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Chapter 8: Problem 1197

A Carnot engine having a efficiency of \(\mathrm{n}=(1 / 10)\) as heat engine is used as a refrigerators. if the work done on the system is \(10 \mathrm{~J}\). What is the amount of energy absorbed from the reservoir at lowest temperature ! (A) \(1 \mathrm{~J}\) (B) \(90 \mathrm{~J}\) (C) \(99 \mathrm{~J}\) (D) \(100 \mathrm{~J}\)

Short Answer

Expert verified
The short answer is: The amount of energy absorbed from the reservoir at the lowest temperature is \(90\, J\).

Step by step solution

01

Understand the Carnot engine as a refrigerator

For a refrigerator, the equation of energy conservation is given by: \[Q_{absorbed} = Q_{rejected} - W\] Where, - \(Q_{absorbed}\) is the amount of heat absorbed from the cold reservoir, - \(Q_{rejected}\) is the amount of heat expelled to the hot reservoir, - \(W\) is the work done on the system. Since we have the work done on the system and are trying to find \(Q_{absorbed}\), we need to find \(Q_{rejected}\).
02

Use the efficiency formula to find \(Q_{rejected}\)

Recall that the efficiency of a heat engine is defined as: \[n = \frac{W}{Q_{rejected}}\] Given in the problem statement, the efficiency is \(n = \frac{1}{10}\) and the work done on the system is \(W = 10\, J\). We can now solve for \(Q_{rejected}\): \[\frac{1}{10} = \frac{10}{Q_{rejected}}\] Rearranging: \[Q_{rejected} = 100\, J\]
03

Calculate the energy absorbed from the cold reservoir

Now, we have the value for \(Q_{rejected}\) and \(W\). We can use the energy conservation equation for a refrigerator to calculate \(Q_{absorbed}\): \[Q_{absorbed} = Q_{rejected} - W\] \[Q_{absorbed} = 100\, J - 10\, J\] \[Q_{absorbed} = 90\, J\] Thus, the amount of energy absorbed from the reservoir at the lowest temperature is 90 J. The correct answer is (B) \(90\, J\).

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