The efficiency of a Carnot engine operating between temperatures of `100^(@)C` and `-23^(@)C` will be

Options

A. `(100 - 23)/(273)`
B. `(100 + 23)/(373)`
C. `(100 + 23)/(100)`
D. `(100 - 23)/(100)`

Answer

To find the efficiency of a Carnot engine operating between two temperatures, we can use the formula for the efficiency of a Carnot engine:

\[ \text{Efficiency} (η) = 1 - \frac{T_C}{T_H} \]

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

**Step 1: Convert the temperatures from Celsius to Kelvin.**

The conversion from Celsius to Kelvin is done using the formula:
\[ T(K) = T(°C) + 273.15 \]

- For the hot reservoir \( T_H = 100°C \):
\[ T_H = 100 + 273.15 = 373.15 \, K \]

- For the cold reservoir \( T_C = -23°C \):
\[ T_C = -23 + 273.15 = 250.15 \, K \]

**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:
\[ η = 1 - \frac{T_C}{T_H} = 1 - \frac{250.15}{373.15} \]

**Step 3: Calculate the fraction.**

Calculating the fraction:
\[ \frac{250.15}{373.15} \approx 0.671 \]

**Step 4: Calculate the efficiency.**

Now substitute this value back into the efficiency formula:
\[ η = 1 - 0.671 \approx 0.329 \]

**Step 5: Convert the efficiency to a percentage.**

To express the efficiency as a percentage, multiply by 100:
\[ η \approx 0.329 \times 100 \approx 32.9\% \]

Thus, the efficiency of the Carnot engine operating between temperatures of \(100°C\) and \(-23°C\) is approximately **32.9%**.

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