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

Answer: The efficiency of the engine in its original form is 1/3 or 33.33%, and the ratio between the temperatures of the two reservoirs in their original form is 2/3.

Step-by-step solution

Write down the efficiency equation for a Carnot Engine

A Carnot engine's efficiency is given by the equation: \[e =1 - \frac{T_{2}}{T_{1}}\]

Setup initial and final efficiency equations

We are given that the temperature of the warmer reservoir T1 is increased by a factor of 2, so the new temperature will be 2T1. Let's denote the initial efficiency as e and the final efficiency as e'. We now have: \[e' = 2e\] Now, using the efficiency equation for the engine with the original temperature and the increased temperature, we get: \[e = 1 - \frac{T_{2}}{T_{1}}\] \[e' = 1 - \frac{T_{2}}{2T_{1}}\]

Substitute expressions and solve for the desired variables

We have two equations with two unknowns, e and T2/T1. We can substitute e from one equation to the other, so we get: \[2(1-\frac{T_2}{T_1})=1-\frac{T_2}{2T_1}\] Now, solve the equation for the ratio \(\frac{T_{2}}{T_{1}}\): Expanding and simplifying the equation: \[2-2\frac{T_{2}}{T_{1}} = 1 - \frac{T_{2}}{2T_{1}}\] \[2\frac{T_{2}}{T_{1}}-\frac{T_{2}}{2T_{1}} = 1\] \[(4-1)\frac{T_{2}}{T_{1}} = 2\] \[\frac{T_{2}}{T_{1}} = \frac{2}{3}\] We found that the ratio between the temperatures of the two reservoirs in their original form is 2/3. Next, we will plug this value into the efficiency equation to find the efficiency e: \[e = 1 - \frac{2}{3}\] \[e = \frac{1}{3}\]

Present the final answer

The efficiency of the engine in its original form is 1/3 or 33.33% and the ratio between the temperatures of the two reservoirs in their original form is 2/3.