If you’re distilling two liquids, you want to get it hot, but not to the boiling point of the mixture. This way, the two components will evaporate at different rates, so when you recondense the vapor, you end up with a new liquid that is a higher percentage of the liquid with the lower boiling point
As the lower boiling point portion if the mixture boils off the boiling point of the mixture increases. Thus a thermometer tells you what grade of purity your current mixture is and what you are getting out the other side.
Though one has a higher boiling point than the other it doesnt simply wait its turn before evaporating, the bit being boiled off is never totally pure (though with enough heat and height you can get close)
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