Maxwell-boltzmann Distribution

Type of physical science: Classical physics

Field of study: Statistical mechanics

The Maxwell-Boltzmann distribution is a mathematical tool used to predict the macroscopic properties of matter from the properties of the molecules of which it is composed.

Overview

The Maxwell-Boltzmann distribution is a mathematical recipe for calculating the probability that a constituent particle in a large collection of noninteracting particles has a particular position and velocity, or a particular energy. The particles typically are atoms, ions, or molecules, and the collection can be any dilute medium, such as a gas, a liquid solution, or a mixed crystal. The importance of the distribution is that it is the key to being able to calculate many of the macroscopic properties of a system, such as its pressure, temperature, heat capacity, surface tension, dielectric constant, viscosity, electrical conductivity, entropy, and total energy.

The Maxwell-Boltzmann distribution makes a rather simple physical assertion: It states that if a fixed quantity of energy is distributed among a collection of particles, it will be rare to find individual particles with a large fraction of the total energy. Thus, the larger the energy that a particle would need to have in order to be at a particular position in space and to have a particular velocity, the less likely it is that particles will be found to have actually that position and velocity. Mathematically, the Maxwell-Boltzmann distribution is a decaying exponential; it predicts that the larger the energy, the smaller the probability of finding a particle with that energy.

To gain insight into the Maxwell-Boltzmann distribution, it is helpful to look at two limiting cases. The first case describes a gas or fluid in the absence of any external force fields, such as electric and magnetic fields, and gravity. (Although in reality gravity is always present close to the earth, for small samples it has only a uniform effect on the different particle positions.) In the absence of the nonuniform effects caused by force fields, there is an equal probability that particles will be found at any position in a container, and thus the positional dependence of the Maxwell-Boltzmann distribution disappears. The distribution can be simplified further so that it depends only on particle speed rather than on velocity. The "reduced" speed formula is often called Maxwell's formula.

Maxwell's formula gives the probabilities for different molecular speeds in a gas or fluid. Unlike the full Maxwell-Boltzmann distribution, it is not a decaying exponential. Instead, it predicts that for a sample of gas (or fluid) in a container at a fixed temperature, there will be a specific nonzero particle speed that is most probable. That is, the molecules will be moving with a range of speeds, but at any moment there will be more molecules traveling at the most probable speed than at any other speed.

The figure gives a plot of probability versus speed for samples of helium gas at two temperatures: 0 degrees Celsius and 1,000 degrees Celsius. Schematically, the curves are bell-shaped. Their maxima mark the most probable speed at each temperature. At 0 degrees Celsius, the most probable speed is 1,065 meters per second, while at 1,000 degrees Celsius, the most probable speed increases to 2,300 meters per second. The higher temperature curve is "stretched out" to higher speeds all along the way. Speeds with a low probability at 0 degrees Celsius (for example, 2,200 meters per second) will have a much higher probability at 1,000 degrees Celsius. The temperature dependence of Maxwell's formula reflects the fact that at a higher temperature, more energy is available to excite the molecules to faster speeds. Maxwell's formula explains why the rates of simple chemical reactions increase when the temperature is raised: The faster-moving molecules will collide more violently with one another, breaking apart more readily to recombine into a new product.

Although particles move faster at higher temperatures, Maxwell's formula still places limits on their possible speeds. There is zero probability that particles will be found "standing still" with zero speed, and close to zero probability that they will be found having speeds much greater than the most probable speed for the given temperature. Despite their small number, however, the particles with the highest speeds can be important in acting as the initiators of chemical reactions.

The Maxwell-Boltzmann distribution reduces to Maxwell's formula in the limit where a particle's energy is independent of its position. The distribution also simplifies in the opposite limit where only the positional dependence of the energy needs to be considered. An example is the case of a vertical column of oxygen gas acted upon by gravity. The Maxwell-Boltzmann distribution (called the barometric formula for this case) gives the probabilities of finding the oxygen molecules at different heights in the column. Because oxygen molecules near the top of the column have a higher potential energy (because of the gravitational force acting on them) than those lower down, they are less probable. Thus, as one climbs up a mountain, the air gets "thinner." The barometric formula predicts that air pressure and concentration both decrease exponentially with height.

In general, a particle's energy can depend on more than merely its speed and position.

For example, water molecules not only travel in straight lines characterized by a velocity but also compress and expand in coordinated movements called vibrations. In addition, they can rotate at various speeds about their center of mass. Finally, the electrons within an atom or molecule can arrange themselves in different configurations about the atomic nuclei, and these configurations all have different energies. To apply the Maxwell-Boltzmann distribution to these more complicated situations, it is expressed in terms of the possible total energies accessible to the molecules of a sample. It can then be used to calculate the fraction of molecules engaged in any particular energy state at a given time.

