[CHM 331] Steiner Problem 85 in Chapter 4

[Andrew J. Pounds]

This is a classic example of something that initially looks hard, looks 
doable if you understand the concept behind the problem, and then is 
almost trivial if you have  a program like Mathematica (or a calculator 
) that will do symbolic calculus.

So you are looking at the Boltzmann distribution function which you 
should remember from CHM 332.

$f(v)= 4 \pi \left(\frac{m}{2 \pi kT}\right)^{3/2} v^2 e^{-mv^2/2kT}$

The most probable speed occurs at the top of the Boltzmann distribution 
curve (its maximum), so we want to take the derivative of the function 
above with repect to V, set it equal to zero, and determine the value of v.

When I do this in Mathematica I get three answers.

$m=0$, $v=0$, and $m=\frac{2kT}{v^2}$.

only one of these is physically correct -- and it has to be the third 
one.  Rearrange it for $v$

$v = \sqrt{\frac{2kT}{m}}$

and that is the answer to the problem.

I have attached my Mathematica notebook so you can see what I did.

-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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