Final answer:
The Carnot engine's work done is calculated using the efficiency formula based on the temperatures of the hot and cold reservoirs. After conversion to Kelvin and conversion of heat input to Joules, the work done by the engine is found to be 8.4 × 10^6 J. Option B is correct.
Step-by-step explanation:
The student is asking about the work done by a Carnot engine that takes in heat from a high-temperature reservoir and expels some of it to a lower temperature sink. To calculate the work done, we need to use the efficiency formula of a Carnot engine, which depends on the temperatures of the heat reservoirs.
The efficiency (e) of a Carnot engine is given by:
e = 1 - (Tc/Th)
where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. The temperatures must be in Kelvin.
First, we convert the temperatures from Celsius to Kelvin:
Th (hot reservoir) = 627°C = 627 + 273 = 900 K
Tc (cold reservoir) = 27°C = 27 + 273 = 300 K
Now, we use the formula to calculate the efficiency:
e = 1 - (300/900) = 1 - 1/3 = 2/3
The efficiency is 2/3, meaning the engine converts two-thirds of the heat input into work. The heat input (Qh) is 3000 kcal, and we need to convert it into Joules:
1 kcal = 4184 J, so Qh = 3000 kcal * 4184 J/kcal = 12,552,000 J
To find the work (W) done by the engine, we multiply the total heat input by the efficiency:
W = Qh * e = 12,552,000 J * 2/3 = 8,368,000 J
So, the work done by the engine is 8.4 × 106 J, which corresponds to option B.