Example 15.7 from College Physics, 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy

Less Work is Produced by a Given Heat Transfer When Entropy Change is Greater

(a) Calculate the work output of a Carnot engine operating between temperatures of 600 K and 100 K for 4000 J of heat transfer to the engine. (b) Now suppose that the 4000 J of heat transfer occurs first from the 600 K reservoir to a 250 K reservoir (without doing any work, and this produces the increase in entropy calculated above) before transferring into a Carnot engine operating between 250 K and 100 K. What work output is produced? (See Figure 15.35.)

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
In both parts, we must first calculate the Carnot efficiency and then the work output.
Solution (a)
The Carnot efficiency is given by
${Eff}_{C} = 1 - \frac{T_{c}}{T_{h}} .$
15.53

Substituting the given temperatures yields

${Eff}_{C} = 1 - \frac{100 K}{600 K} = 0 . 833 .$
15.54

Now the work output can be calculated using the definition of efficiency for any heat engine as given by

$Eff = \frac{W}{Q_{h}} .$
15.55

Solving for $W$ and substituting known terms gives

$\begin{array}{l} W & = & {Eff}_{C} Q_{h} \\ = & (0.833) (4000 J) = 3333 J. \end{array}$
15.56
Solution (b)
Similarly,

${Eff'}_{C} = 1 - \frac{T_{c}}{{T′}_{c}} = 1 - \frac{100 K}{250 K} = 0.600,$
15.57

so that

$\begin{array}{l} W & = & {Eff'}_{C} Q_{h} \\ = & (0.600) (4000 J) = 2400 J. \end{array}$
15.58
Discussion
There is 933 J less work from the same heat transfer in the second process. This result is important. The same heat transfer into two perfect engines produces different work outputs, because the entropy change differs in the two cases. In the second case, entropy is greater and less work is produced. Entropy is associated with the unavailability of energy to do work.

Figure 15.35 (a) A Carnot engine working at between 600 K and 100 K has 4000 J of heat transfer and performs 3333 J of work. (b) The 4000 J of heat transfer occurs first irreversibly to a 250 K reservoir and then goes into a Carnot engine. The increase in entropy caused by the heat transfer to a colder reservoir results in a smaller work output of 2400 J. There is a permanent loss of 933 J of energy for the purpose of doing work.

0 of 4 steps open

How did it go?

Strategy

Solution (a)

Solution (b)

Discussion

Strategy

Solution (a)

Solution (b)

Discussion