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

Options

A. (a)`22%`
B. (b)`34%`
C. (c)`55%`
D. (d)`13.5%`

Answer

To find the efficiency of an ideal Carnot engine working between two temperatures, we can use the formula for efficiency (η) of a Carnot engine:

\[ \eta = 1 - \frac{T_2}{T_1} \]

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 the temperatures from Celsius to Kelvin
1. The temperature of the hot reservoir \(T_1\) is given as \(227^\circ C\):
\[ T_1 = 227 + 273 = 500 \, K \]

2. The temperature of the cold reservoir \(T_2\) is given as \(57^\circ C\):
\[ T_2 = 57 + 273 = 330 \, K \]

### Step 2: Substitute the values into the efficiency formula
Now that we have both temperatures in Kelvin, we can substitute them into the efficiency formula:
\[ \eta = 1 - \frac{T_2}{T_1} = 1 - \frac{330}{500} \]

### Step 3: Calculate the fraction
Calculating the fraction:
\[ \frac{330}{500} = 0.66 \]

### Step 4: Calculate the efficiency
Now, substituting back into the efficiency formula:
\[ \eta = 1 - 0.66 = 0.34 \]

### Step 5: Convert efficiency to percentage
To express efficiency as a percentage, we multiply by 100:
\[ \eta = 0.34 \times 100 = 34\% \]

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

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