Page 310 - Modern Analytical Chemistry

1400-CH09 9/9/99 2:12 PM Page 293

Chapter 9 Titrimetric Methods of Analysis 293

for which
+
-
[ HO ][ A ]
3
K a =
[ HA]
Before the equivalence point, and for volumes of titrant in the titration curve’s
–
buffer region, the concentrations of HA and A are given by the following equations.
-
aa -M V
moles HA - moles OH added MV b b
[HA] = =
total volume V a +V b
-
moles OH added MV
bb
[A - ] = =
total volume V a +V b
Substituting these equations into the K a expression for HA, and rearranging gives
+
[ HO ]( MV b )/( a V + V b )
3
b
K a =
( MV - M V b )/( V a + )
b
aa
V b
+
[ HO ] MV b
3
b
=
MV - M V b
aa
b
+
K M V a - K MV b =[ HO ] MV b
b
b
3
a
a
a
a
a
KM V a +
ab =[
- KV HO ] V b
3
M b
Finally, recognizing that the equivalence point volume is
MV
aa
V eq =
M b
leaves us with the following equation.
+
V b ´[H 3 O ]= K a ´V eq – K a ´V b
+
For volumes of titrant before the equivalence point, a plot of V b ´[H 3O ] versus V b
is a straight line with an x-intercept equal to the volume of titrant at the end point
and a slope equal to –K a.* Results for the data in Table 9.5 are shown in Table 9.6
and plotted in Figure 9.13d. Plots such as this, which convert a portion of a titration
curve into a straight line, are called Gran plots.
Finding the End Point by Monitoring Temperature The reaction between an acid Gran plot
and a base is exothermic. Heat generated by the reaction increases the temperature A linearized form of a titration curve.
of the titration mixture. The progress of the titration, therefore, can be followed by
monitoring the change in temperature.
An idealized thermometric titration curve (Figure 9.14a) consists of three
distinct linear regions. Before adding titrant, any change in temperature is due to
the cooling or warming of the solution containing the analyte. Adding titrant ini-
tiates the exothermic acid–base reaction, resulting in an increase in temperature.
This part of a thermometric titration curve is called the titration branch. The
*Values of K a determined by this method may be in substantial error if the effect of activity is not considered.