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State the first law of thermodynamics for the closed system undergoing a cycle.
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The first law of thermodynamics can be simply stated as follows: during an interaction between a system and its surroundings, the amount of energy gained by the system must be exactly equal to the amount of energy lost by the surroundings.

A closed system can exchange energy with its surroundings through heat and work transfer. In other words, work and heat are the forms that energy can be transferred across the system boundary.

Both heat transfer and work transfer may cause the same effect on a system. They both are different forms of energy in transit. Energy that enters a system as heat may leave as work or vice versa.

Consider a closed system (dm=0;$\frac{dm}{dt}$=0) undergoing a cycle between two states 1 and 2.

Let it be brought to state 2 by adding some work to it by rotating the paddle wheel, and returned to state 1 again by transferring heat to the surrounding. It has been found that the amount of this work transferred is always proportional to heat transferred. If the cycle involves many heat and work quantities, same result will be found. So,

($\sum$W)$_cycle$ = J($\sum$Q)$_cycle$

or, $\oint\delta$W = $\oint\delta$Q

Where, J is called Joule’s equivalent. When heat and work both are measured in same unit, value of J will be 1.

$\oint\delta$W = $\oint\delta$Q

This is the first law of thermodynamics for a closed system undergoing a cycle. It is a general law of nature since no violation of it has ever been demonstrated.