Apply the efficiency formula to a heat engine.

Problem During one cycle, an engine extracts $2\mathrm{.}00103$ J of energy from a hot reservoir and transfers $1\mathrm{.}50103$ J to a cold reservoir. $($a$)$ Find the thermal efficiency of the engine. $($b$)$ How much work does this engine do in one cycle? $($c$)$ What average power does the engine generate if it goes through four cycles in $2\mathrm{.}50$ s$?$

Strategy Apply the equation below to obtain the thermal efficiency, then use the first law, adapted to engines, to find the work done in one cycle. To obtain the power generated, divide the work done in four cycles by the time it takes to run those cycles.

e $=$

Weng$|$Qh$|$

$=$

$|$Qh$||$Qc$||$Qh$|$

$=1$

$|$Qc$||$Qh$|$

SOLUTION

$($a$)$ Find the engine's thermal efficiency.

Substitute Qc and Qh into the thermal efficiency equation.

e $=1$

$|$Qc$||$Qh$|$

$=1$

$1\mathrm{.}50103$ J$2\mathrm{.}00103$ J

$=0\mathrm{.}250,$ or $25\mathrm{.}0\%$

$($b$)$ How much work does this engine do in one cycle?

Apply the first law in the form of Weng $=|$Qh$||$Qc$|$ to find the work done by the engine.

Weng$=|$Qh$||$Qc$|=2\mathrm{.}00103$ J $1\mathrm{.}50103$ J$=5\mathrm{.}00102$ J

$($c$)$ Find the average power output of the engine.

Multiply the answer of part $($b$)$ by four and divide by time.

P $=$

W$$t

$=$

$4\mathrm{.}00(5\mathrm{.}00102$ J$)2\mathrm{.}50$ s

$=8\mathrm{.}00102$ W

LEARN MORE

Remarks Problems like this usually reduce to solving two equations and two unknowns, as here, where the two equations are the efficiency equation and the first law and the unknowns are the efficiency and the work done by the engine.

Question Can the efficiency of an engine always be improved by increasing the thermal energy put into the system during a cycle? Explain.

Yes, increasing the work input always increases the efficiency.No$,$ extra energy input always decreases efficiency.$$No$,$ it depends on how the energy output is affected.Yes, extra energy input decreases the amount of energy wasted.

PRACTICE IT

Use the worked example above to help you solve this problem. During one cycle, an engine extracts $1\mathrm{.}99103$ J of energy from a hot reservoir and transfers $1\mathrm{.}52103$ J to a cold reservoir.

$($a$)$ Find the thermal efficiency of the engine.

Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully.

$($b$)$ How much work does this engine do in one cycle?

J

$($c$)$ How much power does the engine generate if it goes through four cycles in $2\mathrm{.}50$ s$?$

Your response differs from the correct answer by more than $10\%.$ Double check your calculations. W

EXERCISEHINTS:$$GETTING STARTED $|$ I$\text{'}$M STUCK!

The energy absorbed by an engine is three times as great as the work it performs.

$($a$)$ What is its thermal efficiency?

e $=$

There are three forms for the efficiency. Which is most appropriate for the given information?

$($b$)$ What fraction of the energy absorbed is expelled to the cold reservoir?

$|$Qc$||$Qh$|$

$=$

The energy expelled to the cold reservoir must be smaller than the energy absorbed from the hot reservoir.

$($c$)$ What is the power output of the engine if the energy input is $1520$ J each cycle and it goes through two cycles every $3$ seconds?

P $=$ W