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## Homework Statement

Consider two systems which together comprise an isolated system, but are initially not in equilibrium with each other. The temperatures of the two systems are [itex]T_1[/itex] and [itex]T_2[/itex] and the internal energies are [itex]E_1[/itex] and [itex]E_2[/itex]. The systems are separated by a diathermal wall and only allowed to exchange energy by heat exchange. By writing the entropy as a function of the internal energy and the volume, [itex] S = S(E,V) [/itex] and the fact that energy is conserved, show that energy flows from the hotter to the colder body.

## Homework Equations

[itex] dE = TdS - PdV [/itex]

[itex] \Delta S > 0 [/itex] (non-reversible process)

## The Attempt at a Solution

I guess to begin with I'm confused about how to actually write [itex] S(E,V) [/itex] without knowing an equation of state. I'm also unsure if the term [itex] PdV [/itex] is zero or not, because I see no reason the systems can't expand, yet they "only exchange energy through heat" which implies there is no work...

I would begin by writing,

[itex] dS = \frac{dE}{T} + \frac{P}{T}dV [/itex] but from there I don't know how to integrate, because presumably [itex] T = T(E,V) [/itex] and [itex] P = P(E,V) [/itex].

If someone could help me get started or provide some hints as to where to go, I'd greatly appreciate it.