Consider the velocities u, of molecules contained in a box. The number of mole- cules moving in the positive x direction must be equal to the number of molecules moving in the negative x direction. This conclusion is easily seen by examining the consequences of the contrary assumption. If the number of molecules moving in each direction were not the same, then the pressure on one side of the box would be greater than on the other. Aside from violating experimental evidence that the pressure is the same wherever it is measured in a closed system, our common obser- vation is that the box does not spontaneously move in either the positive or nega- tive x direction, as would be likely if the pressures were substantially different. We conclude that the distribution function for the velocity in the x direction, or more generally in any arbitrary direction, must be symmetric; i.e., F(v,) = F(-v,). Func- tions possessing the property that Ax) = A-x) are called even functions, while those having the property that f(x) = -f(-x) are called odd functions. We can ensure that F(v,) be an even function by requiring that the distribution function depend on the square of the velocity: F(v,) = f(v,2). As shown in Section 1.5.3, this condition is also in accord with the Boltzmann distribution law.e
eOther even functions, for example, F =flu:) would be mathematically acceptable, but would not sat- isfy the requirement of Section 1.5.3.
Section 1.5 The Maxwell Distribution of Speeds 1.5.2 The Velocity Distributions Are Independent and Uncorrelated We now consider the relationship between the distribution of x-axis velocities and y- or z-axis velocities. In short, there should be no relationship. The three compo- nents of the velocity are independent of one another since the velocities are uncor- related. An analogy might be helpful. Consider the probability of tossing three hon- est coins and getting "heads" on each. Because the tosses ti are independent, uncorrelated events, the joint probability for a throw of three heads, P(tl = heads, t, = heads, t3 = heads), is simply equal to the product of the probabilities for the three individual events, P(tl = heads) X P(t2 = heads) X P(t3 = heads) =
$ X $ X $ . In a similar way, because the x-, y-, and z-axis velocities are independent and uncorrelated, we can write that
F(ux7uy9uz) = F(ux)F(vy)F(uz). (1.21) We can now use the conclusion of the previous section. We can write, for exam- ple, that F(v,) = f(u2) and similarly for the other directions. Consequently,
What functional form has the property that f(a + b + c) = f(a)f(b)f(c)? A lit- tle thought leads to the exponential form, since exp(a + b + c) = eaebec. It can be
shown, in fact, that the exponential is the only form having this property (see Appendix 1.1), so that we can write
F(vx) = f(u;) = K exp(?~u:), (1.23) where K and K are constants to be determined. Note that although K can appear mathematically with either a plus or a minus sign, we must require the minus sign on physical grounds because we know from common experience that the probabil- ity of very high velocities should be small.
The constant K can be determined from normalization since, using equation 1.17, the total probability that u, lies somewhere in the range from -m to + w
should be unity:
00
J-/(ux)dux = 1. (1.24)
Substitution of equation 1.23 into equation 1.24 leads to the equation
where the integral was evaluated using Table 1.1. The solution is then K = (~l.rr)l".
1.5.3 <v2> Should Agree with the Ideal Gas Law
The constant K is determined by requiring <u2> to be equal to 3kTlm, as in equa- tion 1.6. From equation 1.16 we find
The integral is a standard one listed in Table 1.1, and using its value we find that
As a consequence, the average of the square of the total speed, <v2> = <v:>
+ <v;> + <u:> = 3<v:>, is simply
From equation 1.6 we have that <v2> = 3kTlm for agreement with the ideal gas law, so that 3kTlm = 3 / ( 2 ~ ) , or K = ml(2kT). The complete one-dimensional dis- tribution function is thus
d u , .
This equation is known as the one-dimensional Maxwell-Boltzmann distribution for molecular velocities. Plots of F(v,) are shown in Figure 1.2.
Note that equation 1.29 is consistent with the Boltzmann distribution law, which states that the probability of finding a system with energy E is proportional to exp(--~lkT). Since E, = kmu: is equal to the translational energy of the mole- cule in the x direction, the probability of finding a molecule with an energy E,
should be proportional to exp(-~,lkT), as it is in equation 1.29. In Section 1.5.1 we ensured F(v,) to be even by choosing it to depend on the square of the velocity, F(v,) =flu:). Had we chosen some other even function, say F(u,) =flu:), the final expression for the one-dimensional distribution would not have agreed with the Boltzmann distribution law.
Equation 1.29 provides the distribution of velocities in one dimension. In three dimensions, because F(u,,u,,v,) = F(v,)F(v,)F(u,), and because v2 = v: + u; + v;,
we find that the probability that the velocity will have components v, between v, and vx + dv,, v, between v, and v, + dv,, and u, between u, and v, + dv, is given by
F(vx, u,, vZ) dv, dv, du, = F(v,) F(vy)F(vZ) dv, dv, dv,
(1.30)
= ( 2rrkT ~ y e x ~2kT dv,dv,dv,. ( - ~ )
Section 1.5 The Maxwell Distribution of Speeds
II Figure 1.2
One-dimensional velocity distribution for a mass of 28 amu and two temperatures.