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First law for control mass in cyclic process

Short answer

For a control mass undergoing a cycle, the net heat transfer equals the net work transfer: ∮ δQ = ∮ δW, i.e. Q_net = W_net. A perpetual motion machine of the first kind is impossible.

Explain the first law of thermodynamics for a control mass undergoing a cyclic process.

First law for a control mass (closed system)

Energy cannot be created or destroyed; the net energy supplied to a closed system as heat and work equals the change in its energy:

Q − W = ΔE = ΔU + ΔKE + ΔPE

(Q into system +ve; W done by system +ve.)

Cyclic process

A cycle consists of processes which bring the system back to its initial state, so every property returns to its initial value:

ΔE = E_final − E_initial = 0

Hence
∮ δQ = ∮ δW or Q_net = W_net

Statement

For a control mass undergoing a cycle, the net heat transfer to the system is equal to the net work done by the system. It is impossible to construct a machine that gives work without any energy supplied (perpetual motion machine of the first kind, PMM1).

Consequences

  • Power cycle: W_net = Q_H − Q_L (positive).
  • Refrigerator or heat pump: W_in = Q_H − Q_L (the system receives net work and rejects net heat).
 P|    ___
  |   /   \   clockwise: W_net = + (power cycle)
  |   \___/   area = W_net = Q_net
  +--------- V

✓ Answer checked by Claude Opus 5.5 (high effort)

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