Calculate the Heat Supplied for a Complete Cycle — Thermodynamics

First Law: ΔU = Q − W (or Q = ΔU + W)

For a complete cycle:

• The system returns to its initial state
• Therefore, ΔU (change in internal energy) = 0

Applying the first law: 0 = Q_net − W_net Q_net = W_net

This means:

• Net heat absorbed by the system = Net work done by the system
• If the cycle is clockwise on a P-V diagram → net work done is POSITIVE → heat is absorbed
• If the cycle is anticlockwise on a P-V diagram → net work done is NEGATIVE → heat is rejected

Conclusion: To find the total heat supplied in a complete cycle, calculate the area enclosed by the cycle on the P-V diagram.

Work done by gas during a process = Area under the P-V curve

For a complete cycle:

• Work done during expansion (A→B, top path) = positive
• Work done during compression (B→A, bottom path) = negative
• Net work = Area enclosed by the cycle = Area of the loop

For a rectangular cycle (simple example): If pressure goes from P₁ to P₂ and volume goes from V₁ to V₂: W_net = (P₂ − P₁)(V₂ − V₁)

For a circular or irregular cycle: W_net = Area of the closed loop on P-V diagram

Since Q_net = W_net for a complete cycle: Heat supplied = Area of the P-V loop

In a complete cycle:

• Q_absorbed = heat absorbed during expansion/heating processes
• Q_rejected = heat rejected during compression/cooling processes
• Q_net = Q_absorbed − Q_rejected = W_net

For a heat engine (clockwise cycle): Efficiency η = W_net / Q_absorbed = 1 − Q_rejected/Q_absorbed

For a refrigerator (anticlockwise cycle): COP = Q_cold / W_net = Q_cold / (Q_hot − Q_cold)

Important: 'Heat supplied' usually refers to Q_absorbed (heat added to the system in the cycle).