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.

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April 8, 2010 at 8:11 am

CharleySo… are you saying thermodynamics is fun or… have I missed the point? lol