Physics

Problem

A designer wishes to extract the maximum useful work from a heat source at

T_H = 700\,\text{K}

while discharging heat to the environment at

T_C = 350\,\text{K}

. In each cycle the source can supply at most

4000\,\text{J}

of heat. Consider the following points. a. (i) Derive an expression for the maximum theoretical work obtainable per cycle and evaluate its numerical value for the given data. b. (ii) The designer claims to have built an engine that produces

3000\,\text{J}

of work per cycle while rejecting

1000\,\text{J}

of heat to the

350\,\text{K}

sink. Determine whether this claim violates the second law and give reasons. c. (iii) If the claim in part (ii) were true, describe how such an engine could be combined with other standard devices to create a perpetual-motion machine of the second kind.

Intuitive explanation

The key to maximum work extraction is recognizing that a Carnot engine sets the theoretical upper limit—it's the most efficient heat engine possible between two temperatures. Rather than guessing at a design, you can use the Carnot efficiency formula to find what fraction of the input heat must become useful work, then calculate how much work you can squeeze out of the 4000 J available. The temperature ratio is what matters; the actual temperatures tell you how much waste heat flows to the environment.

Intuitive explanation

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