Question

Two Carnot heat engines (E1 and E2) are operating in series as shown in figure. Engine E1 receives heat from a reservoir at \(T_H = 1600 \, K\) and does work \(W_1\). Engine E2 receives heat from an intermediate reservoir at \(T\), does work \(W_2\), and rejects the rest to a reservoir at \(T_L = 400 \, K\). Both the engines have identical thermal efficiencies. The temperature \(T\) (in K) of the intermediate reservoir is ............ (answer in integer).

Hint:

When two Carnot engines in series have equal efficiencies, the intermediate temperature is the geometric mean: \[ T = \sqrt{T_H \cdot T_L} \]

Answer 1

Correct Answer: 800

Step 1: Efficiency of a Carnot engine.
Carnot efficiency is given by: \[ \eta = 1 - \frac{T_{low}}{T_{high}} \]

Step 2: Efficiencies of the two engines.
- For Engine E1: \[ \eta_1 = 1 - \frac{T}{T_H} = 1 - \frac{T}{1600} \] - For Engine E2: \[ \eta_2 = 1 - \frac{T_L}{T} = 1 - \frac{400}{T} \]

Step 3: Condition for identical efficiencies.
Since \(\eta_1 = \eta_2\): \[ 1 - \frac{T}{1600} = 1 - \frac{400}{T} \] \[ \frac{T}{1600} = \frac{400}{T} \] \[ T^2 = 1600 \times 400 \] \[ T = \sqrt{640000} = 800 \, K \] Final Answer:
\[ \boxed{800 \, K} \]