# Relation Between Entropy and Temperature

## Homework Statement

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Strap in, this one's kind of long. (This problem is from 'Six Ideas That Shaped Physics, Unit T' by Thomas A Moore, 2nd edition. Problem T6R2.)

Imagine that aliens deliver into your hands two identical objects made of substances whose multiplicities increase linearly with thermal energy, something like

$$\Omega = \frac{aNU}{\epsilon}$$

where ε is some energy unit and a is some constant. Answer the following questions about these objects.

a) Do they have a well-defined temperature? If so, how does this temperature depend on the objects' thermal energy?

b) If these objects are placed in thermal contact, will energy spontaneously flow from hot to cold? Will the objects eventually come into equilibrium at a certain common temperature? (Hint: I suggest drawing a graph of ΩAB versus macropartition. This will also help with the next part.)

c) How will the size of the random fluctuations in the energies of these objects compare to those for two normal objects placed in thermal contact?

## Homework Equations

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$$\Omega = \frac{aNU}{\epsilon}$$ From the problem statement

$$\frac{1}{T} = \frac{\partial{S}}{\partial{U}}$$ (definition of temperature)

$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$ (q = U/ε) (definition of multiplicity given in text)

3. The Attempt at a Solution

Using the equation for the definition of temperature gives me T = U/k. (I can reproduce this if needed but don't want to clutter things too much. I want to focus on the information in the table.) In a 'normal' object, the temperature and thermal energy are related by T = U/(3Nk). (The book uses 'Einstein solids' as its model for these things. So that's what I'm comparing these 'alien' objects to.)

Let's just say N = 1 for simplicity from here on. So this just tells me that the temperature increase for these 'alien' objects, in response to a given increase of thermal energy, is different from that of an Einstein solid by a factor of 1/3. So fundamentally, these objects should behave much like normal ones. I.e. they would have well-defined temperatures, they would come to equilibrium by exchanging heat energy, energy would spontaneously flow from the warmer one to the colder one, etc.

The author hints that I should make a graph or table of macropartition vs ΩAB. Unfortunately I have no idea how to present a table in this post, so that will make things difficult. But the basic idea is that if I assume each object consists of 1 atom (N = 1, again for simplicity) and there are 6 'atoms' of energy ε, so that U = 6ε, then for each macro-distribution of those 'atoms' (in the left column), I would list the multiplicity ΩAB of the macrostate as a whole. And this will be the product of the individual multiplicities of objects A and B, as computed from the formula for multiplicity given in the problem statement. So for example, for the distribution 1:5 (where 1 bit of energy is in object A and 5 are in object B), I would compute the multiplicity of A, the multiplicity of B, then multiply them to get the multiplicity of this macrostate.

(Hopefully that's clear. I wish I could just make a table, but I don't know how.)

But right away I run into a problem. Consider the macropartition 0:6, where object A has none of the energy atoms and object B has all 6 of them. If I use the expression for multiplicity given in the problem, then ΩA is zero, thus making the multiplicity of the whole macropartition zero. This would mean there is no way for one of the objects to have all the energy and the other one to have none. But I don't see any reason why that should be the case here any more than for an Einstein solid.

So I figure maybe I'm using the wrong multiplicity equation, but if I use the other one (the one given in the text, for an Einstein solid), then I just get the same table I would for an Einstein solid, which doesn't seem right either.

Is my problem maybe that I'm only using 1 atom, and the difference between the two kinds of object ('alien' vs 'normal') only comes out when using multiple atoms? But that seems wrong, because even with only one atom, there is still the factor of 1/3 that I mentioned above.

So I guess my question is:

Why is it that when I calculate the temperature/thermal energy relationship, I'm getting a straightforward answer that seems physically plausible (U = kT, versus U = 3kT for a 'normal' solid), but when I try to calculate the multiplicity of the 0:6 macropartition, I get an answer that makes no physical sense? Am I just using the wrong multiplicity equation? Or is this just to show that the given multiplicity equation makes no sense physically? Or something else?

UPDATE: I'm now seeing that I need to use the multiplicity equation given in the text, for both objects, in each macropartition. But now my trouble is that this simply reproduces the macropartition vs ΩAB table of an Einstein solid, in which case the problem becomes profoundly uninteresting. But I presume there's something I'm supposed to learn here. I don't see how the multiplicity expression given in the problem statement

$$\Omega = \frac{aNU}{\epsilon}$$

comes into play viz a viz the table. I see that I can derive an expression for temperature from it. But I don't see how temperature and the table are related, but the author seems to want us to spot that connection.

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stevendaryl
Staff Emeritus

## Homework Equations

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$$\Omega = \frac{aNU}{\epsilon}$$ From the problem statement

$$\frac{1}{T} = \frac{\partial{S}}{\partial{U}}$$ (definition of temperature)

$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$ (q = U/ε) (definition of multiplicity given in text)
For this homework assignment, they are telling you NOT to use the second formula for ##\Omega##, but instead to use the formula ##\Omega = \frac{aNU}{\epsilon}##.

The relationship that you're missing is this: ##S = k log(\Omega)##. (I'm assuming that your ##\Omega## means the density of states for a given value of ##U##)

For this homework assignment, they are telling you NOT to use the second formula for ##\Omega##, but instead to use the formula ##\Omega = \frac{aNU}{\epsilon}##.

The relationship that you're missing is this: ##S = k log(\Omega)##. (I'm assuming that your ##\Omega## means the density of states for a given value of ##U##)

So using ##S = kln(\Omega)## seems to still leave me in a quandary, because for the macropartition ##0:6## (where object A has no bits of energy, object B has all 6), ##\Omega_{A} = 0## and therefore ##\Omega_{AB} = \Omega_{A} \Omega_{B} = 0##, which gives me an undefined/unphysical entropy, since there is no ##ln(0)##.

As to which expression I use for multiplicity, my understanding was that the expression

$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$

is essentially just a combinatorics result. In fact, we derived it as part of the homework, and that's exactly what it was: how many ways to distribute q bits of energy amongst 3N oscillators of an Einstein solid. So I was (am) confused by this problem, because I don't see the point of redefining multiplicity and then seeing what happens, since multiplicity just is what it is. It is math, not physics (though it does give rise to physics). I mean, it makes no sense, purely from a combinatorics standpoint, that there is not exactly 1 way of distributing 0 bits of energy among the oscillators of object A. Yet that's what the problem's definition of 'multiplicity' gives me. That just seems absurd and not worth thinking about, but at the same time I trust the textbook author to give me meaningful problems, so I conclude I must be missing something.

Anyway, thus far I have two conclusions about these 'alien objects':

1) They are normal in the sense that they have well-defined temperatures, etc. (which I suspect is incorrect, but if I graph S vs U, I essentially just get an ln function with decreasing slope, i.e. decreasing temperature, as for a normal object

2) The system of the two objects in contact has no entropy for the macropartitions where one of them has zero energy, which is different from Einstein solids.