Question

A Carnot engine takes \( 3 \times 10^8 \) cal of heat from a reservoir at 627°C and gives it to a sink at 27°C. The work done by the engine is

• A. \( 4.2 \times 10^6 \, {J} \)
• B. \( 8.4 \times 10^6 \, {J} \)
• C. \( 16.8 \times 10^6 \, {J} \)
• D. \( 2 \times 10^6 \, {J} \)

Hint:

The work done by a Carnot engine depends on the efficiency, which is determined by the temperature difference between the hot and cold reservoirs.

Answer 1

The Correct Option is B

The efficiency of a Carnot engine is given by: \[ \eta = 1 - \frac{T_2}{T_1} \] where \( T_1 = 627 + 273 = 900 \, {K} \) and \( T_2 = 27 + 273 = 300 \, {K} \). The efficiency is: \[ \eta = 1 - \frac{300}{900} = \frac{2}{3} \] The heat absorbed from the reservoir is \( Q_1 = 3 \times 10^8 \, {cal} \), which can be converted to joules: \[ Q_1 = 3 \times 10^8 \times 4.18 \times 10^3 = 1.254 \times 10^{12} \, {J} \] The work done by the engine is: \[ W = \eta Q_1 = \frac{2}{3} \times 1.254 \times 10^{12} = 8.4 \times 10^6 \, {J} \] Hence, the correct answer is (b).