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First law for control mass; cycle; power and refrigeration

Short answer

For a control mass, Q - W = ΔU + ΔKE + ΔPE. In a cycle ΔE = 0, so ∮δQ = ∮δW (net heat equals net work). A power cycle gives W_net = Q_H - Q_L, η = W_net/Q_H; a refrigeration cycle needs W_in = Q_H - Q_L, with COP_R = Q_L/W_in.

Write down expression for first law of thermodynamics for a control mass. Reduce it for a cycle process and write down statement of first law of thermodynamics for power and refrigeration cycles.

First law for a control mass

For a closed system (control mass), the net energy transferred as heat and work equals the change in its total energy:

Q − W = ΔE = ΔU + ΔKE + ΔPE

In rate form: dE/dt = Q̇ − Ẇ. For a stationary system, Q − W = ΔU. Sign convention: Q into system +ve, W by system +ve.

For a cycle

In a cycle the system returns to its initial state, so ΔE = 0 (E is a property):

∮ δQ = ∮ δW or Q_net = W_net

Statement for power and refrigeration cycles

  • Power cycle: the net work output equals the net heat input: W_net = Q_H − Q_L > 0, where Q_H is heat received from the high-temperature source and Q_L is heat rejected to the sink. Efficiency η = W_net/Q_H.
  • Refrigeration (heat pump) cycle: the work input equals the net heat rejected: W_in = Q_H − Q_L, where Q_L is heat removed from the cold space and Q_H is heat rejected to the hot surroundings. COP_R = Q_L/W_in, COP_HP = Q_H/W_in.

In words: for a closed system undergoing a cycle, the algebraic sum of heat transfers equals the algebraic sum of work transfers. (Energy cannot be created or destroyed.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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