CHM 5414 Lecture Notes
12 September 1996

While the individual molecules of water take chaotic paths through this winter waterfall, this time-lapse photo (Kodak PhotoCD #27098) shows what's important. That is where the average water flow goes over the falls. Just so, we would be mad to attempt to chart the classical (or quantum mechanical) progress of moles of reactants becoming products. We should take advantage of statistics to arrive at meaningful averages.

This time-average photo, with its foggy probability distribution, shows the likelihood of traversal of the falls. Statistical Mechanics substitutes for this picture an instantaneous superposition of the falls in a myriad of identical universes called an ensemble. It is an article of faith that the time-average and ensemble-average pictures are the same!

We're teaching this course in a classroom which is 30'x15'x8' or 97,200 L (ignoring the volume of your bodies). At 293 K, that corresponds to 4043 moles of (ideal) air or 2.4x10²⁷ molecules.

Those molecules (if ideal) are bouncing back and forth at random between the walls at roughly the speed of sound (760 mph). We feel no such wind since they're not all going in the same direction. But what's to prevent that if they're truly bouncing at random?

In fact, what's to prevent a random build-up of more than half the molecules in less than half of the room, resulting in a pressure fluctuation? Nothing. In fact, pressure fluctuations are to be expected from such random motion. But they're never manifest. Why not?

Let's take an egregious case: all the molecules "choosing" at once to be in the left half of the room. Sinister, no? Half of you would asphyxiate while the other half imploded. At least briefly, at any rate since we'd expect the compressed gas to expand in a thunderclap which would take out the wall!

No one's running for the door because you've all done the mental math.

If the probability of one molecule being on the left is (1/2) then the probability of two on the left is (1/2)(1/2). Probability of independent events (ideal gas molecules are independent) multiply.

That means that the likelihood of all 4,043 moles being on the left is (1/2) raised to the power 2.4x10²⁷. A little logarithmetic shows that to be 1 in 10^(7.2x10²⁶)...that's roughly 1 followed by 1200 times Avogadro's Number of zeroes!! Such numbers are inconceivable...except by mathematicians who call 10¹⁰⁰ a googol and 10 ^googol a googolplex. So we're something in between.

So it's NOT going to happen. Not in the lifetime of this universe or in a googol of universes!

While such a gross fluctuation ISN'T going to occur, what magnitude of fluctuation will?

It's clear that the probability calculation above was just like flipping (2.4x10²⁷) coins. Heads you're left; tails you're right. In preparation for this class, I flipped over 1,000,000 coins (on the computer) to simulate this problem. And here are the results:

10 trials with  10  coins.
 7             5             5             8             4             
 3             3             5             8             2            
Average was  5  ±  2               or  0.5 ± 0.2

10 trials with  100  coins.
 42            45            50            47            53            
 48            49            41            55            48           
Average was  47.8  ±  4.166536     or  0.478 ± 0.0417

10 trials with  1000  coins.
 499           502           519           483           499           
 487           504           498           473           523          
Average was  498.7  ±  14.44281    or  0.4987 ± 0.0144

10 trials with  10000  coins.
 4926          4925          5000          5012          5050          
 5019          5016          5094          5073          5037         
Average was  5015.2  ±  52.50715   or  0.5015 ±- 0.0053

10 trials with  100000  coins.
 49827         49961         50091         50021         50001         
 49851         49874         50064         49842         50054        
Average was  49958.6  ±  95.42494  or  0.4996 ± 0.00095

Two things stand out from these trials. The first is that the deviations from 50:50 grow (2 to 95 on average). The second is that the relative deviations diminish (from 0.2 to 0.00095 on average).

In other words, 50:50 is a better bet the more coins you toss. The deviation grows more slowly than the average. In fact, in the limit of large numbers, the deviation grows on the order of the square root of the number.

# of Heads  (# of Heads)1/2  Deviation Above

      5            2.2              2
     48            7.0              4.1
    499           22.4             14.4
   5015           70.2             52.5
  49959          223.6             95.4

So if we had the expected number of molecules in the left half of the room (exactly half the total or 1.2x10²⁷), we'd expect a deviation of roughly square root of that or 3.5x10¹³ extra molecules on one side or the other. Nonchemists might be concerned about "so many," but we realize that the trivial overpressure implied by that is only (3.5x10¹³) / (1.2x10²⁷) = 2.9x10 ^-14 atm! Unmeasurable.

And so it is with virtually any variation from thermodynamic equilibrium of any sort. There are variations, but they're down below the 12th decimal place! Thermodynamic tables are secure. The statistics of large numbers makes for very trustworthy averages.

Were we to deal with molecules a few at a time, all bets would be off. For example, water droplets grow easily in saturated air in spite of the increased number of surface molecules which (we'll see near the end of the course) are under surface tension which raises the average energy of the droplet with their addition. This is because the bulk molecules (which, unaffected by surface tension, diminish the average energy) grow faster (as radius cubed) than surface molecules (as only radius squared). But when a droplet is so small that it's all surface and no interior, it lowers its energy by evaporating even into saturated air!

The breakeven point (thermodynamically) between growing and shrinking is on the order of 10 molecules! We see from the table above that the statistics of large numbers isn't going to ensure small deviations from the average under such circumstances! So equilibrium arguments should be made with caution unless one has a comforting Avogadro's Number of molecules to back one up.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688

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