PV Diagrams — How to Read Work, Heat, and Process Type from Graph

Question

A gas undergoes a cyclic process shown on a PV diagram. How do we read work done, heat exchanged, and identify the process type directly from the graph?

Solution — Step by Step

Each thermodynamic process has a distinct shape on a PV diagram:

Isobaric (constant pressure): horizontal line
Isochoric (constant volume): vertical line
Isothermal (constant temperature): smooth hyperbola ( $PV = \text{constant}$ )
Adiabatic: steeper hyperbola than isothermal ( $PV^\gamma = \text{constant}$ )

Why does the adiabatic curve fall faster? Because $\gamma > 1$ always, so the slope $\frac{dP}{dV} = -\gamma \frac{P}{V}$ is steeper than the isothermal slope $-\frac{P}{V}$ .

Work done BY the gas equals the area under the curve on a PV diagram:

W = \int P\, dV = \text{Area under the curve}

Expansion (V increases, moving right): $W > 0$ (gas does work)
Compression (V decreases, moving left): $W < 0$ (work done ON the gas)
For a cyclic process: $W_{net}$ = area enclosed by the loop. Clockwise loop = positive work; anticlockwise = negative.

Once you know $W$ (from area) and identify the process, use the first law:

Q = \Delta U + W

For specific processes:

Isochoric: $W = 0$ , so $Q = \Delta U = nC_v\Delta T$
Isobaric: $Q = nC_p\Delta T$
Isothermal: $\Delta U = 0$ , so $Q = W = nRT\ln\left(\frac{V_2}{V_1}\right)$
Adiabatic: $Q = 0$ , so $W = -\Delta U$

graph TD
    A[Look at the PV curve] --> B{Is V constant?}
    B -->|Yes| C[Isochoric: W=0, Q=nCvDeltaT]
    B -->|No| D{Is P constant?}
    D -->|Yes| E[Isobaric: W=P.DeltaV, Q=nCpDeltaT]
    D -->|No| F{Hyperbola shape?}
    F -->|Gentle curve| G[Isothermal: DeltaU=0, Q=W]
    F -->|Steep curve| H[Adiabatic: Q=0, W=-DeltaU]
    F -->|Straight line at angle| I[Polytropic: Use PV^n=const]

Why This Works

The PV diagram is essentially an energy map. The area under any process curve gives work because $W = \int P\,dV$ by definition. The shape of the curve encodes which state variable stays constant, which in turn tells us the relationship between $Q$ , $W$ , and $\Delta U$ .

This is why PV diagrams are so powerful for JEE and NEET — one picture contains all the thermodynamic information. You just need to know how to read it.

JEE Main 2023 and 2024 both had PV diagram questions. The trick is usually comparing areas under two different processes connecting the same states — isothermal vs adiabatic work, for instance. Practise reading areas visually.

Alternative Method

For a cyclic process, instead of computing work for each segment, just calculate the enclosed area. If the loop is a simple shape (rectangle, triangle), use geometry:

Rectangle on PV diagram: $W = \Delta P \times \Delta V$
Triangle: $W = \frac{1}{2} \times \text{base} \times \text{height}$

This saves time compared to integrating each segment separately.

Common Mistake

Students often forget the sign convention for work on a PV diagram. Moving RIGHT (expansion) = positive work done BY the gas. Moving LEFT (compression) = negative. For cyclic processes, clockwise = net positive work (heat engine), anticlockwise = net negative work (refrigerator). Mixing these up flips the entire answer.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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