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How should I determine the total number of phases that are at equilibrium with each other in a multi-component mixture?

For example, this problem is from the 2007 GRE published by ETS (Problem 24):

What is the maximum number of phases that can be at equilibrium with each other in a three component mixture?

I initially thought the answer should be 3 since no matter what kind of mixture it is there should always be solid, liquid, and gas. But then I understood that there can be situations like immiscible liquids which increases the total number of phases. The actual answer is 5. What does this number entail?

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  • $\begingroup$ In equilibrium, solid quantity produced no effect on equilibrium. $\endgroup$ – Jack Rod Jun 20 '20 at 6:19
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    $\begingroup$ Does this answer your question? melting and boiling ...really equilibrium? $\endgroup$ – Jack Rod Jun 21 '20 at 4:16
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    $\begingroup$ The linked post (supposed dupe) does not address the question in this post so I vote to reopen. In addition the linked post has also been closed. $\endgroup$ – Buck Thorn Jul 4 '20 at 14:07
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The solution is a direct application of the phase rule

$$F=C+2-P$$

with the number of components C=3 and F the number of degrees of freedom (the number of intensive variables that can be varied while phases continue to coexist). The phase rule can be derived by considering all the conditions of balance (chemical, pressure, temperature) that must be satisfied by the phases in mutual equilibrium.

Solving for P (the number of phases).

$$P=5-F$$

The maximum number of coexisting phases is observed when there are no degrees of freedom left (a coexistence point), that is when

$$P=5$$

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