Question

Two reversible heat engines are operated with same heat input and heat output. One of the engine is operated between the temperature limits of \(T_2\) and 200 K & another is between 800 K and \(T_2\). Then the value of \(T_2\) is

• A. \(100 \text{ K} \)
• B. \(200 \text{ K} \)
• C. \(400 \text{ K} \)
• D. \(500 \text{ K} \)

Hint:

When two reversible engines have the same efficiency (implied here by equal heat inputs and outputs), the intermediate temperature $T_2$ is always the geometric mean of the extreme temperatures: $T_2 = \sqrt{T_{\text{max}} \cdot T_{\text{min}}}$. Here, $T_2 = \sqrt{800 \times 200} = \sqrt{160000} = 400 \text{ K}$.

Answer 1

The Correct Option is C

Step 1: Express the ratio of heat rejected to heat supplied for each engine.
For any reversible engine, \(Q_{out}/Q_{in} = T_L/T_H\), where \(T_L\) and \(T_H\) are the sink and source temperatures. For engine 1, working between \(T_2\) and 200 K: \[ \frac{Q_{out}}{Q_{in}} = \frac{200}{T_2} \] For engine 2, working between 800 K and \(T_2\): \[ \frac{Q_{out}}{Q_{in}} = \frac{T_2}{800} \]

Step 2: Use the fact that both engines share the same heat input and output.
Since \(Q_{in}\) and \(Q_{out}\) are identical for both engines, the two ratios above must be equal: \[ \frac{200}{T_2} = \frac{T_2}{800} \]

Step 3: Solve for \(T_2\), which turns out to be the geometric mean of the two extreme temperatures.
\[ T_2^2 = 200 \times 800 = 160000 \] \[ T_2 = \sqrt{160000} = 400 \text{ K} \] \[ \boxed{T_2 = 400 \text{ K}} \] This matches option 3.