A SIMPLE KINETIC MODEL FOR SINGLE AND CONSECUTIVE FIRST ORDER REACTIONS
0
MILTON KAHN University of New Mexico, Albuquerque, New Mexico
Tm following kinetic scheme A-B-C
is commonly found. I t may, for example, represent a radioactive series in which substance A decays to form substance B, which in turn decays to form substance C; C may be radioactive or stable. If, for instance, N represents the number of atoms of a radioactive substance a t any time t, it can be shown' that and
vhere No is the number of atoms of a radioactive substance present a t time t = 0 ; X is the decay constant and is related to the half-life, by the equation: X = 0.69315/t1,,. If the half-life of A is very long compared with that of B, equation (2) may be approximated by
.."
'
FRIEDLANDER, G., AND J. W. KENNEDY, "Nuclear and Radiochemistry," John Wiley and Sons, Inc., New York, 1955, p. 129. TO MANOMETER
TO MWOMETER
It is the purpose of this paper to describe an experiment whereby the validity of equations (I), (2), and (3) can be demonstrated without resorting to radioactive materials or actual chemical experimentation. Consider the apparatus shown in Figure 1. Stopcock K was constructed from a hollow plug, three-vay Eck and Krebs 3-mm. stopcock. The short bore was drilled to expose the entire volume of the plug to the air. The long bore and the greater part of the plug was filled with Apiezon wax W until a space of only about 1 ml. remained in the plug. . This volume which corresponds t o AV in the following discussion is accessible only through a hole that marks the origiual point of exit of the short bore. Initially the entire system is filled with air a t atmospheric pressure. By proper manipulation of stockcocks S, S', and G, sections B and Care evacuated with A remaining at atmospheric pressure. Section B is then isolated from C by manipulation of S'. It is clear that continued rotation of K in a clockwise direction will result in a transfer of air from A to B and from B to the pump (C). The manometers were constructed of 2-mm. bore capillary tubing in order to minimize the slight change in volume of sections A and B as the pressure changed. Assuming room temperature to be constant during the experiment, it follo~vsthat after one 360" rotation PlVl
=
+
(PA A P A ) ( V A
+ AT')
(4)
where PA is the pressure of air in section A prior to exposure to the evacuated space AV; V Ais the volume of A excluding AV; PAis the change in P A . Because AV is small compared with V,, rearrangement of equation (4) yields
WOML BULB
AP, = --(AV/I~,)PA
TO AIR
(5)
Now in the limit AP, = dP,/dt, if one 360' rotation represents an interval of time, dt. Therefore,
-3
Integration of equation (6) yields Q, three-way stopcock; K,3-mm. 2-mm. straight-bore stopcmlta.
three-way hollow-plug stopcock:
S, S'.
+!,
PA = pneV*
(7)
JOURNAL OF CHEMICAL EDUCATION
IIb, 0 2
, , , , 4
6
\\.here INAis the pressure in A at time t = 0. In the esperiments described below, twenty-five 360' rotations represented an interval of time t. The similarity between equations (1) and (7) is apparent. Thus, AV/VA in the gas system is equivalent to X , in the radioactive system, and PAcorresponds to N,. The analogy between AV/V and A can he seen by considering that since AV/V
