Page 304 - Physical Chemistry

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v
l
z
where exp z e . Since m° depends on T and P, and m° depends on T, the right side Section 9.8
i
i
of (9.61) is a function of T and P. Defining K as Thermodynamic Properties
i
of Ideally Dilute Solutions
v
l
K 1T, P2 P° exp31m° m° 2>RT4 where P° 1 bar (9.62)
i i i
we have for (9.61)
l
P K x solute in ideally dil. soln., ideal vapor (9.63)*
i i i
Henry’s law (9.63) states that the vapor partial pressure of solute i above an ideally
dilute solution is proportional to the mole fraction of i in the solution.
The Henry’s law constant K is constant with respect to variations in solution
i
composition over the range for which the solution is ideally dilute. K has the dimen-
i
l
sions of pressure. Since the standard-state chemical potential m° of solute i in the so-
i
lution depends on the nature of the solvent (as well as the solute), K differs for the
i
same solute in different solvents.
l
The pressure dependence of K arises from the dependence of m° on pressure. As
i
i
noted previously, the chemical potentials in condensed phases vary only slowly with
pressure. Hence, K depends only weakly on pressure, and its pressure dependence
i
can be neglected, except at quite high pressures. We thus take K to depend only on T.
i
This approximation corresponds to a similar approximation made in deriving Raoult’s
law (9.51).
What about the solvent vapor pressure? Equation (9.59) for the solvent chemical
potential m in an ideally dilute solution is the same as Eqs. (9.42) and (9.43) for the
A
chemical potential of a component of an ideal solution. Therefore, the same derivation
that gave Raoult’s law (9.51) for the vapor partial pressure of an ideal-solution com-
ponent gives as the vapor partial pressure of the solvent in an ideally dilute solution
l
P x P* solvent in ideally dil. soln., ideal vapor (9.64)*
A
A
A
Of course, (9.64) and (9.63) hold only for the concentration range of high dilution.
In an ideally dilute solution, the solvent obeys Raoult’s law and the solutes obey
Henry’s law.
At sufficiently high dilutions, all nonelectrolyte solutions become ideally dilute.
For less dilute solutions, the solution is no longer ideally dilute and shows deviations
from Raoult’s and Henry’s laws. Two systems that show large deviations are graphed
in Fig. 9.21.
The solid lines in Fig. 9.21a show the observed partial and total vapor pressures
above solutions of acetone (ac) plus chloroform (chl) at 35°C. The three upper dashed
lines show the partial and total vapor pressures that would occur for an ideal solution,
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where Raoult’s law is obeyed by both species (Fig. 9.18a). In the limit x chl → 1, the
solution becomes ideally dilute with chloroform as the solvent and acetone as the
l
solute. For x chl → 0, the solution becomes ideally dilute with acetone as the solvent
l
and chloroform as the solute. Hence, near x chl 1, the observed chloroform partial
l
pressure approaches the Raoult’s law line very closely, whereas near x chl 0, the
observed acetone partial pressure approaches the Raoult’s law line very closely. Near
l
x chl 1, the partial pressure of the solute acetone varies nearly linearly with mole frac-
l
tion (Henry’s law). Near x chl 0, the partial pressure of the solute chloroform varies
nearly linearly with mole fraction.
The two lower dotted lines show the Henry’s law lines extrapolated from the ob-
l
l
served limiting slopes of P chl near x chl 0 and P near x chl 1. The dotted line that
ac
starts from the origin is the Henry’s law line for chloroform as solute and is drawn tan-
l
l
gent to the P chl curve at x chl 0. This dotted line plots P id-dil versus x , where P id-dil is
chl
chl
chl
the chloroform partial vapor pressure the solution would have if it were ideally dilute.
l
l
The equation of this dotted line is given by (9.63) as P id-dil K x , so at x chl 1 we
chl chl
chl