Page 24 - Partition & Adsorption of Organic Contaminants in Environmental Systems

RAOULT’S LAW 15

than to solutes in dilute solution. When the solute and solvent are similar sub-
stances that mix with a minimal thermal effect, the activity or partial pressure
of the solute at a given mole fraction may also be approximated, usually with
less accuracy, by applying the ideal-solution assumption. When the beha-
vior of the solute deviates significantly from this ideal state, it is expressed by
an activity coefficient, which is attributed to a mixing thermicity. In essence,
Raoult’s law gives no account of the effect of the molecular-size disparity
between the components in a solution on individual component activities.
By Raoult’s law, the partial pressure of component i in solution may be
expressed in the following form:
o o
P i = P x i g i or a i = P P i = x i g i (2.1)
i
i
where P i is the partial pressure of component i at mole fraction x i, P° i the
reference-state vapor pressure of pure substance i at the same temperature T,
g i the activity coefficient of component i at x i, and a i, as stated earlier, is the
activity of component i at x i. Here the reference-state vapor pressure for a
liquid or a solid substance at temperature T is simply the saturation vapor
pressure of the pure liquid or the supercooled liquid at T. For the solvent, in
which x i is close to 1, the solvent g i should approach 1 according to the law, as
is generally found when the solute and solvent have similar molecular sizes. If
the solute behaves ideally, then g i = 1 and hence a i = x i. For a solid solute, one
can thus calculate its ideal mole fraction solubility in a solvent based on the
s
s
s
calculated activity of the pure solid, a i = P i /P° i , where P i is the vapor pressure
of the solid at T. Although the model holds when the solute and solvent have
similar sizes and compositions, it does not hold, as shown later, if one of them
is a macromolecular substance, even in the absence of a thermal effect.
To the extent that Raoult’s law is obeyed,a graphic illustration of the behav-
ior of a component (i) in a solution is depicted in Figure 2.1, in which the
partial pressure (P i) or fugacity (f i) of the component is plotted against its
mole fraction (x i) in solution. It is assumed here that component i is a liquid
completely miscible with the solvent at system temperature T. We shall con-
sider the case for a solid solute later. In Figure 2.1, the straight line between
the origin and the P° i in the ordinate (at x i = 1) is the ideal-solution line for
component i (i.e., where g i = 1 at all x i). The upper curve is for a nonideal
system, where component i exhibits a positive deviation from ideality (i.e.,
g i > 1). The lower curve is for another nonideal system, where the component
i exhibits a negative deviation from ideality (i.e., g i < 1). If g i > 1, as for most
systems, the compatibility between molecule i and other molecules is consid-
ered to be less than that between i molecules; if g i < 1, the reverse is the case.
The latter condition applies for rare systems where specific interactions (e.g.,
complexation) occur between i and other components.
As depicted in Figure 2.1, whereas g i = 1 applies over all x i if the solution is
ideal, one sees that g i always approaches 1 as x i Æ 1 if component i is com-
pletely miscible, whether the solution is ideal or not at other x i . In other words,