Question

A carnot's engine whose sink is at a temperature of \(300 \mathrm{~K}\) has an efficiency of \(40 \%\) By what amount should the temperature of the source change to increase the efficiency to \(60 \%\) (A) \(275 \mathrm{~K}\) (B) \(325 \mathrm{~K}\) (C) \(300 \mathrm{~K}\) (D) \(250 \mathrm{~K}\)

Short answer

The change in temperature of the source required to increase the efficiency to \(60\%\) is \(250K\), which corresponds to answer choice (D).

Step-by-step solution

Calculate the Initial Temperature of the Source

The initial efficiency (η) is given as \(40\%\), and the sink temperature (Tsink) is given as 300K. We will use the formula for the efficiency of a Carnot engine: \(η = 1 - (T_{sink} / T_{source})\) We will now solve for the initial temperature of the source (Tsource_initial): \(0.4 = 1 - (300 / T_{source\_initial})\) To find the initial temperature of the source, isolate Tsource_initial: \(0.6T_{source\_initial} = 300\), \(T_{source\_initial} = 300 / 0.6 = 500 K\)

Find the New Temperature for 60% Efficiency

Now we want to find the new source temperature (Tsource_new) required for a 60% efficiency. Using the efficiency formula, we have: \(η_{new} = 1 - (T_{sink} / T_{source\_new})\) Plug in the new efficiency (0.6) and the sink temperature (300K): \(0.6 = 1 - (300 / T_{source\_new})\) Solve for Tsource_new: \(0.4T_{source\_new} = 300\), \(T_{source\_new} = 300 / 0.4 = 750K\)

Calculate the Change in Temperature of the Source

Now we can find the difference between the new source temperature and the initial source temperature: ΔT = Tsource_new - Tsource_initial ΔT = 750K - 500K ΔT = 250K The change in temperature of the source is 250K, which corresponds to answer choice (D).