But it’s clear what’s gone wrong in this calculation – we’ve counted the state where the first oscillator has the first unit of energy and the second oscillator has the second and third as distinct from the state where the first oscillator has the second unit of energy and the second oscillator has the first and third, when in actuality these are just the same state – the first oscillator has oscillations that are slightly larger than the base state, the second has oscillations that are twice as large, and the third oscillator is in the base state. For instance, one might have supposed the following – there are three units of energy, and three possibilities for the location of each of them, so there are 3 3=27 possible states of the system. Now, there are other ways one might have tried to calculate this value, and gotten the wrong answer. It’s easy to see that these are all the possibilities, because there are no ways for any oscillator to have negative energy, or non-integer energy, and we’re ignoring states in which there is any more energy, or some other aspect of the system has any relevant energy. If you don’t want to try to write them all out for yourself, note that there are 3 possible states where one of the oscillators has energy 3 and the other two have energy 0 6 possible states where one of the oscillators has energy 2, one has energy 1, and the other has energy 0 and 1 possible state where all three of the oscillators have energy 1. With a bit of counting, one can see that the total number of different available states of such a system is 10. 4 of the book, is figuring out how many different states such a system can be in if the total energy of the oscillations is 3. Now consider a set of three identical such oscillators, arranged in a fixed lattice, so that the three can be distinguished by their position in the lattice. By adjusting the units of energy if necessary, we can arrange that the possible states of such an oscillator have exactly 0, 1, 2, … units of energy.
The energy of the system is proportional to the size and frequency of the oscillations, and thus it too is quantized. (I believe a natural example would be a molecule of H 2, where the bond between the two hydrogen atoms can roughly be seen as a vibrating spring, so the two atoms are bouncing closer together and farther apart – if this example is bad for some reason, please let me know in the comments so I can understand this stuff better!) An important discovery in quantum mechanics is the fact that the oscillations in such a system can’t be of just any size, but that the sizes are separated by a constant amount. In particular, I think that there may be ways in which thermodynamics can be used to give arguments for or against the existence of tropes (or substrata or bundles or other metaphysical posits).Ĭonsider a submicroscopic simple harmonic oscillator with a characteristic frequency. From considering the first example discussed on page 4, I’m already starting to consider connections between this epistemic principle and the underlying metaphysics. Appropriate for introductory college chemistry courses, it further lends itself to use as a supplementary text for independent study by more advanced students.I recently picked up Leonard Nash’s 1973 Elements of Statistical Thermodynamics as some light airplane reading (well, it’s light in the sense of being a 138 page paperback printed on thin paper), because I’ve been interested in figuring out more about the applications of the Principle of Indifference (roughly, that one’s credences in various propositions should be proportional to the number of ways that the propositions can be true). The concepts of equilibrium and entropy thus acquire new significance, and readers discover how thermodynamic parameters may be calculated from spectroscopic data.Encompassing virtually all of the forms of statistical mechanics customary to undergraduate physical chemistry books, this brief text requires prior acquaintance with only the rudiments of the calculus and a few of the simplest propositions of classical thermodynamics.
Starting with an analysis of some very simple microcanonical ensembles, it proceeds to the Boltzmann distribution law and a systematic exploration of the proper formulation, evaluation, and application of partition functions.
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