It’s a good book, Jon, Kittel and Kroemer was the book I used as an undergraduate. I went back and looked at it many years later, with more experience of trying to explain things, and was struck at how didactically-well it is constructed. Well-written too.
But it starts from the outgrowth of mechanics, as almost all these treatments do, which makes the entropy seem like a mysterious extra, and its path-independence seem inexplicable. These things are presented as “empirical regularities” that one accepts rather than understands. A nice example was when Pieter wrote down the ideal gas equation of state: pV = nRT and said it probably would be of limited help to Nick because it doesn’t involve the entropy. Of course, I look at it, and I see terms that _only_ exist because there is entropy, and that produce this particular equation of state in the case that the entropy has a certain functional form (the one it has for the ideal gas). So to me it is all entropy on both sides. But since the habit is to think of variables as ad hoc records of measurements, and not as standing for meaningful concepts, the way it is written would never reflect that it is all about the entropy. It seems that 999/1000 times you will encounter it, that will be how it is presented. The first thing I would do with the equation above is stop writing it in energy units and write it non-dimensionally, as it should be: V (p/T) = 2/3 U (1/T) (for the monatomic ideal gas). Then I would ask “who ordered this thing that phenomenally we encounter as pressure-over-temperature (p/T)?” Why is it a thing, a phenomenon, as opposed to some random observational accident, that is not part of any larger category and from which nothing in particular can be derived? To which the answer is: what makes pressure (better, p/T) a thing is the relation (p/T) = \partial S / \partial V And then “Hey, also, why is temperature a thing? (again, better, 1/T)”. To which the answer is (1/T) = \partial S / \partial U (and temperature is properly measured in units of energy, Boltzmann’s constant is a unit-conversion factor to arbitrary degrees adopted historically). I am not asking: why can one measure an average force per unit area? or why can one read a number off a thermometer? I am asking: why is there such a concept as “pressure” that turns out to have properties from which one can derive conclusions? And why that, for temperature, too? So then our equation above becomes V \partial S / \partial V = 2/3 U \partial S / \partial U and that happens to be true because, for a fixed number of molecules (remember, this is the simple formalism modeling a monatomic ideal gas), the form of the entropy function of the state variables that scale with the total system size is S = some constants + log(V) + 3/2 log(U). https://en.wikipedia.org/wiki/Sackur%E2%80%93Tetrode_equation Now I can repeat why we should have started non-dimensionally with entropy (entropy is non-dimensional; Boltzmann’s constant belongs with the temperature as a unit conversion; jettison this history), rather than in energy units. The entropy is just a counting measure that depends on what is available to be counted, and that is determined jointly by V and U. It is not “more” a function of U than it is of V. So there is no reason to denominate it — from its own terms — in the units of one of its arguments and not the other. There might be reasons later to _convert_ units to energies for some applications, but for thinking about what we are doing it is a diversion. (To anybody:) Don’t be scared that somebody had to do some math to derive something (like the specific functional form for the ideal-gas entropy above); that’s just work, like many things are just work. The question that concerns us is: why do concepts turn out to be explanatory of phenomena, and what, as the real source, is the commitment we make that fixes the relations among those concepts? In this case, the derivation that the entropy function for a simple model-gas has a certain form is the reason that gas has a certain equation of state. With the same entropy concept, the same _definitions_ of temperature and pressure from it, and some other system, we could work hard, compute some other functional form for the entropy-state-function, and obtain some other equation of state than the ideal-gas-form. (LANL has spent billions of dollars and tens of thousands of man-hours (and many computer hours) computing such equations of state for various systems.) But we would not be lost, and the cases wouldn’t be “just one damned thing after another”. If we want the fewest symbols to write down, and a quick conversion to mechanical units, my mild symbol re-arrangement above is an annoying digression. If, instead, we want to know at what point we committed to some property of a system that is the actual source of all of its other properties, what I have above is the direct, unidirectionally organized display of how those commitments cascade to consequences. But, Roger has freed me, for which I am grateful. The paradox of the honest student is indeed being played out. A pitcher plant is not trying to get the insect to go pollinate something. And yet, somehow, it’s all good fun anyway. Eric > On Jun 20, 2025, at 6:38, Jon Zingale <[email protected]> wrote: > > I should not put effort in, but is this page useful? Is there something here > that can help Nick to make a well-formed question? 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