Maxwell–Boltzmann distribution

From Wikipedia, the free encyclopedia

Categories: Articles to be merged since October 2007 | Continuous distributions | Gases | Particle statistics | Statistical mechanics

It has been suggested that this article or section be merged with Maxwell speed distribution. (Discuss)

The Maxwell–Boltzmann distribution is a probability distribution with applications in physics and chemistry. The most common application is in the field of statistical mechanics. The temperature of any (massive) physical system is the result of the motions of the molecules and atoms which make up the system. These particles have a range of different velocities, and the velocity of any single particle constantly changes due to collisions with other particles. However, the fraction of a large number of particles within a particular velocity range is nearly constant. The Maxwell distribution of velocities specifies this fraction, for any velocity range, as a function of the temperature of the system. It is named after James Clerk Maxwell and Ludwig Boltzmann.

The distribution can be thought of as the magnitude of a 3-dimensional vector if its components are distributed as a normal distribution with standard deviation Failed to parse (Missing texvc executable; please see math/README to configure.): a . If Failed to parse (Missing texvc executable; please see math/README to configure.): X_i

are distributed as Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim N(0, a^2)

, then

Failed to parse (Missing texvc executable; please see math/README to configure.): Z = \sqrt{X_1^2+X_2^2+X_3^2}

is distributed as a Maxwell–Boltzmann distribution with parameter Failed to parse (Missing texvc executable; please see math/README to configure.): a .

1 Properties
2 Physical applications of the Maxwell–Boltzmann distribution
3 See also
4 External links

Properties

The Maxwell-Boltzmann distribution curve shows how particle velocities are distributed in an average sample of particles.

At any given temperature very few particles will possess very low or very high energy (most will possess an energy level somewhere in between the two extremes); this is called the mean energy. The activation energy barrier needs to be overcome if a reaction is to take place. If we increase the number of particles—i.e. increase the concentration of reactants—we produce more particles with energy greater than the activation energy (see collision theory).

The Maxwell–Boltzmann distribution with Failed to parse (Missing texvc executable; please see math/README to configure.): a=1

is equivalent to the chi distribution with three degrees of freedom. Additionally, if Z is distributed as a Maxwell–Boltzmann distribution with parameter a, then

Failed to parse (Missing texvc executable; please see math/README to configure.): W = \frac{Z}{a}

will be distributed as a chi distribution with three degrees of freedom.

The root-mean-square of a Maxwell–Boltzmann distribution is Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{3}a . Since Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} < 2\sqrt{2/\pi} < \sqrt{3} , it follows that the mode is less than the mean, which is always less than the root-mean-square.

Physical applications of the Maxwell–Boltzmann distribution

The Maxwell–Boltzmann distribution 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 of molecular speeds in a gas, but it can also refer to the distribution of 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.

The Maxwell–Boltzmann distribution can be derived using statistical mechanics (see Maxwell–Boltzmann statistics). It corresponds to the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles in which quantum effects are negligible. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas.

There are many cases (such as elastic collisions) where these conditions do not apply. For example, the physics of the ionosphere and space plasmas where recombination and collisional excitation (i.e. radiative processes) are important: especially for electrons. If you applied the Maxwell distribution and its assumptions here, you would get the wrong numbers, and miss the basic physics of the problem. Another example where applying the Maxwell-Boltzmann Distribution would give incorrect results is in cases where the quantum thermal wavelength of the gas is not small compared to the distance between particles. There, the theory would fail to account for significant quantum effects. Also, as it is based on nonrelativistic assumptions, the Maxwell-Boltzmann distribution does not predict zero probability for molecular velocities in excess of the speed of light.

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{N_i}{N} = \frac{g_i \exp\left(-E_i/kT \right) } { \sum_{j}^{} g_j \,{\exp\left(-E_j/kT\right)} } \qquad\qquad (1)

where N_i is the number of molecules at equilibrium temperature T, in a state i which has energy E_i and degeneracy g_i, 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 g_i. In this case the index i will specify an individual state, rather than a set of g_i states having the same energy E_i.) 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 of 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

Failed to parse (Missing texvc executable; please see math/README to configure.): E=\frac{p^2}{2m}

where p² is the square of the momentum vector p = [p_x, p_y, p_z]. We may therefore rewrite Equation 1 as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{N_i}{N} = \frac{1}{Z} \exp \left[ \frac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT} \right] \qquad\qquad (3)

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 N_i/N is proportional to the probability density function f_p for finding a molecule with these values of momentum components, so:

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathbf{p} (p_x, p_y, p_z) = \frac{c}{Z} \exp \left[ \frac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT} \right]. \qquad\qquad (4)

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 p_x, p_y, and p_z must be 1.

