increase and decrease of the nnmher of particles in the volume element d3u d:'r due to collisions in the interval dt. The loss term can he written out readily in terms of the collision cross-section c. If v is the velocity of a molecule in the volume element, and w is the velocity of any other molecule, their relative speed is lv - wl = g and the rate of collisions between them is gof(v)f(wld"u d2w The number of collisions in d+ during dt which involve vmolecules and any value of w is
( )
ERII
dtdud"
= dtd"ud2rf(v)Jf(w)gcd3w
The integral is simply the collision rate of one v-molecule averaged over the velocity distrihution of w-molecules, so we can write
af
- d t d "d+ = ddtd3ud3rr(v)n(go)
(z),,,
-
dtd:'ud:'r P(v)n(gn)
Using these expressions to find dN,,,a, substituting the result in eqn. (71, and cancelling differentials leads us to ~
~
which is the approximate form of the Boltzmann transport equation we were seeking. The quantity n(gn), which appears on the right side of eqn. (lo), has the units of reciprocal time. Indeed, the right side of eqn. (10) is most frequently written as (P- f)/r, where r = l l n (go) is the average time between collisions of a molecule with velocity v, and is known as the relaxation time of the system. The origin of the term becomes more obvious if we consider a gas of uniform density which is not subject to external forces, but which has a nonequilibrium velocity distribution. Then eqn. (10) becomes
which can he integrated to give where f, is the distrihution function a t zero time. We see that iis the time constant for the exponential relaxation of the system towards equilihrium. Accordingly, eqn. (10) is known as the relaxation approximation to the Boltzmann transport equation. The Transport Coefficients
We shall now use eqn. (10) to determine an approximate expression or the diffusion coefficient. For simplicity, we assume that steady-state conditions have been reached, so that aflat = 0, there are no external forces, and that a concentratioh gradient of an isotopically labeled atom exists in the z direction. The Boltzmann equation becomes 24 / Journal of Chemical Education
f = P(r) + I'
The first term P ( r )represents a "local" equilihrium. That is, @(r)is the equilibrium Maxwell-Boltzmann function evaluated for the conditions which hold a t each differential volume element in the gas. In the case of diffusion, this merely means that the f" of eqn. (1) is used, hut with the concentration n taken as a function of position. The term f1 represents the small departures from local equilihrium. We substitute for the distribution function in eqn. (11) and ignore the very small quantity afl/&. The result is uz
afo(z)= -
a2
i
(9)
In principle, the collision rate n ( g 0 ) is still a function of u , but in our applications we shall ignore this. The formulation of the rate of increase of molecules in d3u is more difficult, since collisions involving all types of initial conditions of both collision partners must he considered in order to find which lead to scattering into d%. Use of the conservation laws allows the initial conditions for these inverse collisions to he found.4 Rather than pursue this exact but somewhat lengthy analysis, we shall adopt a heuristic approach. We note that the rate of increase of particles in d:'u must involve an averare collision rate Der . . .article n ( ~ 0 )and that ar e q u i i i l ~ r ~ rile ~m OI :sin vquals I ~ mt;. C \.itemi. not i . ~Irceln r e~tuilil~rium. rile rare d gain must he very close to the equilihri;m rate ofloss. ~ h e r e k r e , we shall approximate the numher gained in d"u d"r during dt under all conditions by eqn. (9)evaluated a t equilihrium; that is, with f replaced by f . Thus
($)+ "llll dtd:'ud:'r
-We shall assume that the distribution function f can he written as the sum of two terms
where n is the concentration of the diffusing species. We see that the perturbation to the local equilibrium distrihution function is proportional to the relaxation time and to the concentration gradient. With the distrihution function availahle, we can calculate the diffusive flux
+
J , = J'u,fd:'u = Jul(F1 fl)d%
The integral which involves f vanishes, since F is an even function of velocity, and u, is an odd function. Thus the flux simplifies to
The right side has the form of the usual phenomenological diffusion equation, with D = ? [JP~,~d:'ul
as the diffusion coefficient. The integral is just equal to the local concentration of diffusing species n times the mean square velocity component. Thus D = ~ k T l m= kTl(nt(grr)rnl (13) We note that the relaxation time involves the total gas concentration nt. It is of interest to compare eqn. (13) to the conventional expression from mean free path theory D='G/ 3
where 1 is the mean free path. Since 1 = TG, we have
Thus the two approaches give very similar results, with the relaxation expression larger by a factor of 3a/8 = 1.18. We can now treat the electrical conductivity which results from a dilute solution of univalent positive ions in an excess of background gas. The system is in a steady state, the charge concentration is uniform, hut each charge carrier is suhject to an external force ec,, where & is the electric field. The Boltzmann equation becomes e t , af - fO - f m du, r
+
Again we assume that f can he written as f" fl, where P is the equilihrium distrihution function, and f1 is small enough so that its velocity derivative is negligible. Then we find re(, a f n f' = -m a", The current density is
