions for each of the three directions:

where the distribution for a single direction is

This distribution has the form of a normal distribution, with variance . As expected for a gas at rest, the average velocity in any particular direction is zero.

Distribution for the speed

The speed probability density functions of the speeds of a few noble gases at a temperature of 298.15 K (25 °C). The y-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless. Usually, we are more interested in the speeds of molecules rather than their component velocities. The Maxwell-Boltzmann distribution for the speed is written as

where speed, v, is defined as

Note that the units of f(v) in equation are probability per speed, or just reciprocal speed as in the graph at the right.

Since the speed is the square root of the sum of squares of the three independent, normally distributed velocity components, this distribution is a Maxwell–Boltzmann distribution, with

.

We are often more interested in quantities such as the average speed of the particles rather than the actual distribution. The mean speed, most probable speed (mode), and root-mean-square can be obtained from properties of the Maxwell–Boltzmann distribution.