Thermodynamic systems: spot the invalid fundamental equation

[andrewkirk]

Homework Statement

In Herbert Callen's text 'Thermodynamics and an introduction to thermostatistics', 2nd edition, Problem 1.10-1 on page 32 presents ten potential fundamental equations of thermo systems, labelled (a) - (j), and asks the reader to identify the five that are invalid because they violate one of his postulates 2-4 of Thermodynamics.

Homework Equations

Postulates are:
2. There exists a function S, called entropy, of the extensive parameters of a composite system, defined for all possible equilibrium states of the system, with the property that adding a constraint cannot increase the entropy.

3.a Entropy of a composite system is additive over constituent sub-systems.
3.b Entropy is a differentiable function of the extensive parameters.
3.c Entropy is a monotone increasing function of energy.

4. $\frac{\partial U}{\partial S}=0\Rightarrow S=0$

The Attempt at a Solution

I have identified four of the invalid equations, being:
(c) fails postulate 4.
(d), (h) and (j) fail postulate 3.

This leaves six equations, of which one must be invalid. However I have checked all of them against 3 and 4 and found them to be compatible. I cannot see how one could check against postulate 2 without being given additional equations for the component subsystems.

The six remaining equations are:

(a)$\ \ \ S=\left(\frac{R^2}{v_0\theta}\right)^{1/3}(NVU)^{1/3}$
(b)$\ \ \ S=\left(\frac{R}{\theta^2}\right)^{1/3}(\frac{NU}{V})^{2/3}$
(e)$\ \ \ S=\left(\frac{R^3}{v_0\theta^2}\right)^{1/5}\left(N^2VU^2\right)^{1/5}$
(f)$\ \ \ S=NR\ \log(\frac{UV}{N^2R\theta v_0})$
(g)$\ \ \ S=\left(\frac{R}{\theta}\right)^{1/2}(NU)^{1/2}\exp\left(\frac{-V^2}{2N^2{v_0}^2}\right)$
(i)$\ \ \ U=\left(\frac{v_0\theta}{R}\right)\frac{S^2}{V}\exp\left(\frac{S}{NR}\right)$

Any suggestions would be appreciated.

 

[wabbit]

Are you sure that (c) fails postulate (4) ?

Also, did you check postulate 3.a ?

 

[andrewkirk]

Are you sure that (c) fails postulate (4) ?

Also, did you check postulate 3.a ?

Thank you wabbit for your reply. I will check my working on the first one and report back.

Regarding the second one, how could we check 3.a when we do not have the fundamental equation for any constituent sub-systems? Conversely, if we put several systems obeying this formula together, how could we check additivity without knowing what the fundamental equation of the composite system would be?

 

[wabbit]

how could we check 3.a when we do not have the fundamental equation for any constituent sub-systems?

There is a hint about that in the book, you might want to review his discussion about postulate III.

 

[andrewkirk]

Oh darn, you are right wabbit! I did my differentiation wrong in calculating $\frac{\partial U}{\partial S}$ for equation (c). It does satisfy (4) after all.

So now I'm even further from a solution, as I need to find two invalid equations from seven.

The extra one (c) is:
Here is the equation for (c):
$S=\left(\frac{R}{\theta}\right)^{1/2}\left(NU+\frac{R\theta V^2}{{v_0}^2}\right)^{1/2}$

The three invalid equations that I have identified are all invalid because of 3c. They all satisfy 3b and 4. I cannot see any way of testing against 3a or 2 without additional information about the systems.

 

[wabbit]

Did you go back.and review the discussion in the book about III ?

 

[andrewkirk]

I have now reviewed the discussion. I think you might have in mind the following. If not, the rest of this post will be irrelevant and I'll need to try again.

Callen p28:
'The additivity property applied to spatially separate subsystems requires the following property: the entropy of a simple system is a homogeneous first-order function of the extensive parameters.'

Prima facie it might appear that we can then just test the candidate equations to see if they have that property. However that is not necessarily applicable because Callen states that the property is only required for simple systems, and the problem does not state that the systems are simple.