It can be shown that:

Failed to parse (Missing texvc executable; please see math/README to configure.): c = \frac{Z}{(\sqrt{2 \pi mkT}) ^ 3}. \qquad\qquad (5)

Substituting Equation 5 into Equation 4 gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathbf{p} (p_x, p_y, p_z) = \sqrt{\left( \frac{1}{2 \pi mkT} \right)^3} \exp \left[ \frac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT} \right]. \qquad\qquad (6)

The distribution is seen to be the product of three independent normally distributed variables Failed to parse (Missing texvc executable; please see math/README to configure.): p_x

Failed to parse (Missing texvc executable; please see math/README to configure.): p_y

, and Failed to parse (Missing texvc executable; please see math/README to configure.): p_z , with variance Failed to parse (Missing texvc executable; please see math/README to configure.): mkT . Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with Failed to parse (Missing texvc executable; please see math/README to configure.): a=\sqrt{mkT} .

Distribution of the energy

Using p² = 2mE we get the energy distribution:

Failed to parse (Missing texvc executable; please see math/README to configure.): f_E\,dE=f_p\left(\frac{dp}{dE}\right)\,dE =2\sqrt{\frac{E}{\pi(kT)^3}}~\exp\left[\frac{-E}{kT}\right]\,dE. \qquad \qquad(7)

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): f_E(E)\,dE=\chi^2(x;3)\,dx

where

Failed to parse (Missing texvc executable; please see math/README to configure.): x=\frac{2E}{kT}.\,

The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a quantum gas.

Distribution of the velocity vector

Recognizing that the velocity probability density f_v is proportional to the momentum probability density function by

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathbf{v} d^3v = f_\mathbf{p} \left(\frac{dp}{dv}\right)^3 d^3v

and using p = mv we get

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathbf{v} (v_x, v_y, v_z) = \sqrt{ \left(\frac{m}{2 \pi kT} \right)^3} \exp \left[ \frac{-m(v_x^2 + v_y^2 + v_z^2)}{2kT} \right], \qquad\qquad (8)

which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element [dv_x, dv_y, dv_z] about velocity v = [v_x, v_y, v_z] is

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathbf{v} \left(v_x, v_y, v_z\right)\, dv_x\, dv_y\, dv_z.

Like the momentum, this distribution is seen to be the product of three independent normally distributed variables Failed to parse (Missing texvc executable; please see math/README to configure.): v_x

Failed to parse (Missing texvc executable; please see math/README to configure.): v_y

, and Failed to parse (Missing texvc executable; please see math/README to configure.): v_z , but with variance Failed to parse (Missing texvc executable; please see math/README to configure.): kT/m . It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity [v_x, v_y, v_z] is the product of the distributions for each of the three directions:

Failed to parse (Missing texvc executable; please see math/README to configure.): f_v \left(v_x, v_y, v_z\right) = f_v (v_x)f_v (v_y)f_v (v_z)

where the distribution for a single direction is

Failed to parse (Missing texvc executable; please see math/README to configure.): f_v (v_i) = \sqrt{\frac{m}{2 \pi kT}} \exp \left[ \frac{-mv_i^2}{2kT} \right]. \qquad\qquad (9)

This distribution has the form of a normal distribution, with variance Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{kT}{m} . As expected for a gas at rest, the average velocity in any particular direction is zero.

Distribution of speeds

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) will be unitless.

Usually, we are more interested in the speeds of molecules rather than their component velocities. The Maxwell-Boltzmann distribution of speeds is written as

Failed to parse (Missing texvc executable; please see math/README to configure.): f (v) = 4 \pi \left(\frac{m}{2 \pi kT}\right)^{3/2}\!\!v^2 \exp \left[ \frac{-mv^2}{2kT} \right] \qquad (10)

where speed, v, is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): v = \sqrt{v_x^2 + v_y^2 + v_z^2}

Note that the units of f(v) in equation (10) 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 Failed to parse (Missing texvc executable; please see math/README to configure.): a=\sqrt{2kT/m} .

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.

Typical speeds

Although the above equation gives the distribution of speeds or in other words the fraction of molecules having a particular speed, we are often more interested in quantities such as the average speed of the particles rather than the actual distribution.

The most probable speed, v_p, is the speed most likely to be possessed by any molecule in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate df/dv, set it to zero and solve for v:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{df(v)}{dv} = 0

which yields:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_p = \sqrt { \frac{2kT}{m} } = \sqrt { \frac{2RT}{M} }

Where R is the gas constant and M = N_A m is the molar mass of the substance.

The mean speed is the mathematical average of the speed distribution

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle v \rangle = \int_0^{\infin} v \, f(v) \, dv= \sqrt { \frac{8kT}{\pi m}}= \sqrt { \frac{8RT}{\pi M}}

The root mean square speed, v_rms is the square root of the average squared speed:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_\mathrm{rms} = \sqrt{\int_0^{\infin} v^2 \, f(v) \, dv}= \sqrt { \frac{3kT}{m} }= \sqrt { \frac{3RT}{M} }

The typical speeds are related as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_p < \langle v \rangle < v_\mathrm{rms}.