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Step 1 — Identify the process: Look for keywords — "insulated" (adiabatic), "slowly" (quasi-static/isothermal if in thermal contact), "rigid container" (isochoric), "piston free to move" (often isobaric). If $PV^n = C$ is given, it is polytropic.

Step 2 — Apply the First Law: $Q = \Delta U + W$. For ideal gases, $\Delta U = nC_v\Delta T$ always. Calculate $W$ using the process-specific formula. Find the third quantity.

Step 3 — Cyclic processes: $\Delta U = 0$ (state function returns to start), so $Q_{\text{net}} = W_{\text{net}} =$ area enclosed on P-V diagram. Clockwise = positive work (engine), anticlockwise = negative work.

Step 4 — Carnot problems: Always convert to Kelvin first. Use $\eta = 1 - T_C/T_H$. For combined engine-refrigerator problems, the work output of one drives the other.

Common traps: (1) Using Celsius in Carnot formula. (2) Confusing $W$ done "by" vs "on" the gas. (3) Applying $\Delta U = nC_v\Delta T$ only to isochoric — it works for all ideal gas processes. (4) Forgetting that free expansion has $W = 0$ (no opposing pressure). (5) In mixing problems, use $T_f = (n_1C_{v1}T_1 + n_2C_{v2}T_2)/(n_1C_{v1} + n_2C_{v2})$ for gases in an insulated container.