Concise Physical Chemistry

c10 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

146 CHEMICAL KINETICS

Some chemical reactions follow the same mathematical laws as radioactive decay.

The rate equation in these cases is afirst-order differential equation, so the reactions

that follow it are calledfirst-order reactions. A case in point is the decomposition of

a dilute solution of hydrogen peroxide H 2 O 2 in contact with finely divided platinum

catalyst. The probability arguments given above apply equally to radioactive decay

and to first-order chemical reactions. In the case of the decomposition of a dilute

solution of hydrogen peroxide in contact with Pt, the reaction is first order with a

half-time of about 11 minutes.

The half-time can be determined for a first-order reaction in a simple and illus-

trative way. In the logarithmic form of the first-order equation above, setX=^12 X 0 ,

which is the definition of the half-time. Now

ln

X

X 0

=ln

1

2

1

=−kt (^12)
kt 12 =−ln
1
2
1
=ln

1

1

2

=ln 2= 0. 693

and

k= 0. 693 /t (^12)
The half-time is found by drawing a horizontal halfway between the base line and
X 0. The horizontal intersects the experimental curve att^12. Notice that the con-
centration does not enter into calculation of the rate constant. Knowing either the
half-time or the rate constant gives you the other one. Generally the half-time is
easier to observe.
The exponential decay ofXwithtis seen in Fig. 10.1. The half-time is seen there
as well. The linear decrease of lnXwithtis seen in Fig. 10.2.
The linear plot of lnXis usually used to obtain the rate constant because it
averages many experimental points and is, presumably, more reliable than any one
X
X 0
t t
1/2
FIGURE 10.1 First-order radioactive decay.