I also do not see why Callen thinks that this property follows from the additivity requirement. The only way that I can get it to follow is if we adopt the following additional postulate:

Postulate V: The fundamental equation of a composite system made up of a number of spatially separated identical systems is the same as the fundamental equations of each component system.

Perhaps he has implicitly assumed this without realising it. Or am I missing something?

Thanks again for your continued help.

 

[wabbit]

Yes the homogeneity was what I was referring to - I don't know if you need to explicitly add postulate V, but something along those lines may indeed be implicit when you split a system in halves or such to relate homogeneity to additivity.

 

[andrewkirk]

I've decided to put the question of whether homogeneity follows from additivity into a new thread in the non-homework section. For the time being let's assume that homogeneity does apply for all systems, rather than just for simple systems as Callen says - or alternatively, that the systems in the problem are simple systems and he forgot to mention that.

That then allows us to identify (b) as invalid, as the RHS is proportional to the (4/3)th power of the extensive parameters. I've checked the remaining six, and they all seem to comply with the homogeneity requirement. So I still need to find one invalid equation out of the following:

(a)$\ \ \ S=\left(\frac{R^2}{v_0\theta}\right)^{1/3}(NVU)^{1/3}$
(c)$S=\left(\frac{R}{\theta}\right)^{1/2}\left(NU+\frac{R\theta V^2}{{v_0}^2}\right)^{1/2}$
(e)$\ \ \ S=\left(\frac{R^3}{v_0\theta^2}\right)^{1/5}\left(N^2VU^2\right)^{1/5}$
(f)$\ \ \ S=NR\ \log(\frac{UV}{N^2R\theta v_0})$
(g)$\ \ \ S=\left(\frac{R}{\theta}\right)^{1/2}(NU)^{1/2}\exp\left(\frac{-V^2}{2N^2{v_0}^2}\right)$
(i)$\ \ \ U=\left(\frac{v_0\theta}{R}\right)\frac{S^2}{V}\exp\left(\frac{S}{NR}\right)$

I daresay I'm missing something obvious, but I just can't find the invalid one. Any suggestions will be gratefully accepted.

 

[Kibble]

Late reply, but try checking postulate 4 on (i). When I put it into mathematica, I got a non-zero solution for S.

$$u1 = \frac{v0 \theta}{R} \frac{S^2}{V} e^{\frac{S}{n R}};$$
Solve[D[u1, S] == 0, S]

{{S -> 0}, {S -> -2 n R}}

 

[MathematicalPhysicist]

I am a late bloomer;

There should be a solutions manual for this textbook, but I don't find any place to purchase it from, it seems like top secret of lecturers, I guess.

 

[MathematicalPhysicist]

How did you tackle this problem eventually @andrewkirk ?

 

[andrewkirk]

@Kibble, I suppose it depends on whether negative entropy is 'allowed', ie whether a physical state can have negative entropy. I thought it could not, but a quick glance over the postulates doesn't show any prohibition on negative entropy. If there is no such prohibition then it seems that Postulate 4 is violated.

If there is a prohibition, Postulate 4 is not violated because we will still have the solution S=0, which is consistent with Postulate 4.

Perhaps somebody more experienced in thermo can comment.

 

[andrewkirk]

@MathematicalPhysicist I haven't looked at this for a couple of years. I think I got discouraged and went to do something else instead - a luxury one has when one is self-studying. I'm still interested in the solutions though. On a quick review of the posts it looks like

• d, h and j all fail as per the OP;
  
• b fails Postulate V (see Post 7 above), which is not in Callen's book, but we have surmised that he intended that to be a postulate despite not actually stating it. Or perhaps there's some clever way of proving it from the other postulates. But I have yet to find such a proof;
• i fails provided there is nothing anywhere that forbids negative entropy (so far I've found nothing that forbids it, even though it sounds weird)

That would give us the five requested violations.

 

[MathematicalPhysicist]

I have a solution to this problem; as I am studying for exams in stat mech and thermodynamics; I will let you know of the solution next week.

Stay tuned, or search the web... :-)

 

[MathematicalPhysicist]

P.S I wish I could find the official solution manual to purchase it, but alas; life isn't a piece of chocolate cake...

