Eutectic systems behavior near 100%-0% composition and low temperature I can't understand why in this phase diagram 
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in the area where the system is approaching the 100% $\alpha$ or 100%  $\beta$ composition, the state is just a pure solid. I mean, where does the other component go? If I am interpreting it right, there's no liquid-phase mixture left, and there's neither any solid rest from the other substance.
 A: This used to confuse me too! The key is that the $\alpha$ and $\beta$ phases are not necessarily pure component A or B. For example $\alpha$ is a single phase which consists of mostly component A with small amounts of component B dissolved in it - it is a single "A-rich" phase. The B atoms are distributed throughout the A crystal but do not form into separate regions. If you looked into a microscope you would see only one kind of crystal throughout. In contrast, in the  $\alpha + \beta$ region you will see two different kinds of crystal one where there is mostly A with B dissolved in it and one where there is B with a little A dissolved in it. 
A: At 100 \% there is only one component present, right?  So there is no missing component.
I am guessing that your problem lies in the "approaching 100 %" part of the question.  Now how can that happen.  Either one is fishing one component out of the system, leaving the other alone, or one is adding huge amounts of the other component making the percentage of the first smaller and smaller.  In either case only one component is left and the other component has either been removed from the solution or is present at zero relative concentration.
