Why Q1/T1 = Q2/T2?
Clausius in his 1867 text on the mechanical theory of heat writes (p. 120, Fourth Memoir): “This compression is continued until K_2 [heat sink] has received the same quantity of heat Q1 as was before furnished by K1 [furnace].”
Perhaps this is so because Q1 is measured in terms of T1; similarly Q2 is measured in terms of T2. Equality Q1/T1 = Q2/T2 in effect says: the two points of the heat cycle, at Q1 and Q2, are equalized when their respective degrees of freedom relative to the prevailing temperature are equal. Equality of dimensional capacity is required for the cycle. In that way at the two extremes the capacity of the chamber to hold heat (say at T1 if that is the higher temperature ) is maintained through the cycle down to lower temperature T2.
How do you explain the roles in thermodynamical entropy of the fractions Q1/T1 = Q2/T2 and of T1 and T2?
According to the Second Law of Thermodynamics, a heat engine can do no better than to minimize the entropy production due to its operation at the value zero. This obtains if the entropy increase of the cold reservoir +Q2/T2 equals the entropy decrease of the hot reservoir -Q1/T1. Then the heat engine's work output W = Q1 - Q2 is maximized, as is its efficiency (useful output)/input = W/Q1.
This question turns out to be more interesting than I had imagined, and also susceptible of an answer. Clausius in his paper developing the entropy concept (the fourth memoir in his 1867 text book in English) does not give a physical justification for dividing by temperature. I have posted an article about this question on RG: Preprint Why not just Q 1 = Q 2 ?
An article with this question as title, mentioned in my January 13 reply, developed some ideas that led to my writing,
Preprint Entropy 1865, 2019 update
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