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  $\partial S/\partial V = -p/T$? What has gone wrong here?

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Consider two systems, 1 and 2, which can exchange heat and do only the mechanical p-V work. They are isolated from the rest of the world. At equilibrium they have the common pressure, say $\bar p$. Then if we nudge them slightly from equilibrium, then we must have, because of isolation, that (I denote by $E$ the internal energies)
\text{total increase in energy } &=(dE_1 - \bar p\;dV_1) + (dE_2 - \bar p\;dV_2)\\
&= d(E_1 + E_2 -\bar p(V_1 + V_2))
be zero, and hence
E_1 + E_2 = \bar p(V_1 +V_2) + \text{const.}\tag{1}

Now, assuming that the systems are independent,
S_\text{tot}(E_1, E_2, V_1, V_2) = S_1(E_1, V_1)+S(E_2, V_2).\tag{2}

Now, maximizing (2) subject to (1) leads that, at equilibrium,
\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2}
which we identify as $1/T$, and
\frac{\partial S_1}{\partial V_1} = -\frac{\bar p}{T},
which contradicts the oft-stated result with the minus sign flipped.

In hindsight, there must be a minus sign on the RHS of the constraint (1), but I don't see why.

Question: What exactly has gone awry in the above reasoning of mine?

asked Mar 8, 2021 in Theoretical Physics by Atom (5 points) [ revision history ]
edited Mar 8, 2021 by Atom

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