Page 162 - Essentials of physical chemistry

124 Essentials of Physical Chemistry

another way to express the G i value per mole. However, in an open system we can include the
change in the moles of a given substance in the change in G i in a partial molal sense using a more
general formula.

qG qG qG
dG ¼ SdT þ VdP þ dn 1 þ dn 2 þ dn 3 þ
qn 1 qn 2 qn 3
T, P, n i6¼1 T, P, n i6¼2 T, P, n i6¼3
This shows that the change in G depends on the changing amount of moles of each component.

qG
P
Thus, we have dG ¼ SdT þ VdP þ i dn i for an open system.
qn i
T, P, n j6¼i
This can have far reaching effects in the analysis of an on-stream process but we will just
show the implications here and leave further treatment to engineering texts. Recall that
A U TS ¼ U þ (PV PV) TS ¼ H TS PV ¼ G PV

" #

qG
X
Thus dn i PdV VdP
i
dA ¼ dG PdV VdP ¼ SdT þ VdP þ
qn i
T,P,n j6¼i

qG
P
dn i . Also using H ¼ G þ TS we find that
Then dA ¼ SdT PdV þ i
qn i
T,P,n j6¼i

qG
P
dn i and along with the use of U ¼ H PV we find that
i
dH ¼ TdS þ VdPþ
qn i
T,P,n j6¼i
" #

qG qG
P P
dn i . In each case, the dn i terms accom-
dU ¼ TdS PdV þ i i
qn i qn i
T,P,n j6¼i T,P,n j6¼i
pany the state variables in an open system. This can be reinterpreted to realize that these terms are
part of the differentials of each of the state variables. However, the amazing thing is that

qG
i
m ¼ in each case! We can also write the general case of the total differential in
qn i
T,P,n j6¼i
each case for H, U, G and A as follows:

qH qU
X X
dn i dn i
dH ¼ TdS þ VdP þ dU ¼ TdS PdV þ
qn i qn i
i S, P, n j6¼i i S, V, n j6¼i

qG qA
X X
dn i dn i :
dG ¼ SdT þ VdP þ and dA ¼ SdT PdV þ
qn i qn i
i T, P, n j6¼i i T, V, n j6¼i
P
Comparing the total differentials to the derived equations in terms of i m dn i we find that
i

qG qA qH qU
. Now the basic equations for
i
m ¼ ¼ ¼ ¼
qn i qn i qn i qn i
T, P, n j6¼i T, V, n j6¼i S, P, n j6¼i S, V, n j6¼i
open systems are modified to allow for the change in moles of species involved as
X
m dn i ,
dH ¼ TdS þ VdP þ i
i
X
m dn i ,
dU ¼ TdS PdV þ i
i
X
m dn i ,
dG ¼ SdT þ VdP þ i
i
X
m dn i :
dA ¼ SdT PdV þ i
i