How can you show that the area under a P-V graph is equal to the work done?

Question

Sir Yahya · Accepted Answer

### **Questions I read:**  
1. Show that the area under a P-V graph is equal to the work done.

### **Group 1 - P-V Graph and Work Done (Proof) - Covers Q1**  
**Answer:**  
- **Work done by a gas** is given by:  
  $$  
  W = \int_{V_i}^{V_f} P \, dV  
  $$  
  (where $P$ = pressure, $dV$ = small change in volume).

- **On a P-V graph:**  
  - The **area under the curve** represents the integral of pressure with respect to volume.  
  - For a finite process, this area equals the **total work done** by the gas during expansion/compression.

- **Example:**  
  - In a **rectangular process** (constant pressure), work $W = P \Delta V$ → area = height (P) × width ($\Delta V$).  
  - For **non-constant pressure**, the curve's shape gives the exact work via integration.

**Also suits:**  
- Why is the area under a P-V graph equal to work?  
- How is work calculated from a P-V diagram?

**Memory tip:**  
- **"P-V Area = Work"** → Just like distance = area under a v-t graph, work = area under P-V graph.

---  
**Quick check:**  
- If a gas expands at constant pressure (P = 2 Pa) from $V_i = 1 m³$ to $V_f = 3 m³$, what is the work done?  
  *(Ans: $W = P \Delta V = 2 \times (3-1) = 4 J$)*

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