Can we mathematically show that water flows from high-pressure region to low-temperature region and why particles flow from a region of higher chemical potential to lower on the basis of the principle of increase of entropy?
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asked May 17, 2020 at 13:47
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1 Answer
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Starting from the chemical potential denfinition:
\begin{equation} \frac{\partial S(E,V,N)}{\partial N} = - \frac{\mu(E,V,N)}{T(E,V,N)}, \end{equation}
For two systems in contact $\mu_1 = \mu_2$ should be satisfied and for same temperatures of the system, the change in entropy until equilibrium will be (certainly positive, since 2nd therm. law):
\begin{equation} dS = dS_1 + dS_2 = -(\mu_1 - \mu_2)dN_1 > 0, \end{equation}
with $dN_2 = - dN_1$, since particles do not get lost. Now assume $\mu_2>\mu_1$, then it must be that $dN_1 > 0 $, meaning that particles flow to system 1, where the chemical potential is lower. Maybe you can find something similar for your pressure and temperature related questions.
Cheers
