Thermodynamics Cycle Shortcut ⚡ JEE Concept

 

❓ Concept Question

In a cyclic thermodynamic process, how do we determine the work done by the gas from a P–V diagram?

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🖼 Concept Image

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✍️ Short Concept

In thermodynamics, when a gas undergoes a complete cycle, the work done equals the area enclosed in the P–V graph.

Geometry directly gives the answer.

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🔷 Step 1 — Cyclic Process Golden Rule 💯

For cyclic process:

Initial state = Final state

So,

$$\Delta U = 0$$

From First Law:

$$Q = W$$

👉 Net heat supplied equals net work done.

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🔷 Step 2 — Work Done in a Cycle

Work done by gas:

$$W = \oint P\,dV$$

Graphically:

$$\boxed{W = \text{Area enclosed in P–V diagram}}$$

⚠️ Path calculation usually not needed.

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🔷 Step 3 — Direction of the Cycle

Direction determines the sign of work.

Clockwise cycle →

$$W > 0$$

Work done by gas.

Anticlockwise cycle →

$$W < 0$$

Work done on gas.

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🔷 Step 4 — Shape of the Loop

Area depends on the shape of loop.

Rectangle:

$$A = l \times b$$

Triangle:

$$A = \frac{1}{2} \times base \times height$$

Circle / ellipse:

Use geometry formula.

📌 JEE often hides this as a geometry question.

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🔷 Step 5 — Units Check

Pressure:

$$P \text{ in Pascal (Pa)}$$

Volume:

$$V \text{ in } m^3$$

Work unit:

$$1\, \text{Pa} \cdot m^3 = 1\, \text{Joule}$$

⚠️ Convert kPa, cm³ before solving.

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✅ Final Takeaway

For cyclic processes:

$$\boxed{W = \text{Area enclosed in P–V diagram}}$$

And

$$\Delta U = 0$$

So heat supplied equals work done.

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⭐ Golden JEE Insight

Whenever closed loop appears in P–V graph:

👉 Immediately think geometry.

Physics becomes area calculation.