Carnot engine is an ideal heat engine, which converts heat energy into mechanical energy. Efficiency of Carnot engine is given by `eta =1 - (T_2 // T_1)`, where `T_1` is temperature of source and `T_1`, is temperature of sink. If `Q_1` is the amount of heat absorbed/cycle from the source, `Q_2` is the amount of heat rejected/cycle to the sink and W is the amount of useful work done/cycle, then
` W= Q_1 -Q_2 and (Q_2)/(Q_1) = (T_2)/(T_1)`
Efficiency of the engine in the above question is .if source and sink temperature respectively 227 and 127 .

Options

A. 0.3
B. 0.1
C. 0.2
D. 0.8

Answer

To solve the problem regarding the efficiency of a Carnot engine operating between two temperatures, we follow these steps:

### Step 1: Understand the Efficiency Formula
The efficiency (\( \eta \)) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
where \( T_1 \) is the temperature of the source (hot reservoir) and \( T_2 \) is the temperature of the sink (cold reservoir).

### Step 2: Convert Temperatures to Kelvin
The temperatures given in the problem are in degrees Celsius. We need to convert them to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]
For the source temperature \( T_1 = 227 °C \):
\[ T_1 = 227 + 273 = 500 \, K \]
For the sink temperature \( T_2 = 127 °C \):
\[ T_2 = 127 + 273 = 400 \, K \]

### Step 3: Substitute the Temperatures into the Efficiency Formula
Now that we have \( T_1 \) and \( T_2 \) in Kelvin, we can substitute these values into the efficiency formula:
\[ \eta = 1 - \frac{T_2}{T_1} = 1 - \frac{400}{500} \]

### Step 4: Calculate the Efficiency
Now we perform the calculation:
\[ \eta = 1 - \frac{400}{500} = 1 - 0.8 = 0.2 \]

### Step 5: Express Efficiency as a Percentage
To express the efficiency as a percentage, we multiply by 100:
\[ \eta = 0.2 \times 100 = 20\% \]

### Final Answer
The efficiency of the Carnot engine is **20%**.

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