Question

Does the cyclic integral of heat have to be zero (i.e., does a system have to reject as much heat as it receives to complete a cycle \() ?\) Explain.

Short answer

Explain your reasoning. Answer: Yes, the cyclic integral of heat must be zero for a system to complete a cycle. This is because, according to the first law of thermodynamics, the heat absorbed by the system (Q) must be equal to the work done by the system (W) in a cyclic process where the internal energy (∆U) does not change. As a result, a system must reject as much heat as it receives in order to maintain a constant internal energy and complete a cycle.

Step-by-step solution

Understand the cyclic integral of heat

A cyclic integral refers to the integral of a quantity in a system undergoing a cyclic process, meaning the system returns to its initial state after the completion of the process. In the context of this exercise, we are specifically looking at the cyclic integral of heat, which refers to the net amount of heat absorbed or released by the system during a complete cycle.

Identify principles for cyclic processes

For cyclic processes, we can rely on the first law of thermodynamics, which states that the change in internal energy (∆U) of a system is equal to the heat (Q) absorbed by the system minus the work (W) done by the system: ∆U = Q - W. Since the system returns to its initial state after completing a cycle, its internal energy would remain the same, therefore ∆U = 0.

Relate the cyclic integral of heat to the first law of thermodynamics

If we rewrite the first law of thermodynamics with respect to the fact that ∆U = 0, we get Q - W = 0, or Q = W. This means that the heat absorbed by the system must be equal to the work done by the system for a complete cycle to occur.

Conclude whether the cyclic integral of heat has to be zero

Since the heat absorbed by the system is equal to the work done by the system during a complete cycle, it means that a system must indeed reject as much heat as it receives in order to complete a cycle. Therefore, the cyclic integral of heat must be zero.