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I have read two statements for first law of a closed system undergoing a cycle. One says that the net work in the cycle must be equal to net heat, so that no energy change takes place. This part i get, but then i read another article, which is obtained from joules experiment, in which the work input in a cycle is proportional to the heat rejected by a factor of J(mechanical equivalent of heat). Aren't these two statements contradictory. One says that, in a cycle net work is EQUAL to net heat, while the other, says there are proportional. Please help, i am confused.

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This is really to do with the acceptance of the law of the conservation of energy.
Using modern day units to simplify matters Joule's experiment was used to find how much work had to be done in joules to produced a certain amount of heat in the unit of the day, one calorie.

The calorie was defined as the amount of heat of heat required to raise the temperature of one gramme of water by one degree Celsius and work was defined in terms of a force moving through a distance.

Looking back one now realises that Joule was measuring the specific heat capacity of water but using what he thought were two independently defined units, the joule and the calorie to measure it.

Your constant of proportionality is the numerical link between these two units, $4.2\;\rm J \approx 1 \;calorie$.

If you google "mechanical equivalent of heat" you will find many articles on the subject and for some light relief I suggest the lyrics and the actual song by Flanders and Swann who "explain" everything.

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