Biological Physics: Energy, Information, Life

6.5. Open systems[[Student version, January 17, 2003]] 189

Equation 6.17 makes precise how hard it’s willing to push. Equation 6.17 has intentionally been
writen to emphasize its similarity to the corresponding formula from ordinary mechanics,f =
−dU/dL.
Wecan now also find theworkwhich our subsystem can do against a load. We see that the
subsystem will spontaneously expand, even if this requires opposing an external load, as long as
the opposing force is less than the value in Equation 6.17. To get the maximum possible work, we
should continuously adjust the load force to be always just slightly less than the maximum force
the system can exert. Integrating Equation 6.17 overLgives a sharpened form of Idea 6.14:

If a subsystem is in a state of greater than minimum free energy, it can do work

on an external load. The maximum possible work we can extract isFa−Fa,min. (6.18)

Gilbert says: By the way, where did the work come from? The internal energy of the gas molecules
didn’t change, becauseTdidn’t change.
Sullivan: That’s right: The cylinder has drawn thermal energy from the reservoir (system “B”)
and converted it intomechanical work.
Gilbert: Doesn’t that violate the Second Law?
Sullivan: No, our system sacrificed someorderbyletting the gas expand. After the expansion
wedon’t know as precisely as before where the gas molecules are located. Something—order—did
get used up, exactly as foreshadowed in Chapter 1 (see Section 1.2.2). The concept of free energy
makes this intuition precise.
In a nutshell:

The cost of upgrading energy from thermal to mechanical form is that we must

give up order.

(6.19)

This is just the obverse of the slogan given in Idea 6.13 above.

6.5.3 Free energy transduction is most efficient when it proceeds in small, controlled steps

Idea 6.18 tells us about the maximum work we can get from a small subsystem in contact with a
thermal reservoir. To extract this maximum work, we must continuously adjust the load force to
bejust slightly less than the force the system can exert. This is generally not practical. We should
explore what will happen if we maintain a load that is somewhere between zero (free expansion,
Figure 6.3) and the maximum. Also, most familiar engines repeat acycleover and over. Let’s
construct such an engine in our minds, and see how it fits into the framework of our ideas.

Example

Let subsystem “a” be a cylinder of gas with a piston of areaAat one end, held down

byweightsw 1 andw 2 to maintain initial pressurepi=(w 1 +w 2 )/A(Figure 6.5).

Forsimplicity suppose there’s no air outside; also, take the weight of the piston

itself to be zero. The cylinder is in thermal contact with a reservoir “B” at fixed

temperatureT. Suddenly remove weightw 2 from the piston (slide it off sideways

so this doesn’t require any work). The piston pops up from its initial heightLi

to its final heightLf,measured from the bottom, and the pressure goes down to

pf=w 1 /A.Find the change in the free energy of the gas, ∆Fa,and compare it to

the mechanical work done lifting weightw 1.

Solution:The final temperature is the same as the initial, so the total kinetic energy

Ekin=^32 NkBT of the gas doesn’t change during this process. The pressure in