Deriving the Maxwell Distribution Robert C. Dunbar Case Western Reserve University, Cleveland, OH 44106
Many contemporary physical chemistry texts offer an appealing derivation of the Maxwell distrihution of molecular velocities in a gas. Based on the assumptions (a) that the velocity distribution is isotropic, and (b) that the distributions of velocities f(u), f(u) and f(w) in the x, y, and 2 directions are independent, the equality where c is the molecular speed, can he written immediately, and i t follows through elementary manipulations that the distribution must have the form
Maxwell1 presented this derivation in 1860 in his first published work on the kinetic theory of gases (2). This might seem like a sound authority for a textbook derivation, but in fact Maxwell regarded this as an inadequate derivation: he never used it again and spent much effort in considering more secure proofs. The problem lies with assumption (h), that the one-dimensional distributions are independent of each other. The tvnical text takes this as obvious. often without ex~lanation.~ But is it actually obuious? Why should not amoleiule which is moving"exce~tionallv fast in the x direction be more (or less) . likely to be moving u&sually fast in the y direction? This has never seemed obvious to me. and it has not seemed obvious to many who have thought mire deeply about this subject (3, 4). Worse yet, it is not even true. For if we take the exact expression for the energy of a molecule including relativistic effects, the velocity distribution takes the form ( 5 ) " A
I
-mc2 Idududw (3) kTJ1-(u2 + u2 + w2)/c2, (C here is the speed of light and A' is a normalization constant) which obviously does not factor into independent u, u, and w distributions. Therefore, including relativistic effects, ass u m ~ t i o n(h) is in fact untrue (althouph such effects are insignificant except perhaps for electron gases in stellar interiors). A molecule moving. rapidly . . in the x direction moves slower on the average in the y direction than one moving slowly in the x direction. The logically unsatisfactory nature of Maxwell's first derl Maxwell was interested in this subject through much of his career, oublishina several maior works over the vears 1860 to 1879. This in&st .-.in ihe kinetic thkarv . of oases mav well have orioinated with his essay on the r ngs of Satdm for the aims Prize in 1 k 7 , wn'ch for% kought h m to scienlif c prommence. ano wh cn gave elaborate consideration to the rings as composed of a large number of freely moving, colliding pieces of material ( 1). No one has improved on Maxwell's own words, ". . .the existence of the velocity xdoes not in any way affect that of the velocities yand r, since these are all at right angles to each other and independent, . . ." (Ref.2, p. 380). Neglecting the complication of Fermi-Dirac or Bose-Einstein statistics. These arguments can be extended to non-spherical molecules, although not without some complexity. (See Ref. (6). ~
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ivation is interesting, hut the important point, from a teaching point of view, is that it gives no u a l ~ dinsight into where the Maxwell distrihution comes from and why it must he true. Such a great milestone in the development of the molecular picture of the world deserves a better presentation. Efforts to prove the Maxwell distrihution played an imnortant historical role in the develonment of statisticalmechanical concepts. Actually at least three points need proof; first. that e m . 12) . . is a stable. time-invariant state of the eas: secohd, that i t is the only (or the most likely) such state; and third, that a gas having some other distribution approaches eqn. (2) with time. Probably the most satisfactory derivation for first-time understanding of this subject is through quantum statistics (6-8). Derivation of the Boltzmann distribution, eqn. (4) A
(where p(EJ is the prohability of being in the ith energy level, which has degeneracy g, and energye,) as the most probable distribution, from the statistical mechanics of quantized energy levels3, is standard fare in physical chemistry texts, and rests on verv fundamental assumvtions. Eauation (2) then follows at once. This makes it clea; that the independence of the one-dimensional velocitv distributions is a consequence of the fact that the (non-relativistic) kinetic energy is-a sum of independent u, u, and w terms, corresponding to independent x , y, and t quantum numbers in the Hamiltonian. Historically and logically the derivation through quantum statistics grows naturally out of the corresponding development through classical statistical mechanics (3,6, 7,9-11). Both the cl&sical and the quantum derivations show that eqn. (2) is the most probable distrihution and that distributions deviating significantlv from it are vastlv less orohable. The derivations through statistical mechanics give no insight into whether and how the initiallv non-eauilibrium eas an~roaches do, however, show t h i i eqn. (2) a Maxwell distribution. and ean. 14) do not denend on the nature of the intermolecular collisions, and are valid for mixtures, for non-spherical molecules, and for those with internal degrees of freedom. The earliest satisfactory way of thinking about these questions began with Maxwell's paper of 1867, in which he considered the detailed nature of collisions hetween spherical molecules (12). He showed explicitlv that a steady state of the
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assumption thatthe rate at which collisi&s carry molecules from a particular point v in velocity space to another point V is just balanced by the rate of collisional transfer from V to v; and he showed that the steady state distribution that this imnlies is in fact the Maxwell distribution. eon. 12) . , 13.6.7). ,, , . ~ o k z m a n nmade the remaining key step in 18j2 (131, showing that any distrihution other than eqn. (2) does in fact move toward eqn. (2). Boltzmann formulated hi famous H-theorem (3,6) which led to the statistical interpretation of entropy, for the purpose of showing this. Putting these arguments into the language of distrihution functions. Maxwell's theorem of detail& balancing says that if v and v' are two points in velocity space, and V and V' are two corresponding points which can result from the collision of molecules in v and v', then
f(v) fr(v') = f(V) f(V')
Literature Cited
H-theorem says that the function H
=
Sf(") log f(v) dv
where the integral is over all velocities, always tends to decrease toward a minimum, and, that when H is a minimum, eqn. ( 5 ) is true, so that the gas evolves to the Maxwell distrihution from anv startine woint. This last apGroach to deriving the Maxwell distribution through detailed collision theory was improved, refined, and extended by Boltzmann, Lorentz, and others, but most important, it led on into the great development in the late 19th and early 20th centuries of the behavior of non-uniform gases and transport properties. ~~~
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1940,App. I. ~ ~E ~~ . ,S~ h, yRs ~. Uznd , ser6,slo ( I Q ~ ~ ) ; TR.Oc .~, ~~h .i i . ~ o g28,583 . (1914);~uttner,AmriPhy~. 34,856 (1911).~emarkuhiyonough,as shown by Juttner. the ideal gas lawpu = nRT is obeyed by the relativistiegas, even though the M ~ X W distdbutim ~ I is not. (6) T ~ IR. ~c., ~he~~rincipies , of statistical ~ ~ ~ h ~ oxford ~ i c suniversity : P ~ S , London, 1938. (7) Kennard. E. H.,'Kinetic Theory of G a s d McGraw~Hiil,New York, 1928. (8) ~ ~ * N.,d‘ ' s ~~a t i~s t i d~echanics: ~ , M C G I ~ Vill, New York. 1962. (91 Jeans. J. H., "Dynarnical Theory of Gases," Cambridge University Press, 1904. 2, p, 713, (,,, ,axwe ll,J,c,, ..scientificPaperr: (11) R ~ I ~ Z ~ ~ ~ k.~~ , k L~W . d~, ~. S, B~~ .~I . V ~ I ~I80ctobe1 I .. 1868. (12) Maxwell, J. C., Phil. Trmm Roy. Sue, 157.48 (18671 ((iScientifi~Papers," Val. 2, p. 26). (13) ~ o l t z m s n nL., . w i r n ~ e i . 6 6 . 2 7 5(18721. (5)
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Volume 59
Number 1 January 1982
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