I spent the summer of 2009 essentially doing two things: reading and waiting for the birth of my daughter.  Although that is not entirely true (e.g., I also collected some awesome EPR data on a confined system and I spent an inordinate amount of time at the library with my two-year old son), those are the things that most stand out to me.  My reading list was surprisingly long for a typical summer.  I read Robert Ludlam’s book The Sigma Protocol, which was very good right up to the end, and Donald McQuarrie’s Statistical Thermodynamics.  McQuarrie’s text was excellent.  My research interests generally lie in the area of vibrational spectroscopy, and I have only used thermodynamics in my research a few times.  Moreover, I had not read anything on statistical mechanics since graduate school, and this was an excellent refresher.  I learned quite a few things from that book, but one has remained with me over the past few months: relating the various thermodynamic functions through the Legendre transformation.  In short, the Legendre transform changes a function f(x) into a new function g(p) by f(x) – g(p) = xp where p = f'(x).  For example, the internal energy (U) is a function of entropy (S), volume (V), and number of particles (N).  The enthalpy (H), which depends on pressure (P), S, and N, may be found from U using the Legendre transform.  In this formalism, the appropriate equation is U – H = V(dU/dV).  Using the thermodynamic relation dU/dV = -P, we arrive at the familiar expression for the enthalpy, H = U + PV.  Other functions can be found in a similar manner.  This is an elegant way of relating the various thermodynamic values, and I have not yet seen an undergraduate textbook on physical chemistry present it this way.

I have also seen that the Legendre transform may be used to derive the Hamiltonian formulation of classical mechanics from the Lagrangian formulation; however, I did not spend much time on those sections.  Perhaps next summer?

See here for more information on the use of the Legendre transform in thermodynamics.