 

[MathematicalPhysicist]

It's seems like really simple question, when you translate the English into maths;
One condition that you are missing in your assessment is that $\frac{\partial S}{\partial U} >0$.
Anyway,f) isn't physical since $\partial U / \partial S \ne 0$ whenever $S=0$ as you can check.

 

[MathematicalPhysicist]

Late reply, but try checking postulate 4 on (i). When I put it into mathematica, I got a non-zero solution for S.

$$u1 = \frac{v0 \theta}{R} \frac{S^2}{V} e^{\frac{S}{n R}};$$
Solve[D[u1, S] == 0, S]

{{S -> 0}, {S -> -2 n R}}

Never trust your programming skills, you have $U \approx S^2 \exp{S}$ so $\partial U / \partial S \approx Se^S+S^2e^S$ which is zero whenever $S=0$.

 

[Kibble]

Yeah! The issue is that it would require entropy less than zero.. which may be problematic for lots of other reasons. I got carried away by math and stopped thinking about the physical implications.

In chatting with some of my peers, we eventually came to the conclusion that i) was okay but:

$$ f) \hspace{10 mm} S=NR Log( \frac{UV}{N^2 R θ v_0})$$

is not allowed because:

$$ U \hspace{2 mm} \alpha \hspace{2 mm} e^{S}$$
Thus:
$$ \frac{dU}{dS} \hspace{2 mm} \alpha \hspace{2 mm} e^{S}$$

Which does not go to zero when s goes to zero. Our teacher seemed to indicate that was correct.. which would mean that postulate 4 as it is written at the very top should have a double-sided arrow.

 

[MathematicalPhysicist]

Yeah! The issue is that it would require entropy less than zero.. which may be problematic for lots of other reasons. I got carried away by math and stopped thinking about the physical implications.

In chatting with some of my peers, we eventually came to the conclusion that i) was okay but:

$$ f) \hspace{10 mm} S=NR Log( \frac{UV}{N^2 R θ v_0})$$

is not allowed because:

$$ U \hspace{2 mm} \alpha \hspace{2 mm} e^{S}$$
Thus:
$$ \frac{dU}{dS} \hspace{2 mm} \alpha \hspace{2 mm} e^{S}$$

Which does not go to zero when s goes to zero. Our teacher seemed to indicate that was correct.. which would mean that postulate 4 as it is written at the very top should have a double-sided arrow.

Yes, I wrote it in post number 17 here.

When you think about it these questions in Thermodynamics are the easy part, the tough questions are in Ising, Potts models etc in Stat Mech.

 

[andrewkirk]

One condition that you are missing in your assessment is that $\frac{\partial S}{\partial U} >0$.

I don't have my text with me right now so I can't check. Can you please point out where that condition is laid down in Callen. I'll follow it up when I get home.

Anyway,f) isn't physical since $\partial U / \partial S \ne 0$ whenever $S=0$ as you can check.

Why does that make it non-physical?

 

[MathematicalPhysicist]

@andrewkirk as for your last question, this is postulate 4; you quoted it incorrectly in your first post, it should be $\partial U / \partial S =0$ whenever $S\to 0$.

BTW it's better to write it as a limit and not equality, cause the derivative may not be defined at $S=0$ but it will have a limit there.

I read now postulate number 4 from the book by Callen, it seems your interpretation of it is correct; the problem is that the derivative may vanish even if $S\not \to 0$.

I believe what is written in the book is wrong, it should be $\partial U /\partial S |_{N,V}(S\to 0) = 0$.

Otherwise the solutions I have are wrong.

I upload the solution to this problem, it's problem 4; I don't understand the reason for everyone to use computers when you can use your pen and paper quite easily. :-)

 

[MathematicalPhysicist]

@andrewkirk or others, what are your thoughts regarding this problem?

 

[mncmahankali134]

I need a clarification regarding the third postulate
Why can't I write it as S(U1,V1,N1,...)+S(U2,V2,N2,...) = S(U1+U2,V1+V2,N1+N2,...)