# Flow of the component of a hidrostatic system

I have a problem that asks me to prove, from notions of thermodynamic chemichal potential, that at constant temperature, air (or any component of a hidrostatic system) will flow to lower pressures, from Gibbs-Duhem, if there's only one component: $$d\mu=-\bar{S}dT+\bar{V}dP$$ If $T=constant$, then $d\mu=\bar{V}dP$, and $\bar{V}$ is always positive, so if pressure goes down, the chem. potential will have to go down as well, and so it will flow to lower pressures. Is the opposite true? Supposing constant pressure and a gradient of temperature, then: $$d\mu=-\bar{S}dT$$ By the third principle, $S=0$ in $T=0$ and it goes up for higher $T$, so $S$ is always positive so the air will flow to higher temperatures. Is this right?

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