BioPHYSICAL chemistry

Electric motors draw a current when the motor is turning the transmis-
sion (which turns the wheels). With regenerative braking, the electric motor
turns backwards when the car is slowing down, generating a current that
regenerates the battery. Regenerative braking allows hybrid cars to con-
vert what would normally be simple heat loss into electrical energy that
is stored until it is needed to power the car.

CHAPTER 3 SECOND LAW OF THERMODYNAMICS 63

Derivation box 3.1 Entropy as a state function

Entropy is defined in terms of state functions, and so entropy should also be a state func-

tion. All state functions have the property that the functions are independent of path and

are dependent only on the initial and final states. To prove the assertion that entropy is a

state function, it is necessary to establish the path independence. One way to establish this

property is to demonstrate that the integral of the function around an arbitrary path is zero,

as this establishes that the entropy is the same at the initial and final states regardless of

path. For the Carnot cycle, the integral of the entropy change is given by the contributions

of the first and third steps:

(db3.1)

If entropy is a state function, then this integral is equal to zero. In that case it is necessary

to establish that the ratio of heat contributions is equal to the ratio of the temperatures:

(db3.2)

To establish this relationship, consider an ideal gas. In this case, the heat has been related

to the ratio of the final and initial volumes:

(db3.3)

So, the heat terms in the Carnot cycle can be written in terms of the volume changes:

(db3.4)

qwnRT

V

V

nal

initial

=− = ln^3

If then

q

T

q

T

q

q

hot

hot

cold

cold

hot

cold

+= 0 =

TT

T

hot

cold

dS

q

T

q

T

hot

hot

cold

cold

x∫ =+

qnRT

V

cold cold V

d

c

= ln

qnRT

V

hot hot V

b

a

= ln