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The equal sign in the Carnot cycle efficiency equation is related to the fact that the total entropy doesn't change at the end of the whole cycle (being related to the fact that the heat exchanges occur in stages in which the auxiliary system is at same temperature that the respective thermal reservoir)? In other words, if the auxiliary system were at a different temperature from the hot reservoir, for example, would there be an additional growing of the entropy in the whole system, collaborating for the decreasing of the efficiency?

Thank you