CHAPTER 4 THE SECOND LAW
4.4 DERIVATION OF THEMATHEMATICALSTATEMENT OF THESECONDLAW 122
zero if the process is adiabatic. It gives the same value ofÅSfor any reversible change
between the same two reversible adiabatic surfaces, and it makes the sum of theÅSvalues
of several consecutive reversible processes equal toÅSfor the overall process.
In Eq.4.4.6,ÅSis the entropy change when the system changes from one arbitrary
equilibrium state to another. If the change is an infinitesimal path element of a reversible
process, the equation becomes
dSD
∂q
Tb(4.4.7)
(reversible process,
closed system)It is common to see this equation written in the form dSD∂qrev=T, where∂qrevdenotes
an infinitesimal quantity of heat in a reversible process.
In Eq.4.4.7, the quantity1=Tbis called anintegrating factorfor∂q, a factor that makes
the product.1=Tb/ ∂qbe the infinitesimal change of a state function. The quantity
c=Tb, wherecis any nonzero constant, would also be a satisfactory integrating factor;
so the definition of entropy, usingcD 1 , is actually one of an infinite number of possible
choices for assigning values to the reversible adiabatic surfaces.4.4.3 Some properties of the entropy
It is not difficult to show that the entropy of a closed system in an equilibrium state is an
extensiveproperty. Suppose a system of uniform temperatureT is divided into two closed
subsystems A and B. When a reversible infinitesimal change occurs, the entropy changes of
the subsystems are dSAD∂qA=Tand dSBD∂qB=Tand of the system dSD∂q=T. But
∂qis the sum of∂qAand∂qB, which gives dSDdSACdSB. Thus, the entropy changes
are additive, so that entropy must be extensive:S=SA+SB.^8
How can we evaluate the entropy of a particular equilibrium state of the system? We
must assign an arbitrary value to one state and then evaluate the entropy change along a
reversible path from this state to the state of interest usingÅSD
R
.∂q=Tb/.
We may need to evaluate the entropy of anonequilibrium state. To do this, we imagine
imposing hypothetical internal constraints that change the nonequilibrium state to a con-
strained equilibrium state with the same internal structure. Some examples of such internal
constraints were given in Sec.2.4.4, and include rigid adiabatic partitions between phases of
different temperature and pressure, semipermeable membranes to prevent transfer of certain
species between adjacent phases, and inhibitors to prevent chemical reactions.
We assume that we can, in principle, impose or remove such constraints reversibly with-
out heat, so there is no entropy change. If the nonequilibrium state includes macroscopic
internal motion, the imposition of internal constraints involves negative reversible work to
bring moving regions of the system to rest.^9 If the system is nonuniform over its extent, the
(^8) The argument is not quite complete, because we have not shown that when each subsystem has an entropy of
zero, so does the entire system. The zero of entropy will be discussed in Sec.6.1.
(^9) This concept amounts to defining the entropy of a state with macroscopic internal motion to be the same as
the entropy of a state with the same internal structure but without the motion, i.e., the same state frozen in time.
By this definition,ÅSfor a purely mechanical process (Sec.3.2.3) is zero.