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Long before statistical mechanics, entropy was introduced as:

$dS = \frac{dQ}{T}$

At the time when entropy was introduced in this manner, was it known that entropy represents how "disordered" a system is? If so, how can one tell that entropy represents the disorder of a system from the above definition?

If however it wasn't known that $S$ was related to the disorder of a system, what was the interpretation of $S$ at the time of formulation of the above formula?

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I'm not an expert on the history, but wiki has a page on the history of entropy that may be helpful (I couldn't tell you if it's accurate). Note that the early work in thermodynamics was almost entirely experimental, with people working on the efficiency of heat engines and so on. So it was in that context that entropy emerged as a useful quantity, without any special "microscopic" interpretation attached to it. In any case the interpretation of entropy as "disorder" makes no sense before statistical mechanics, and it's innacurate and sometimes misleading even with statistical mechanics.

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