Thermodynamics and Chemistry

CHAPTER 4 THE SECOND LAW

4.4 DERIVATION OF THEMATHEMATICALSTATEMENT OF THESECONDLAW 120

0:02 0:03 0:04 0:05

300

400

500

V/m^3

.

T

/K

Figure 4.9 A family of reversible adiabatic curves (two-dimensional reversible adi-

abatic surfaces) for an ideal gas withVandTas independent variables. A reversible

adiabatic process moves the state of the system along a curve, whereas a reversible

process with positive heat moves the state from one curve to another above and to the

right. The curves are calculated fornD 1 mol andCV;mD.3=2/R. Adjacent curves

differ in entropy by 1 J K^1.

are two-dimensional curves. Each curve is a contour of constantS. At this stage in the
derivation, our assignment of values ofSto the different curves is entirely arbitrary.
How can we assign a unique value ofSto each reversible adiabatic surface? We can
order the values by letting a reversible process withpositiveone-way heat, which moves the
point for the state to a new surface, correspond to anincreasein the value ofS. Negative
one-way heat will then correspond to decreasingS. We can assign an arbitrary value to the
entropy on one particular reversible adiabatic surface. (The third law of thermodynamics
is used for this purpose—see Sec.6.1.) Then all that is needed to assign a value ofSto
each equilibrium state is a formula for evaluating thedifferencein the entropies of any two
surfaces.
Consider a reversible process withpositiveone-way heat that changes the system from
state A to state B. The path for this process must move the system from a reversible adiabatic
surface of a certain entropy to a different surface of greater entropy. An example is the path
A!B in Fig.4.10(a) on the next page. (The adiabatic surfaces in this figure are actually
two-dimensional curves.) As before, we combine the experimental system with a Carnot
engine to form a supersystem that exchanges heat with a single heat reservoir of constant
temperatureTres. The net heat entering the supersystem, found by integrating Eq.4.4.1, is

q^0 DTres

ZB

A

∂q

Tb

(4.4.4)

and it is positive.
Suppose the same experimental system undergoes a second reversible process, not nec-
essarily with one-way heat, along a different path connecting the same pair of reversible
adiabatic surfaces. This could be path C!D in Fig.4.10(a). The net heat entering the