Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 306

UmEDÅUm(mix) (11.1.17)

VmEDÅVm(mix) (11.1.18)

By substitution from Eqs.9.5.14and11.1.4in Eq.11.1.14, we can relate the excess molar
Gibbs energy to the activity coefficients of the mixture constituents based on pure-liquid
reference states:
GmEDRT

X

i

xiln (^) i (11.1.19)
It is also possible to derive the useful relation
"
@

nGmE

@ni

T;p;nj§i

DRTln (^) i (11.1.20)
To derive Eq.11.1.20, consider infinitesimal changes in the mixture composition at
constantTandp. From Eq.11.1.19, we write
d

nGEm


DRT
X
i
d.niln (^) i/DRT
X
i
nid ln (^) iCRT
X
i
.ln (^) i/dni (11.1.21)
FromiDiCRTln.
ixi/, we have diDRT .d ln (^) iCdxi=xi/. Substitution in
the Gibbs–Duhem equation,
P
ixidiD^0 , gives
X
i
xid ln (^) iC
X
i
dxiD 0 (11.1.22)
In Eq.11.1.22, we set the sum
P
idxiequal to zero (because
P
ixiequals 1) and
multiply by the total amount,n, resulting in
P
inid ln^ iD^0. This turns Eq.11.1.21
into
d

nGEm


DRT
X
i
.ln (^) i/dni (11.1.23)
from which Eq.11.1.20follows.

11.1.4 The entropy change to form an ideal gas mixture

When pure ideal gases mix at constantT andpto form an ideal gas mixture, the molar
entropy changeÅSmid(mix)DR

P

iyilnyi(Eq.11.1.9) is positive.
Consider a pure ideal-gas phase. Entropy is an extensive property, so if we divide this
phase into two subsystems with an internal partition, the total entropy remains unchanged.
The reverse process, the removal of the partition, must also have zero entropy change. De-
spite the fact that the latter process allows the molecules in the two subsystems to inter-
mingle without a change inT orp, it cannot be considered “mixing” because the entropy
does not increase. The essential point is that thesamesubstance is present in both of the
subsystems, so there is no macroscopic change of state when the partition is removed.
From these considerations, one might conclude that the fundamental reason the entropy
increases when pure ideal gases mix is that different substances become intermingled. This
conclusion would be mistaken, as we will now see.
The partial molar entropy of constituentiof an ideal gas mixture is related to its partial
pressurepiby Eq.9.3.6:
SiDSiRln.pi=p/ (11.1.24)