Biological Physics: Energy, Information, Life

6.4. The Second Law[[Student version, January 17, 2003]] 185

tem energyEtotequals the sum ofEkin,plus the potential energy stored in the

spring,Espring.(By definition, in an ideal gas the molecules’ potential energy can

beneglected; see Section 3.2.1 on page 73). The system is isolated, soEtotdoesn’t

change. Thus the potential energyfδlost by the spring increases the kinetic energy

Ekinof the gas molecules, increasing the temperature and entropy of the gas slightly.

Atthe same time, the loss of volume ∆V=−Aδdecreases the entropy.

Wewish to compute the change in the gas’s entropy (Equation 6.6). Using that

∆lnV=(∆V)/V,and similarly for ∆ lnEkin,gives a net change of

∆S/kB=∆(lnE^3 kinN/^2 +lnVN)=^3

2

N

Ekin

∆Ekin+N

V

∆V.

ReplaceEkin/Nin the first term by^32 kBT.Next use ∆Ekin=fδand∆VV=−δLto

find ∆S/kB=

( 3

2

f

3 kBT/ 2 −

N

L

)

δ,or

∆S=^1

T

(

f−NkBT

L

)

δ.

b. The Statistical Postulate says every microstate is equally probable. Just as in

Section 6.3.1, though, there will be far more microstates withLclose to the value

Leqmaximizing the entropy than for any other value oofL(recall Figure 6.2). To

findLeq,set ∆S=0in the preceding formula, obtaining thatf=NkBT/Leq,or

Leq=NkBT/f.

Dividing the force by the area of the piston yields the pressure in equilibrium, p = f/A=
NkBT/(AL)=NkBT/V.Wehavejust recovered the ideal gas law once again, this time as a
consequence of the Second Law: IfN is large, then our isolated system will be overwhelmingly
likely to have its piston in the location maximizing the entropy. We can characterize this state
as the one in which the spring is compressed to the point where it exerts a mechanical force just
balancing the ideal-gas pressure.

6.4.2 Three remarks

Before proceeding to use the Statistical Postulate (Idea 6.4), some remarks and caveats are in order:

1. The one-way increase in entropy implies a fundamentalirreversibilityto physical processes. Where
   did the irreversibility come from? Each molecular collision could equally well have happened in
   reverse. The origin of the irreversibility is not in the microscopic equations of collisions, but rather
   in the choice of a highly specialized initial state.The instant after the partition is opened, suddenly
   ahuge number of new allowed states open up, and the previously allowed states are suddenly a
   tiny minority of those now allowed. There is no analogous work-free way to suddenlyforbidthose
   new states. For example, in Figure 6.3, we’d have to push the gas molecules to get them back into
   the left side once we let them out. (In principle we could just wait for them all to be there by a
   spontaneous statistical fluctuation, but we already saw that this would be a very long wait.)
   Maxwell himself tried to imagine a tiny ‘demon’ who could open the door when he saw a molecule
   coming from the right, but shut it when he saw one coming from the left. It doesn’t work; upon
   closer inspection one always finds that any physically realizable demon of this type requires an