Step 1: Understanding the Concept:
In a Pressure-Volume (PV) diagram, the net work done by a gas during a cyclic process is equal to the area enclosed by the cycle on the graph.
The sign of the work depends on the direction of the cycle:
- Clockwise cycle: Net work done by the gas is positive ($W>0$).
- Anti-clockwise cycle: Net work done by the gas is negative ($W<0$). Step 2: Key Formula or Approach:
1. Identify the shape and calculate its area: $\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}$.
2. Determine the direction (clockwise vs. anti-clockwise) to assign the correct sign to the work. Step 3: Detailed Explanation:
First, let's analyze the shape of the cycle in the graph. It is a right-angled triangle with vertices at points roughly corresponding to $(V, P)$, $(3V, P)$, and $(3V, 4P)$.
Let's find the lengths of the base and height of this triangle:
- Base (along the V-axis): The process goes from $V$ to $3V$. So, $\text{base} = 3V - V = 2V$.
- Height (along the P-axis): The process goes from $P$ to $4P$. So, $\text{height} = 4P - P = 3P$.
Now, calculate the magnitude of the area enclosed:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ \text{Area} = \frac{1}{2} \times (2V) \times (3P) = 3PV \]
Next, determine the sign. Observe the arrows on the path in the graph:
- The bottom path goes from $(V, P)$ to $(3V, P)$ (Rightwards).
- The rightmost path goes from $(3V, P)$ to $(3V, 4P)$ (Upwards).
- The diagonal path goes from $(3V, 4P)$ back to $(V, P)$ (Down-Left).
Tracing this path reveals that the cycle is moving in an anti-clockwise direction.
An anti-clockwise cycle means work is done ON the gas, so the net work done BY the gas is negative.
\[ W = -\text{Area} = -3PV \] Step 4: Final Answer:
The work done by the gas is $-3\text{ pv}$.
