Biological Physics: Energy, Information, Life

190 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]

areaA

subsystem a

subsystem B

T

L

w 1 w 2

Figure 6.5:(Schematic.) Extracting mechanical work by lifting a weight. After we remove weightw 2 ,the cylinder
rises, lifting weightw 1 .Alarge thermal reservoir (subsystem B) maintains the cylinder at a fixed temperatureT.

system “B” (the outside world) was assumed to be zero. So all that changes in the

free energy is−TSa.Using the Example on page 178, we find the change to be

∆Fa=−NkBTlnLf

Li

.

In the final statepfLf=NkBT/A,bythe ideal gas law, wherepf=w 1 /A. Let

X =(Lf−Li)/Lf,soXlies between 0 and 1. Then the mechanical work done

raising weight #1 isw 1 (Lf−Li)=NkBTX,whereas the free energy changes by

|∆F|=−NkBTln(1−X). ButX<−ln(1−X)whenXlies between 0 and 1, so

the work is less than the free energy change.

One could do something useful with the mechanical work done liftingw 1 bysliding it off the piston,
letting it fall back to its original heightLi,and harnessing the released mechanical energy to grind
coffee or whatever. As predicted by Idea 6.18, we can never get more useful mechanical work out
of the subsystem than|∆Fa|.
What could we do to optimize the efficiency of the process, that is, to get outallof the excess
free energy as work? The ratio of work done to|∆Fa|equals−ln(1X−X).This expression is maximum
for very smallX,that is, for smallw 2 .Inother words,we get the best efficiency when we release
the constraint in tiny, controlled increments—a “quasistatic” process.
Wecould get back to the original state by moving our gas cylinder into contact with a different
thermal reservoir at a lower temperatureT′.The gas cools and shrinks until the piston is back at
positionLi.Nowweslide the weights back onto it, switch it back to the original, hotter, reservoir
(atT), and we’re ready to repeat the whole process ad infinitum.
Wehave just invented a cyclicheat engine.Every cycle converts some thermal energy into
mechanical form. Every cycle also saps some of the world’s order, transferring thermal energy
from the hotter reservoir to the colder one, and tending ultimately to equalize them. Figure 6.6
summarizes these words.