- Physical Chemistry I
Spring 2012, Unique 52135
Lecture Summary, 11 April 2012
Distribution: We have already seen that
under a given set of conditions, a system of atoms or molecules will
have a distribution of velocities described by some kind of
"average." Although knowing the average is useful, it does not
tell us about outliers in the system which may be of great
importance. Therefore we need to know not just the average
behavior of the system, but how it is distributed around that
average. This is worked out in the Maxwell-Boltzmann
Distribution, one of the most important concepts in all of physical
chemistry. The M-D distribution says that the distribution of
velocities, f(v) is
proportional to v^2 and exp[-v^2]. These
two dependencies on v give the M-D distribution an
asymmetric shape that has a long tail to high v.
This means that at a given set of conditions, a substantial fraction
of the population of the system will be moving at high velocity, and
therefore might have enough kinetic energy to cause a reaction to
occur when they collide with something.
Molecular Collisions: So far everything we have done has been for a single molecule colliding with the walls of the container holding the system. What we really need, however, is information about what happens when two separate molecules collide. We therefore define a few new variables: 1) c(rel), relative c; 2) z(coll), collision frequency (z(coll) = (sigma(c(rel))P)/kBT); and 3) mean free path, lambda ( = kBT/(sigmaP). NB: our result for mean free path is slightly different from the result derived in M&S. That is because I am sticking to describing average velocity as c, rather than other options which M&S use interchangeably.