Three engines A, B and C take steam at `130^(@)C` and reject it at `20^(@)C, 40^(@)Cand50^(@)C` respectively. The most efficient engine would be

Options

A. A
B. B
C. C
D. all same

Answer

To determine which of the three engines (A, B, and C) is the most efficient, we will calculate the efficiency of each engine using the formula for thermal efficiency of a heat engine:

\[ \text{Efficiency} = \frac{T_1 - T_2}{T_1} \]

Where:
- \( T_1 \) is the temperature of the heat source (in Kelvin).
- \( T_2 \) is the temperature of the heat sink (in Kelvin).

### Step 1: Convert temperatures from Celsius to Kelvin
The temperatures given in the question are in Celsius. We need to convert them to Kelvin using the formula:

\[ T(K) = T(°C) + 273.15 \]

For engine A:
- \( T_1 = 130 + 273.15 = 403.15 \, K \)
- \( T_2 = 20 + 273.15 = 293.15 \, K \)

For engine B:
- \( T_1 = 130 + 273.15 = 403.15 \, K \)
- \( T_2 = 40 + 273.15 = 313.15 \, K \)

For engine C:
- \( T_1 = 130 + 273.15 = 403.15 \, K \)
- \( T_2 = 50 + 273.15 = 323.15 \, K \)

### Step 2: Calculate the efficiency of each engine

**For Engine A:**
\[ \text{Efficiency}_A = \frac{T_1 - T_2}{T_1} = \frac{403.15 - 293.15}{403.15} = \frac{110}{403.15} \approx 0.273 \text{ or } 27.3\% \]

**For Engine B:**
\[ \text{Efficiency}_B = \frac{T_1 - T_2}{T_1} = \frac{403.15 - 313.15}{403.15} = \frac{90}{403.15} \approx 0.223 \text{ or } 22.3\% \]

**For Engine C:**
\[ \text{Efficiency}_C = \frac{T_1 - T_2}{T_1} = \frac{403.15 - 323.15}{403.15} = \frac{80}{403.15} \approx 0.198 \text{ or } 19.8\% \]

### Step 3: Compare the efficiencies
Now we compare the efficiencies calculated:

- Engine A: 27.3%
- Engine B: 22.3%
- Engine C: 19.8%

### Conclusion
The most efficient engine is Engine A with an efficiency of approximately 27.3%.