CHAPTER 9 MIXTURES
9.7 ACTIVITY OF ANUNCHARGEDSPECIES 273
reference state based on mole fraction or molality, this process brings the system to the
reference state of componentiat pressurep^0. The change ofiin this case is given by
integration of diDVidp:
refi .p^0 / iD
Zp 0
p
Vidp (9.7.14)
The appropriate partial molar volumeViis the molar volumeViorVAof the pure sub-
stance, or the partial molar volumeVB^1 of solute B at infinite dilution.
Suppose we want to use a reference state for solute B based on concentration. Because
the isothermal pressure change involves a small change of volume,cBchanges slightly
during the process, so that the right side of Eq.9.7.14is not quite the correct expression for
refc;B.p^0 / c;B.
We can derive a rigorous expression forrefc;B.p^0 / c;Bas follows. Consider an ideal-
dilute solution of solute B at an arbitrary pressurep, with solute chemical potential
given byBDrefc;BCRTln.cB=c/(Table9.2). From this equation we obtain
@B
@p
T;fnig
D
@refc;B
@p
!
T
CRT
@ln.cB=c/
@p
T;fnig
(9.7.15)
The partial derivative.@B=@p/T;fnigis equal to the partial molar volumeVB(Eq.
9.2.49), which in the ideal-dilute solution has its infinite-dilution valueVB^1. We
rewrite the second partial derivative on the right side of Eq.9.7.15as follows:
@ln.cB=c/
@p
T;fnig
D
1
cB
@cB
@p
T;fnig
D
1
nB=V
@.nB=V /
@p
T;fnig
DV
@.1=V /
@p
T;fnig
D
1
V
@V
@p
T;fnig
DT (9.7.16)
HereTis the isothermal compressibility of the solution, which at infinite dilution is
T^1 , the isothermal compressibility of the pure solvent. Equation9.7.15becomes
VB^1 D
@refc;B
@p
!
T
CRT T^1 (9.7.17)
Solving for drefc;Bat constantT, and integrating fromptop^0 , we obtain finally
refc;B.p^0 / c;BD
Zp 0
p
VB^1 RT T^1
dp (9.7.18)
We are now able to write explicit formulas for ifor each kind of mixture component.
They are collected in Table9.6on the next page.
Considering a constituent of a condensed-phase mixture, by how much is the pressure
factor likely to differ from unity? If we use the valuespD 1 bar andT D 300 K, and
assume the molar volume of pureiisViD 100 cm^3 mol ^1 at all pressures, we find that
iis0:996in the limit of zero pressure, unity at 1 bar,1:004at 2 bar,1:04at 10 bar, and