Question

A Carnot engine operates between a warmer reservoir at a temperature \(T_{1}\) and a cooler reservoir at a temperature \(T_{2}\). It is found that increasing the temperature of the warmer reservoir by a factor of 2 while keeping the same temperature for the cooler reservoir increases the efficiency of the Carnot engine by a factor of 2 as well. Find the efficiency of the engine and the ratio of the temperatures of the two reservoirs in their original form.

Short answer

The efficiency of the Carnot engine in its original form is 3/5, and the ratio of the temperatures of the two reservoirs is 2/5.

Step-by-step solution

Write down the given conditions and efficiency formula

We know that: 1. Initially, \(T_{1}\) is the temperature of the warmer reservoir, and \(T_{2}\) is the temperature of the cooler reservoir. 2. After increasing the temperature of the warmer reservoir to 2\(T_{1}\), the efficiency of the engine also increases by a factor of 2. The efficiency equation for a Carnot engine is given by: η = 1 - \(\frac{T_{2}}{T_{1}}\)

Determine the efficiency after increasing the temperature of the warmer reservoir

When the temperature of the warmer reservoir is doubled, we have: T1' = 2\(T_{1}\) The new efficiency (η') is given by: η' = 1 - \(\frac{T_{2}}{2T_{1}}\) According to the given condition, the efficiency increases by a factor of 2: η' = 2η

Solve for η

Now we can write an equation system with the relationships obtained in step 2: η = 1 - \(\frac{T_{2}}{T_{1}}\) 2η = 1 - \(\frac{T_{2}}{2T_{1}}\) Multiply the second equation by \(T_{1}\): 2η\(T_{1}\) = \(T_{1}\) - \(\frac{1}{2}T_{2}\) Substitute η from the first equation: 2(1 - \(\frac{T_{2}}{T_{1}})T_{1}\) = \(T_{1}\) - \(\frac{1}{2}T_{2}\)

Solve for the ratio \(\frac{T_{2}}{T_{1}}\)

Simplify the equation from step 3: 2\(T_{1}\) - 2\(T_{2}\) = \(T_{1}\) - \(\frac{1}{2}T_{2}\) Now, move all the terms involving \(T_{2}\) to one side and \(T_{1}\) to the other: \(\frac{5}{2}T_{2}\) = \(T_{1}\) Now, we can find the ratio \(\frac{T_{2}}{T_{1}}\): \(\frac{T_{2}}{T_{1}} = \frac{2}{5}\)

Find the efficiency of the engine

Now, we can substitute the ratio we found in step 4 back into the efficiency equation: η = 1 - \(\frac{T_{2}}{T_{1}}\) η = 1 - \(\frac{2}{5}\) η = \(\frac{3}{5}\) The efficiency of the Carnot engine in its original form is \(\frac{3}{5}\), and the ratio of the temperatures of the two reservoirs is \(\frac{2}{5}\).