3.2. Decoding the ideal gas law[[Student version, December 8, 2002]] 79
Wewill refer to this formula as theBoltzmann distribution^3 after Ludwig Boltzmann, who found
it in the late 1860s.
Weshould pause to unpack the very condensed notation in Equation 3.26. To describe a state
of the system we must give the locationrof each particle, as well as its speedv.The probability
to find particle “a” with its first coordinate lying betweenx 1 ,aandx 1 ,a+dx 1 ,aand so on, and
its first velocity lying betweenv 1 ,aandv 1 ,a+dv 1 ,aand so on, equals dx 1 ,a×···×dv 1 ,a×···×
P(x 1 ,a,...,v 1 ,a,...). ForKparticles,P is a function of 6Kvariables and we have a total of 6K
differential factors in front. Equation 3.26 gives the probability distribution as a function of these
6 Kvariables.
Equation 3.26 has some reasonable features: At very low temperatures, orT→0, the expo-
nential is a very rapidly decreasing function ofv:The system is overwhelmingly likely to be in the
lowest energy state available to it. (In a gas, this means that all of the molecules are lying on the
floor at zero velocity.) As we raise the temperature, thermal agitation begins; the molecules begin
to have a range of energies, which gets broader asTincreases.
It’s almost unbelievable, but the very simple formula Equation 3.26 is exact. It’s not simplified;
you’ll never have to unlearn it and replace it by anything more complicated. (Suitably interpreted,
it holds without changes even in quantum mechanics.) Chapter 6 will derive it from very general
considerations.
3.2.4 Activation barriers control reaction rates
Weare now in a better position to think about a question posed at the end of Section 3.2.1: If
heating a pan of water raises the kinetic energy of its molecules, then why doesn’t the water in the
pan evaporate suddenly, as soon as it reaches a critical temperature? For that matter, why does
evaporation cool the remaining water?
Tothink about this puzzle, imagine that it takes a certain amount of kinetic energyEbarrierfor
awater molecule to break free of its neighbors (since they attract each other). Any water molecule
near the surface with at least this much energy can leave the pan; we say that there is anactivation
barrierto escape. Suppose we heat a covered pan of water, then turn off the heat and momentarily
remove the lid, allowing the most energetic molecules to escape. The effect of removing the lid is to
clipthe Boltzmann probability distribution, as suggested by the solid line in Figure 3.8a. We now
replace the lid of the pan and thermally insulate it. Now the constant jostling of the remaining
molecules once again pushes some up to higher energies, regrowing the tail of the distribution as in
the dashed line of Figure 3.8a. We say that the remaining molecules haveequilibrated.But the new
distribution is not quite the same as it was initially. Since we removed the most energetic molecules,
the average energy of those remaining is less than it was to begin with: Evaporation cooled the
remaining water. Moreover, rearranging the distribution takes time: Evaporation doesn’t happen
all at once. If we had taken the water to be hotter initially, though, its distribution of energies
would have been shifted to the right (Figure 3.8b), and more of the molecules would already be
ready to escape: Evaporation proceeds faster at higher temperature.
The idea of activation barriers can help make sense of our experience with chemical reactions,
too. When you flip a light switch, or click your computer’s mouse, there is a minimal energy, or
activation barrier, which your finger must supply. Tapping the switch too lightly may move it a
fraction of a millimeter, but doesn’t click it over to its “on” position. Now imagine drumming your
(^3) Some authors use the synonym “canonical ensemble.”