6.6. Microscopic systems[[Student version, January 17, 2003]] 195
Figure 6.8:(Metaphor.) Where the buffalo hop. [Cartoon by Larry Gonick.]spontaneously flip (orisomerize)between two configurations. We’ll call the states #1 and #2 and
denote the situation by the shorthand
2k+
k1. (6.26)
The symbolsk+andk- in this formula are called the forward and backwardrate constants,
respectively; we define them as follows.
Initially there areN 2 molecules in the higher-energy state #2 andN 1 in the lower-energy state
#1. In a short interval of time dt,the probability that any given molecule in state #2 will flip
to #1 is proportional to dt;call this probabilityk+dt. Section 3.2.4 argued that the probability
pertimek+is a constantCtimes e−E
‡/kBT
,sothe average number of conversions per unit time
isN 2 k+=CN 2 e−E‡/kBT.The constantCroughly reflects how often each molecule collides with a
neighbor.
Similarly, the average rate of conversion in the opposite direction isN 1 k-=CN 1 e−(E
‡+∆E)/kBT
.
Setting the two rates equal (no net conversion) then gives that the equilibrium populations are
related by
N 2 ,eq/N 1 ,eq=e−∆E/kBT, (6.27)
which is just what we found forP 2 /P 1 in the previous subsection (Equation 6.24). Since both
isomers live in the same test tube, and so are spread through the same volume, we can also say
that the ratio of their densities,c 2 /c 1 ,isgiven by the same formula.
Webegan with an idea about thekineticsof molecular hopping between two states, then found
in the special case of equilibrium that the relative occupancy of the state is just what the Boltzmann
distribution predicts. The argument is just like the observation at the end of Section 4.6.3 on page
124, where we saw that diffusion to equilibrium ends up with the concentration profile expected
from the Boltzmann distribution. Our formulas hold together consistently.
Wecan extract a testable prediction from this kinetic analysis. If we can watch a single 2-state
molecule switching states, we should see it hopping with two rate constantsk+andk-.Ifmoreover
wehave some prior knowledge of how ∆E depends on imposed conditions, then the prediction