The hot reservoir for a Carnot engine has a temperature of 890 K, while the cold reservoir has a temperature of 670 K. The heat input for this engine is 4800 J. The 670 K reservoir also serves as the hot reservoir for a second Carnot engine. This second engine uses the rejected heat of the first engine as input and extracts additional work from it. The rejected heat from the second engine goes into a reservoir that has a temperature of 420 K. Find the total work delivered by the two engines.

Options

A. `3500 J`
B. `2500 J`
C. `3000 J`
D. `2000 J`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the efficiency of the first Carnot engine.
The efficiency \( \eta_1 \) of a Carnot engine is given by the formula:
\[ \eta_1 = 1 - \frac{T_L}{T_H} \]
Where:
- \( T_H = 890 \, K \) (hot reservoir)
- \( T_L = 670 \, K \) (cold reservoir)

Substituting the values:
\[ \eta_1 = 1 - \frac{670}{890} = 1 - 0.7528 \approx 0.2472 \]

### Step 2: Calculate the work done by the first engine.
The work done \( W_1 \) by the first engine can be calculated using the efficiency:
\[ W_1 = \eta_1 \times Q_1 \]
Where:
- \( Q_1 = 4800 \, J \) (heat input)

Substituting the values:
\[ W_1 = 0.2472 \times 4800 \approx 1189.56 \, J \approx 1200 \, J \]

### Step 3: Calculate the heat rejected by the first engine.
The heat rejected \( Q_2 \) by the first engine can be calculated using:
\[ Q_2 = Q_1 - W_1 \]
Substituting the values:
\[ Q_2 = 4800 - 1200 = 3600 \, J \]

### Step 4: Calculate the efficiency of the second Carnot engine.
For the second Carnot engine, the efficiency \( \eta_2 \) is given by:
\[ \eta_2 = 1 - \frac{T_L'}{T_H'} \]
Where:
- \( T_H' = 670 \, K \) (hot reservoir from the first engine)
- \( T_L' = 420 \, K \) (cold reservoir for the second engine)

Substituting the values:
\[ \eta_2 = 1 - \frac{420}{670} = 1 - 0.6269 \approx 0.3731 \]

### Step 5: Calculate the work done by the second engine.
The work done \( W_2 \) by the second engine can be calculated using:
\[ W_2 = \eta_2 \times Q_2 \]
Substituting the values:
\[ W_2 = 0.3731 \times 3600 \approx 1342.16 \, J \approx 1343.28 \, J \]

### Step 6: Calculate the total work delivered by both engines.
The total work \( W_{total} \) is the sum of the work done by both engines:
\[ W_{total} = W_1 + W_2 \]
Substituting the values:
\[ W_{total} = 1200 + 1343.28 \approx 2543.28 \, J \]

### Final Answer:
The total work delivered by the two engines is approximately \( 2543.28 \, J \).

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