Question

A certain gasoline engine has an efficiency of 30.0%. What would the hot reservoir temperature be for a Carnot engine having that efficiency, if it operates with a cold reservoir temperature of 200ºC?

a) 362ºC
b) 400ºC
c) 450ºC
d) 500ºC

Answer 1

The hot reservoir temperature for a Carnot engine with an efficiency of 30.0%, and a cold reservoir temperature of 200°C, would be approximately 402.78°C. The closest option is (b) 400°C.

Step-by-step explanation:

The question is about determining the hot reservoir temperature for a Carnot engine with a given efficiency when the cold reservoir temperature is known. The Carnot efficiency formula that relates the temperatures of the hot and cold reservoirs (Th and Tc) to the efficiency (e) is:

e = 1 - (Tc / Th)

Given that the efficiency of the engine is 30.0% (or 0.30 as a decimal), and the cold reservoir temperature (Tc) is 200°C, we need to convert the Celsius temperature to Kelvin before applying the formula. The conversion is:

Tc (K) = Tc (°C) + 273.15

So,

Tc (K) = 200 + 273.15 = 473.15 K

Using the Carnot efficiency formula:

0.30 = 1 - (473.15 K / Th)

Th = 473.15 K / (1 - 0.30)

Th = 473.15 K / 0.70

Th ≈ 675.93 K

Now, to convert Th back to Celsius:

Th (°C) = Th (K) - 273.15 = 675.93 K - 273.15 ≈ 402.78°C

The hot reservoir temperature for the Carnot engine is therefore approximately 402.78°C, which is closest to option (b) 400°C.