# Heat Efficiency of a Carnot Engine

1. Feb 6, 2014

### castrodisastro

1. The problem statement, all variables and given/known data
A Carnot engine operates between a warmer reservoir at a temperature T1 and a cooler reservoir at a temperature T2. It is found that increasing the temperature of the warmer reservoir by a factor of 2.00 while keeping the same temperature for the cooler reservoir increases the efficiency of the Carnot engine by a factor of 5.00. Find the efficiency of the engine and the ratio of the temperatures of the two reservoirs in their original form.

2. Relevant equations
ε=1-(T2/T1)

3. The attempt at a solution

I think we are supposed to get a numerical value for an answer but I don't think I have enough information.

ε=1-(T2/T1)

The question states, "increasing the temperature of the warmer reservoir by a factor of 2.00 while keeping the same temperature for the cooler reservoir increases the efficiency of the Carnot engine by a factor of 5.00"

That means T1 will be doubled and once that happens, ε is quintupled, resulting in...

5ε=1-(T2/2T1)

The question asks for the efficiency, which I am supposed to answer with a numerical value. So I tried solving for ε.

ε=(1/5)-(T2/10T1))

I can't think of a way to solve this with 2 independent variables. Am I ignoring something from the question that would point me towards a value? Maybe I'm forgetting some mathematical property?

Once I have the efficiency then I can solve for the ratio of the two temperatures. Since it asks me for the ratio, I assume that knowing T1 and T2 are not required, so how could we solve for ε and get a numerical value?

2. Feb 6, 2014

### BvU

Use R for $T_1/T_2$. Now you have 2 equations in 2 unknowns!

3. Feb 6, 2014

### castrodisastro

Thanks! I knew it was staring at me in the face.