Applications

Any macroscopic object consists of an unimaginably large number of constituent atoms and molecules. For example, a child's balloon filled with helium gas at room temperature contains approximately 1 x 10²³ (100 billion trillion) helium atoms. One usually thinks of the goal of science as being able to describe a system in the most complete and detailed way possible; however, for a system with so many "parts," a description that is too detailed can be distracting. For example, a tabulation of the position and velocity values for all the helium atoms in the balloon would require a stack of paper 10 trillion kilometers high. Furthermore, all of those values would be accurate only for an instant of time, since the atoms are constantly changing their positions and velocities as they move about and collide with one another and with the walls of their container. Although it is impossible for the human mind to make sense of such a large amount of information, the vastness of the information is itself not a fundamental limitation, since one could (in principle) use a computer to generate and store the numbers. The reason that this description simply is not useful is that the atoms in the balloon are not directly observable; therefore, there are no experimental data to compare to all of those calculated numbers.

Instead of aiming for a microscopic description of the helium atoms in the balloon, it is more productive to try to sum up the macroscopic behavior of the helium gas. The description will appear to leave out the behavior of the individual gas particles, but in fact one cannot ignore them, since the overall behavior of the gas must result from the behavior of its constituent particles. The connection between the behavior of the individual particles and the behavior of the gas as a whole is that the overall properties can be expressed in terms of averages of the behavior of the individual particles. The field of physics that makes this connection is called statistical mechanics.

Gas pressure is a macroscopic property whose microscopic origin is particularly clear.

A gas exerts pressure on the walls of its container resulting from the collisions of the gas particles with the walls. Slow particles exert a small force, and fast particles exert a large force.

Although there may be microscopic fluctuations in the overall force exerted, they will be negligibly small if the sample contains a very large number of particles. Over time, the gas pressure will be equal to the average force per unit wall area exerted by the particles. The average force, in turn, is simply related to the average particle speed. Indeed, it can be shown that the pressure of a gas is proportional to the square of the average speed of its molecules. The key to calculating the macroscopic properties of a system, therefore, is being able to calculate average properties of its constituent particles. How are average properties calculated?

If, for example, one wanted to calculate the average age of the members of a bird-watching group, one would need to know the fraction of the group falling into different age ranges. If one-sixth of the members are between the ages of nineteen and twenty-one, one-third are between twenty-nine and thirty-one, and one-half are between thirty-nine and forty-one, then the average age is approximately (1/6 x 20) + (1/3 x 30) + (1/2 x 40) = 33. The 1/6, 1/3, and 1/2 factors are called distribution numbers because they describe how a certain characteristic (age) is distributed over the members of the group.

The distribution numbers that characterize a group can be obtained by experiment (by polling the members of the bird-watching club) or by calculation, if a mathematical formula is known for the characteristic of interest. The Maxwell-Boltzmann distribution is the formula used to calculate the distribution numbers for position and velocity, or for energy, of the particles in a medium. The distribution numbers are in the form of expressions that depend on the sample temperature, the particle mass, and any external force fields. In a manner analogous to the calculation of the average age of the bird-watching group, the Maxwell-Boltzmann distribution expressions can be used to calculate expressions for the average speed and energy of the molecules in a medium.

The average speed of low-pressure gas molecules, for example, is found to be proportional to the square root of the ratio of the temperature to the particle mass, while their average energy is directly proportional to the temperature but independent of the particle mass.

Further, theoretical quantities, such as the average rate of particle-particle collisions and the average distance traveled by the molecules in between collisions, can be generated from these basic averages.

The theoretical expressions derived from the Maxwell-Boltzmann distribution are then used to understand the observed properties of matter. Many of these properties depend on the movement of particles. These so-called transport phenomena include Graham's law, which states that the rate of escape of a gas through a pin hole is inversely proportional to the square root of the gas density; the processes of adsorption (sticking) and desorption (popping off) of gases onto and off of surfaces; the rates of evaporation and condensation of liquids; the rates of flow and mixing of gases and fluids; the rate of temperature equalization in a medium initially subject to an uneven temperature profile; and the characteristics of the viscosity (resistance to flow) of a medium. An early triumph of Maxwell's speed formula was its explanation of the seemingly nonintuitive fact that the viscosity of a gas increases as the temperature is raised but is independent of the gas density.

Other important applications of the Maxwell-Boltzmann distribution include predicting the arrangement of ions in electrically conducting solutions and the orientation of molecules in electric and magnetic fields. Finally, the Maxwell-Boltzmann distribution is indispensable for understanding the interaction between atoms or molecules and electromagnetic radiation. When atoms and molecules are exposed to radiation, they can be stimulated to absorb or emit energy in patterns that are characteristic of their structure and movements. These patterns, called spectra, provide a glimpse into the microscopic world--if they can be interpreted correctly. A complicating factor is that the molecules absorb and emit from all the many energy states available to them, and in order to analyze the spectra in terms of the individual states, one needs to know the fraction of molecules in each state. Not only can the Maxwell-Boltzmann distribution give the exact populations of the states but also the distribution's use can generate helpful "rules of thumb," such as the knowledge that at room temperature, molecules overwhelmingly prefer only their lowest energy electron arrangement and will be found rotating at a range of different frequencies.

Context

James Clerk Maxwell published his speed formula in 1860. He was the first to describe a physical property of matter (the distribution of velocities in a gas) using a statistical treatment.

