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r 2 = k [ Mn ( II ) ][ MnO x ] .
(5.160)
The overall rate of the auto-catalyzed reaction is
d [ Mn ( II ) ]
d t
= (k + k [ MnO x ] ) [ Mn ( II ) ] .
r =−
(5.161)
Upon integration using the boundary condition [MnO x ] 0 =
0 we obtain
(k [ Mn ( II ) ] 0 + k) e ((k + k [ Mn ( II ) ] 0 )t)
1 .
[ MnO x ]
( [ Mn ( II ) ] 0 −[ MnO x ] ) =
k
(5.162)
If k [Mn(II)] 0 > k , we can approximate the above equation and rearrange to obtain
ln
= ln
[ MnO x ]
[ Mn ( II ) ] 0 −[ MnO x ]
k
k [ Mn ( II ) ] 0
+[ k + k [ Mn ( II ) ] 0 ] t .
(5.163)
Thus, a plot of ln [ MnO x ] /( [ Mn ( II ) ] 0 −[ MnO x ] ) versus t will give at suffi-
ciently large t values, a straight line slope of k + k [Mn(II)] 0 , and an intercept of
ln (k/k [Mn(II)] 0 ) from which the values of k and k can be ascertained. The data for
Mn(II) oxidation at a constant partial pressure of oxygen and different pH values were
reported by Morgan and Stumm (1964), and are plotted in Figure 5.16. The linear fit to
the data shows the appropriateness of the rate mechanism given above.
0
y = 0.0212 x - 2.0956
R 2 = 0.9912
-0.5
-1
-1.5
-2
-2.5
0
20
40
60
80
100
t / h
FIGURE 5.16 Kinetics of autooxidation of Mn(II) in alkaline solutions at pH = 9.8
at 298 K. [ A ] 0 = 8 × 10 5 M; P O 2 = 1 atm. (From Morgan, J.J. and Stumm, W. 1964.
Proceedings of the Second InternationalWater Pollution Research Conference , Tokyo,
Elmsford, NY: Pergamon Press.)
 
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