Page 314 - Physical Chemistry

lev38627_ch10.qxd 3/14/08 1:07 PM Page 295

295
The defining equation (10.3) for a is chosen to resemble (10.2) for ideal and ideally Section 10.1
i
dilute solutions, so as to lead to a nonideal m expression that can be readily compared Activities and Activity Coefficients
i
with (10.1). Taking logs of (10.3), we get ln a (m m°)/RT, or
i
i
i
m m° RT ln a every soln. (10.4)*
i
i
i
Thus, the activity a replaces the mole fraction x in the expression for m in a non-
i
i
i
ideal solution. From (10.1) and (10.4) we see that a x in an ideal or ideally dilute
i
i
solution. When solution component i is in its standard state, m equals m° and, from
i
i
(10.3), its activity a equals 1 (a° 1).
i
i
The difference between the real-solution chemical potential m in (10.4) and the
i
id
corresponding ideal-solution m in (10.1) is
i
id
m m RT ln a RT ln x RT ln 1a >x 2
i
i
i
i
i
i
The ratio a /x is thus a measure of the departure from ideal behavior. We therefore
i
i
define the activity coefficient g (gamma i) of component i as g a /x , so that
i
i
i
i
a g x every soln. (10.5)*
i
i i
The activity coefficient g measures the degree of departure of substance i’s behavior
i
from ideal or ideally dilute behavior. The activity a can be viewed as being obtained
i
from the mole fraction x by correcting for nonideality. In an ideal or ideally dilute
i
solution, the activity coefficients g are 1. From (10.4) and (10.5), the chemical
i
potentials in a nonideal solution of nonelectrolytes are
m m° RT ln g x (10.6)*
i
i i
i
Since m depends on T, P, and the mole fractions, the activity a in (10.3) and the
i
i
activity coefficient g a /x depend on these variables:
i
i
i
a a 1T, P, x , x , . . .2, g g 1T, P, x , x , . . .2
i
i
2
1
2
1
i
i
Note from (10.3) and (10.5) that a and g are dimensionless and nonnegative.
i
i
The task of thermodynamics is to show how a and g can be found from experi-
i
i
mental data; see Sec. 10.3. The task of statistical mechanics is to find a and g from
i
i
the intermolecular interactions in the solution.
The activity a of species i is a e m i >RT m i °>RT [Eq. (10.3)]. If the composition of
e
i
i
the solution is varied at fixed T and P, the factor e m i °>RT remains constant and a varies
i
in proportion to e m i >RT . The activity a is a measure of the chemical potential m in the
i
i
solution. As m increases, a increases. If we add some of substance i to a solution
i
i
at fixed T and P, the chemical potential m must increase [Eq. (4.90)]. Therefore,
i
constant-T-and-P addition of i to a solution must increase the activity a . Like the
i
chemical potential, a is a measure of the escaping tendency of i from the solution.
i
The activity a is more convenient to use in numerical calculations than m i
i
because (a) we cannot determine absolute values of m (only relative values);
i
(b) m → q as x → 0; (c) a can be compared with x (and g with 1) to judge the
i
i
i
i
i
degree of nonideality.
Standard States for Nonideal-Solution Components
To complete the definitions (10.3) and (10.5) of a and g , we must specify the stan-
i
i
dard state of each solution component. Two different standard-state conventions are
used with Eq. (10.6).
Convention I For a solution where the mole fractions of all components can
be varied over a considerable range, one usually uses Convention I. The most common
case is a solution of two or more liquids (for example, ethanol plus water). The