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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. [itex]\frac{\partial U}{\partial S}=0\Rightarrow S=0[/itex]
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.