Question

The efficiency of a Carnot engine working between temperatures $127^\circ\text{C}$ and $27^\circ\text{C}$ is

• A. $25\\%$
• B. $75\\%$
• C. $50\\%$
• D. $20\\%$

Hint:

Remember that for a Carnot engine, the efficiency depends only on the temperatures of the hot and cold reservoirs. Always convert temperatures to Kelvin before applying the formula.

Answer 1

The Correct Option is A

Step 1: Recall the efficiency formula.
A Carnot engine has efficiency \[ \eta = 1 - \frac{T_c}{T_h} \] where $T_h$ is the hot temperature and $T_c$ is the cold temperature, both in kelvin.

Step 2: Convert the temperatures.
Change the celsius values to kelvin by adding $273$. \[ T_h = 127 + 273 = 400 \text{ K} \] \[ T_c = 27 + 273 = 300 \text{ K} \]

Step 3: Put values in the formula.
\[ \eta = 1 - \frac{300}{400} \]

Step 4: Simplify the fraction.
\[ \frac{300}{400} = \frac{3}{4} = 0.75 \] So \[ \eta = 1 - 0.75 = 0.25 \]

Step 5: Write as a percentage.
Multiply by one hundred. \[ \eta = 0.25 \times 100 = 25\% \]

Step 6: State the answer.
The engine turns one quarter of the heat into work. \[ \boxed{\eta = 25\%} \]