In 1868, Ludwig Boltzmann extended the theory to cases where a system is acted upon by a force, obtaining the full Maxwell-Boltzmann distribution. The statistical methods used by Maxwell and Boltzmann were ultimately and elegantly refined into the science of statistical mechanics by Josiah Willard Gibbs in 1902. Gibbs's treatment defines the equilibrium state of a system to be the one that has the greatest probability of occurring. His derivation first constructs an expression for the probability of the occurrence of any state and then looks for the particular state that maximizes the probability. The only physical assumptions that enter the derivation are that the total energy and number of particles are fixed and that all possibilities for the distribution of energy are equally likely. (Rotations of a certain energy are considered to be as probable as vibrations of that energy, for example.) Because Gibbs's argument is independent of a specific physical picture, it can be applied to systems ranging from individual molecules to gases, liquids, and solids.

During the latter half of the nineteenth century, there was spirited debate over whether atoms and molecules actually exist or are simply ideas helpful in interpreting experimental observations. An issue that was not in doubt, however, was the applicability of Newtonian (classical) mechanics to the behavior of the particles, real or imagined. An important tenet of classical mechanics is that energy can take on a continuous range of values. It is a simple matter to apply this classical idea to the Maxwell-Boltzmann distribution to study the energetics of complicated molecules that simultaneously rotate, vibrate, and move in straight lines. The theory predicts that the three energy modes will be equally excited; experimental observation, however, is more consistent with a picture that shows straight-line motion energized first, followed by rotations, and then vibrations as the temperature of a sample is raised. Only at very high temperatures are all three modes simultaneously and completely excited. The breakdown of the equipartition principle of energy uptake at low temperatures was an important clue that the classical treatment of the internal motions of molecules needed to be modified. It thus contributed to the overall impetus toward the creation of a more general physics--quantum mechanics--in the beginning of the twentieth century.

The Maxwell-Boltzmann distribution is a general theoretical equation that can be tailored to apply to different physical situations. There is no way to "prove" that it will hold for all cases; instead, it must be tested on a case-by-case basis. Maxwell's formula, in contrast, has been experimentally verified and may be regarded as fact rather than theory. Starting in 1955, the molecules of a wide variety of gases at a range of temperatures have been directly sampled, and their speeds fall convincingly onto curves, such as those plotted in the figure. The experimental confirmation of Maxwell's formula (and the barometric formula), as well as the indirect verification of the complete distribution function by the agreement of experiment with various derived quantities (effusion rates, viscosity, and the like) convince one that the Maxwell-Boltzmann distribution is a correct and fundamental law of physics and that the statistical model on which it is based is valid.

Principal terms

DISTRIBUTION: a mathematical formula for calculating the probability that a member of a group possesses a certain characteristic

EXPONENTIALLY DECAYING FUNCTION: a relationship between an input and an output such that the larger the input, the smaller the output

GAS: a state of matter which, because of its low density and easy compressibility, completely fills a container of any size and shape; examples include helium, oxygen, and air

MACROSCOPIC: concerning the overall properties of bulk matter

MICROSCOPIC: concerning the structure, properties, and behavior of the constituents of a phenomenon or object's atoms and molecules

STATISTICAL THEORY: a description of reality based on the laws of chance; the overall properties of a system are assumed to be the same as those predicted from the average behavior of its constituents

VELOCITY: a quantity that specifies both the speed and direction of a moving object; a typical value is 10 meters per second in the northeast direction

Bibliography

Atkins, P. W. THE SECOND LAW. New York: Scientific American Books, 1984. The second law of thermodynamics states that spontaneous change always occurs in the direction of increasing randomness. Atkins discusses the second law in a way that is "accessible to any persistent reader, however scientifically unprepared." Boltzmann's contributions to the interpretation of the law in terms of molecular behavior are mentioned.

Gillispie, Charles Coulston, ed. DICTIONARY OF SCIENTIFIC BIOGRAPHY. New York: Charles Scribner's Sons, 1972. The entries on Boltzmann, Maxwell, and Gibbs show the interrelationship of their work on the theory of gases and statistical mechanics. These multipage articles give much detail on both the interactions between these important scientists and on the physics involved in their work.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 2d ed. New York: John Wiley & Sons, 1981. A general first-year college text. Chapter 21 presents an introduction to the simplest statistical theory of gases, kinetic theory.

Haneweaver, Jefferson. THE WORLD OF PHYSICS: A SMALL LIBRARY OF THE LITERATURE OF PHYSICS FROM ANTIQUITY TO THE PRESENT. Vol. 1. New York: Simon & Schuster, 1987. Each section in this physics anthology covers a fundamental concept. The sections contain essays written by physicists with accompanying commentaries. Pages 764-766 describe the Maxwell-Boltzmann distribution function in a particularly clear way and show its importance in the historic debate over the existence of atoms.

Present, R. D. KINETIC THEORY OF GASES. New York: McGraw-Hill, 1958. Although this text is meant for advanced undergraduate science majors, it contains an unusual amount of explanatory prose and historical background. Chapters 1, 2; and 5 are especially relevant.

Maxwell's speed formula: helium gas

Diffusion in Gases and Liquids

The Behavior of Gases

Thermal Properties of Matter

Laws of Thermodynamics