The Boltzmann Distribution - Statistical Mechanics

Boltzmann's Law states that at equilibrium the probability of finding an energy H in some molecular "degree of freedom", such as motion along the x axis or vibration of a particular C-H bond, is proportional to e-H/kT. This is the same as saying that the equilibrium ratio between two states of different energy (ΔH) is given by eΔH/kT.

This law is a result of a very simple statistical idea, but deriving it analytically requires mathematical techniques that are not comfortable for the person on the street. To get the idea we can use the same approach we used in elementary school for getting the idea of multiplication by counting up 3 sets of 4 toothpicks to find that 3 x 4 = 12:

Suppose there are four identical bits of energy to share among four atoms (each with only one degree of freedom).

Suppose that all possible ways of distributing the energy among the four atoms are equally likely.

What is the likelihood that an atom should have 4 bits of energy, or 3, or 2, or 1, or none?

This problem is simple enough that we can just write down all the possibilities and count to see how often the different amounts occur.

Consider the number of permutations that involve putting 4 (or 3, 2, 1, 0) bits of energy in the first atom:
With all 4 in the first atom there must be 0 in each of the others. So there is only one way to do this.

With 3 in the first atom, the remaining bit of energy can be in any of the three other atoms. Hence 3 ways.

With 2 in the first atom, there can be 2 in any of the others (3 ways), or one in each of two others and one vacant atom (any of 3). Thus 6 permutations in all.

With 1 in the first atom, there are 3 bits to permute among the other atoms, 3 in any one (3 ways), 2 in one and 1 in either of the two others (6 ways), or 1 in each of the three atoms (1 way). Thus 10 permutations in all.

With none in the first atom, there are 4 bits to permute among 3 atoms, 4 in any one (3 ways), 3 in one and 1 in one of the others (6 ways), 2 in each of 2 and one of the others vacant (3 ways), 2 in one and 1 in each of the others (3 ways). Thus 15 permutations in all .

This yields the following table of the number of permutations of bits of energy among the other atoms for a certain number of bits of energy in the first atom:

 Number in first atom 0 1 2 3 4 Permutations 15 10 6 3 1

[Incidentally, it is crucial in counting these permutations correctly that one consider the bits of energy indistinguishable, so that you don't care which energy bits are in which atom, just how many are in each atom. If they were distinguishable, one would derive a different distribution - the Poisson distribution - which gives a much higher weight for intermediate occupancies. Boltzmann was certainly not disturbed by this indistinguishability, because he didn't really think that energy came in little bits anyway. He was just using the model as a computational device and would ultimately take the limit of lots of infinitesimal bits to model continuous energy.]

If we then plot the number of instances of having a certain amount of energy against the amount of energy as blue diamonds, we find a distribution that is reminiscent of a decaying exponential (e-x), the dashed red curve. As the number of energy units and containers increases, the distribution approaches an exponential curve more closely, as shown by the large blue diamonds and red curve in the following plot, which were calculated for 16 bits of energy in 16 atoms. In this case we represent the number of permutations as a percentage of all possible permutations rather than as a raw number. Note that 16 energy units in 16 atoms is the same energy density (temperature) as 4 energy units in 4 atoms. The other points/curves in this plot are for 16 energy units in 8 or 4 atoms, that is twice and four times the energy per molecule of the previous plot. Again the points are well described by exponential curves, but they decay less rapidly, and therefore the percentage of the sample represented by the E=0 population is smaller (30% and 16% vs. 48%). Still E=0 is the most likely single value for the energy, for the obvious reason that it allows the maximum number of arrangements of energy bits among the other atoms.
 After looking at these points and curves, it will not surprise you to learn that in 1877 when Boltzmann, who was good at math, found the limiting case for many atoms and infinitesimal energy packets, the distribution turned out to be exactly his decaying exponential (e-H/kT). [You'll learn how to do this for yourself in PChem. Click here if you can't wait.] You can see from the plot above why temperature occurs in the denominator of the exponent. Higher temperature slows the exponential decay. The Boltzmann distribution can be considered just an answer to a counting problem. One way of having a different amount of energy in one molecule as compared to another is for them to have isomeric structures. Thus the Boltzmann formula gives the equilibrium constant between isomers from their energy difference. Since all chemical transformations can be considered isomerizations (rearrangement of a fixed set of atoms), this approach can predict any equilibrium constant. Ludwig Boltzmann (1844-1906)

Entropy

Remember that statistics enters into determining an equilibrium constant in another way as well, one that has in a sense to do with nomenclature. If you're talking about the equilibrium between gauche- and anti-butane, you need to consider not only the Boltzmann energy factor, you also need to consider the fact that there are two gauche butane structures and only one anti-butane. This means there should be a factor of about 2 in favor of gauche that attenuates the room-temperature energy factor (of 10-3/4 0.9 = 5) in favor of anti. Overall you would expect anti to be favored by a factor of about 2.5 at room temperature.

Here we have profitted from the useful fact that at room temperature e-ΔH/RT is the same as 10-3/4 ΔH, if ΔH is in kcal/mol

In the case of cyclohexane, the chair form is favored by about 6 kcal/mole (a factor of about 30,000), but the flexible twist-boat form includes many more structures (about 7 times as many), so the equilibrium constant falls to about 4,000 in favor of the chair.

This kind of statistical factor is very closely related to what entropy is about, though entropy is usually couched in terms that make it seem mystical to most people. To me entropy seems needlessly obscure.

Entropy (S) is typically multiplied by T and combined with H to make up Gibbs free energy G = H - TS. This free energy (G) is then substituted for H in the Boltzmann factor to give eG/RT.

Now a very common value of ΔS is 1.377 cal/mole/K. This value turns out to be R ln 2, the gas constant (Boltzmann constant times Avogadro's Number) times the natural log of 2. Consider what this means to an equilibrium constant:

K = eG/RT = e-(ΔH - TΔS)/RT = eH /RT e TΔS/RT = eH /RT e ΔS/R = eH /RT e R ln 2/R = eH /RT e ln 2 = eH /RT * 2

That is, after all this rigamarole 1.377 cal/mole/K just came out to mean a factor of 2 in the equilibrium constant.

Are you surprised that the entropy difference between gauche and anti butane is about 1.377 cal/mole/K?
Might it be more user friendly to say that gauche-butane has a statistical factor of two favoring it, rather than to say that it has 1.377 cal/mole/K more entropy that the anti-isomer?
You bet!

While I'm on the subject, here's another pet peeve:
Because ΔG = ΔH - TΔS, it is often said that "ΔS becomes more important as temperature increases". This is nonsense, because when you go to use free energy in calculating an equilibrium or rate constant, you divide by T in the Boltzmann expression. A given ΔS contributes a factor to the equilibrium (or rate) constant that is independent of temperature. What happens with increasing temperature is that ΔH becomes less important. This is sensible because at higher temperature there is more energy to go around, so it does not "cost as much probability" to put a little more energy into one place (the exponential curve in the graph above becomes flatter).

This discussion was designed to give you a fundamental idea of what entropy is about without being intimidating. Statistical thermodynamics does involve some pretty sophisticated ideas, and lots of math. Those who created it were not being intentionally obscurantist, they were just trying to be careful and rigorous. How you should do the counting to decide how many "structures" there are for a certain molecule can be a delicate question. For example the more you warm a molecule, the floppier it becomes, and the more structures you should consider for it. Furthermore, the questions of indistinguishability and symmetry restrictions can get really hairy.