The Median Speed and the Error Function John T. Pyle Columbus College, Columbus, GA 31993 This article reviews methods of the calculus which allow the derivation of several molecular speeds of statistical sianificance and develops a method based on a statistical intkgral for determining the median speed. Most physical chemistry texts develop the three-dimensional Maxwell-Boltzmann speed distribution for identical, noninteracting, nonlocalized molecules. This distribution is usually expressed as the fraction of molecules, W I N , with speeds in the range c to c dc.
+
Equation (3) can be evaluated by first integrating by parts. If s = u and se-.' = du, then u = -('/~)e-"' and eqn. (3) becomes
Now
dh'/N = f(c)dc = 4n(m/2rkT)3" c2exp(-mcZ/2kT)dc
where m is the molecular mass, k is the Boltzmann constant, and T is in kelvins. Distribution curves are generated by plotting this fraction as a function of speed for values of T and m. The area under the curve is unity since the sum of all the fractions is equal to 1. Several speeds can be derived from this distribution using methods of the calculus. The most probable speed, c,, can be found by taking the derivative of the function and setting i t equal to zero. This, of course, is the peak in the distribution curve. As a result c, is expressed as e , = (2kTlm)l"
So eqn. (4) becomes
The integral in eqn. (6) and its coefficient constitute what is called the error function, erf(s). Thus,
The average speed, F,is found by the usual method of dealing with a continuous function' and F(s) = 1+ 2se-a'/& The denominator is normalized and equal to unity. Thus,
and by substitution and evaluation of a standard integral F = (8kTlrm)'/2. The root mean square, c,, is found by evaluating (J; c2f(c)dc)'/2. Thus, c,, = (3kTlm)'W. Determination of the median speed, ern&, affords an opportunity to analyze the area under a curve for a desired quantity as well as to explore the use of a statistical integral. In evaluating the median speed, we wish to find the speed above which (and below which) half the molecules travel. We want particularly to find the speed at which the distribution F(c) = %
Since s = dc,,, of s
dc = c,,ds,
and c2 = s2c& Then in terms (3)
(7)
The name "error fundion" is chosen because of its frequent use in probahility calculations involving the standard normal probability distribution. There are numerous sources for error function value tahles. These include books on statistics, mathematical handbooks for physical chemistry, and some physical chemistry texts. A computer program based on the asymptotic expansion of the Gauss error function2 can be used to evaluate erf(s). With values of erf(s) available, one can set eqn. (7) equal to %and solve for s by an iterative process. Thus, for s = 1.0877. Then c = c,d
The problem is more tradable if we find c in terms of cmp. Letting s = clc, and T = mc&jZk in the function of eqn. (I), then
- erf(s)
= 1.0877 c,,
= 1.0877(2kT/m)'" = 1.538(kTlm)1'Z
Other solutions of eqn. (6) allow us to find other fractional areas of speed distributions, or t o find c,,d in terms of other standard speeds. In addition we can find what fraction of the molecules have speeds in excess of some multiple of cmedor -, erf(s) 1. even some other c. Note that a s s
-
-
'
Rosenbaum, Eugene J., "Physical Chemistly:' Meredith Corpcration, New York. 1970, p. 132. Arfken, George, "Mathematical Methods for Physicists." 2nd ed., Academic Press. New York, 1970, p. 295.
Volume 61 Number 11 November 1984
979
