Suppose two phases 1 and 2 of water, say ice and water, are kept in a closed container, at a fixed temperature $T$ and fixed pressure $P$? Then I have the following question:
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Is phase 1 in thermodynamic equilibrium with phase 2 during the first order phase transition?
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If yes, phase 1 must be in chemical equilibrium with phase 2. This implies $\mu_1=\mu_2$. Doesn't this imply that the rate of the transition from ice $\rightarrow$ water is same as the rate of transition from water $\rightarrow$ ice? If yes, it appears to me that, as long as phase 1 and 2 are in equilibrium with each other, the quantity of ice and water remains the same. But this obviously doesn't happen. Then where am I making the mistake?
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To derive the Clausius-Clapeyron equation, do we need phase 1 to be in thermodynamic equilibrium with phase 2 or is it enough to consider the full system to be in thermodynamic equilibrium at temperature $T$ and pressure $P$?
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Is the consequence of chemical equilibrium in case of diffusion different from that of the chemical equilibrium between two phases of a given substance?
Best Answer
The answer to this starts from Gibbs-Duhem relationships that define the number of independent variables you have to completely characterize a thermodynamic systems. You can find this in textbooks or the web. I am happy to write it up later. In any case, for a closed system the number of independent variable you can pick is 2. By putting some ice and water in a closed container (rigid), you have fixed the overall density (number of molecules in the rigid volume), and in this case you have chosen temperature. Therefore, the system will come to phase equilibrium (thermal, mechanical, chemical) by adjusting how much it wants in water form and how much as ice.
This means even if you started out of equilibrium, three things will happen: 1) $T_{ice}=T_{water}$ thermal equilibrium (you already started with that) 2) $P_{ice}=P_{water}$ mechanical equilibrium 3) $\mu_{h20, ice}=\mu_{h20,water}$ chemical equilibrium. Since chemical potential is a function of T, P and composition (in this case purely H2O) these three things will adjust simultaneously to satisfy equality across the phases.
So to answer your questions:
i) You may or may not have started in equilibrium but it will settle to equilibrium, i.e., you will move the right mass split across the phases on the first-order phase transition.
ii) You are right, the phase-split stops when you reach equal chemical potentials.
iii) the complete system is in thermodynamic equilibrium only when no sub-systems are out of equilibrium. Strictly, you cannot define a unique thermodynamic state for the complete system unless all its parts are in equilibrium as well.
iv) No, chemical potential is the derivative of free energy that dictates exchange of matter between phases, reactions, etc.