Page 106 - Physical Chemistry

lev38627_ch03.qxd 2/29/08 3:12 PM Page 87

87
3. We used a perfect gas as the working substance in a Carnot cycle and used the Section 3.4
ideal-gas temperature scale to find that e rev 1 T /T . From statement 2, this Calculation of Entropy Changes
C
H
equation holds for any system as the working substance. Equating this expression
for e rev to that in statement 2, we get q /T q /T 0 for any system that
C
H
H
C
undergoes a Carnot cycle.
4. We showed that an arbitrary reversible cycle can be divided into an infinite num-
ber of infinitesimal strips, each strip being a Carnot cycle. Hence for each strip,
dq /T dq /T 0. Summing the dq/T ’s from each strip, we proved that
C
C
H
H
dq /T 0 for any reversible cycle that any system undergoes. It follows that
rev
the integral of dq /T is independent of the path. Therefore dq /T is the differ-
rev
rev
ential of a state function, which we call the entropy S; dS dq /T.
rev
Don’t be discouraged by the long derivation of dS dq /T from the Kelvin–
rev
Planck statement of the second law. You are not expected to memorize this derivation.
What you are expected to do is be able to apply the relation dS dq /T to calculate
rev
S for various processes. How this is done is the subject of the next section.
3.4 CALCULATION OF ENTROPY CHANGES
The entropy change on going from state 1 to state 2 is given by Eq. (3.21) as S
2
S S dq /T, where T is the absolute temperature. For a reversible process, we
rev
2
1
1
can apply (3.21) directly to calculate S. For an irreversible process pr, we cannot in-
tegrate dq /T to obtain S because dS equals dq/T only for reversible processes. For
pr
an irreversible process, dS is not necessarily equal to dq irrev /T. However, S is a state
function, and therefore S depends only on the initial and final states. We can there-
fore find S for an irreversible process that goes from state 1 to state 2 if we can 2
conceive of a reversible process that goes from 1 to 2. We then calculate S for this Rev.
reversible change from 1 to 2, and this is the same as S for the irreversible change
from 1 to 2 (Fig. 3.6).
In summary, to calculate S for any process; (a) Identify the initial and final states
1 and 2. (b) Devise a convenient reversible path from 1 to 2. (c) Calculate S from Irrev.
2
S dq /T. 1
1
rev
Let us calculate S for some processes. Note that, as before, all state functions
refer to the system, and S means S syst . Equation (3.21) gives S syst and does not Figure 3.6
include any entropy changes that may occur in the surroundings. Reversible and irreversible paths
1. Cyclic process. Since S is a state function, S 0 for every cyclic process. from state 1 to 2. Since S is a
state function, S is the same for
2. Reversible adiabatic process. Here dq rev 0; therefore each path.
¢S 0 rev. ad. proc. (3.23)
Two of the four steps of a Carnot cycle are reversible adiabatic processes.
3. Reversible phase change at constant T and P. At constant T, (3.21) gives
2 dq rev 1 2 q rev
¢S dq rev (3.24)
1 T T 1 T
where, since T is constant, we took 1/T outside the integral. q rev is the latent heat
of the transition. Since P is constant, q rev q H [Eq. (2.46)]. Therefore
P
¢H
¢S rev. phase change at const. T and P (3.25)
T
Since H q is positive for reversible melting of solids and vaporization of
P
liquids, S is positive for these processes.