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It's not so much a matter of justifying it as recognizing that the function δq/T has been assigned the name "entropy" - specifically:

dS = δq/T (by definition)

The quantity δq/T was assigned a name because it is so useful in thermodyanmics for predicting direction of heat flow, efficiency of cycles, and natural (spontaneous) processes.

The idea that entropy is a measure of disorder comes from the proof by by Ludwig Boltzmann in the 1870s who analyzed the statistical behavior of the microscopic components of system. Boltzmann showed that the statistical-mechanical definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant.

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