I am currently lecturing on statistical thermodynamics to our second year students, and I’ve got to the point where I am talking about the calculation of thermodynamic properties, like entropy and Gibbs free energy. Where possible I like to relate calculated properties to experimental results, and I seem to have come up against a problem here, so I thought I would mention it here in case someone with a better understanding of the subject than me reads it, and gives me some pointers. Heat capacities can be compared directly; here one is calculating dU/dT, and that works fine, and the agreement is impressive for rare gases and some diatomics, like nitrogen. Even entropies can be compared with tabulated values, but what about Gibbs free energies? From a chemical viewpoint, we talk about changes in Gibbs energy, not about absolute values. Further, Gibbs free energies of formation of elements in their standard states are defined as zero, so what are we comparing with when we calculate the Gibbs free energy of a nitrogen molecule using statistical thermodynamics? Or is there a subtle point I’m missing? Advice and pointers to reading would be gratefully received!

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3 thoughts on “Statistical thermodynamics questions”

Since posting this, I may be making some progress. It seems that the expression for G from ST gives the Gibbs free energy for an atom, or a molecule, and that the energy from this cannot be compared directly with a value from a thermochemistry calculation, which is usually based on a change in Gibbs free energy. So in order to calculate a comparable quantity, you have to calculate all the terms in the reaction and then combine them appropriately. I won’t have time to do this for my lecture today, but it should be possible for next time!

There are some quantities that can be measured absolutely and some that can only be measured relatively. Entropy is absolute because of the Third Law of Thermo (at 0 degree K, the entropy of a perfect crystalline solid is zero); Other quantities do not have an absolute zero to start from. Take spectroscopy: you only measure a transition between levels, and there is no absolute zero to relate to. So in those cases we must choose an arbitrary reference point. For example, in electrochemistry the Standard Hydrogen potential is chosen to be zero and all others are measured relative to that. Same with Free energies of formation: we make some rules, such as an element has zero energy of formation, but when we calculate the free energy of a reaction, we subtract the LHS from the RHS, so any consistent reference point cancels.
Finally when you do have a reference point, like the H potential, all other potentials are measured relative to that and can then be listed in tables. We also like to list intensive quantities, because these are independent of extensive quantities like volume or mass. Hope this helps.
Stat. mech is a great way to understand thermodynamics and how the microscopic gives rise to macroscopic quantities. I wrote a chapter in our book Physical Chemistry (Chapter 15 of Laidler Meiser and me) (see website) and it takes the approach of starting with the microcanoncial ensemble rather than the canonical. The former gives good insight into entropy, whereas the latter gives insight into the enthalpy.

Since posting this, I may be making some progress. It seems that the expression for G from ST gives the Gibbs free energy for an atom, or a molecule, and that the energy from this cannot be compared directly with a value from a thermochemistry calculation, which is usually based on a change in Gibbs free energy. So in order to calculate a comparable quantity, you have to calculate all the terms in the reaction and then combine them appropriately. I won’t have time to do this for my lecture today, but it should be possible for next time!

There are some quantities that can be measured absolutely and some that can only be measured relatively. Entropy is absolute because of the Third Law of Thermo (at 0 degree K, the entropy of a perfect crystalline solid is zero); Other quantities do not have an absolute zero to start from. Take spectroscopy: you only measure a transition between levels, and there is no absolute zero to relate to. So in those cases we must choose an arbitrary reference point. For example, in electrochemistry the Standard Hydrogen potential is chosen to be zero and all others are measured relative to that. Same with Free energies of formation: we make some rules, such as an element has zero energy of formation, but when we calculate the free energy of a reaction, we subtract the LHS from the RHS, so any consistent reference point cancels.

Finally when you do have a reference point, like the H potential, all other potentials are measured relative to that and can then be listed in tables. We also like to list intensive quantities, because these are independent of extensive quantities like volume or mass. Hope this helps.

Stat. mech is a great way to understand thermodynamics and how the microscopic gives rise to macroscopic quantities. I wrote a chapter in our book Physical Chemistry (Chapter 15 of Laidler Meiser and me) (see website) and it takes the approach of starting with the microcanoncial ensemble rather than the canonical. The former gives good insight into entropy, whereas the latter gives insight into the enthalpy.

Thanks very much for this! I will have a look at that chapter. I’ve given the lectures now, but it will help me improve them for next year.