An ideal Carnot's engine works between `227^(@)C` and `57^(@)C`. The efficiency of the engine will be

Options

A. `22%`
B. `34%`
C. `55%`
D. `13.5%`

Answer

To find the efficiency of an ideal Carnot engine working between two temperatures, we can use the formula for efficiency given by:

\[ \text{Efficiency} (η) = 1 - \frac{T_2}{T_1} \]

where:
- \(T_1\) is the absolute temperature of the hot reservoir (in Kelvin),
- \(T_2\) is the absolute temperature of the cold reservoir (in Kelvin).

### Step 1: Convert Celsius to Kelvin
First, we need to convert the given temperatures from Celsius to Kelvin.

1. For the hot reservoir temperature \(T_1\):
\[ T_1 = 227^\circ C + 273.15 = 500.15 \, K \approx 500 \, K \]

2. For the cold reservoir temperature \(T_2\):
\[ T_2 = 57^\circ C + 273.15 = 330.15 \, K \approx 330 \, K \]

### Step 2: Apply the Efficiency Formula
Now that we have both temperatures in Kelvin, we can substitute them into the efficiency formula.

\[ η = 1 - \frac{T_2}{T_1} = 1 - \frac{330}{500} \]

### Step 3: Calculate the Efficiency
Now we perform the calculation:

1. Calculate the fraction:
\[ \frac{330}{500} = 0.66 \]

2. Substitute back into the efficiency equation:
\[ η = 1 - 0.66 = 0.34 \]

### Step 4: Convert to Percentage
To express the efficiency as a percentage, we multiply by 100:

\[ η = 0.34 \times 100 = 34\% \]

### Final Answer
The efficiency of the Carnot engine is \(34\%\).

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