5. 0. Smith' and I. D. R. Stevens2 University of California Los Angelen
Nomographs for First and Second Order Kinetics
In the study of reaction kinetics it is often necessary and useful to estimate rapidly the relation between a given per cent reaction, reaction time, and rate constant. For this purpose we have found the nomographs in Figures 1 and 2 more useful Research supported by the National Science Foundation. Present address: Noyea Laboratory, University of Illinois, Urbttns, Ill. Present address: TJniversity of Southampton, Southampton, England.
than the more general ones available in the literature.3-' Figure 1 applies to reactions that follow first order kinetics; its use is readily seen from the following example. In the solvolysis of p-methoxyneophyl-ptoluenesulfonate in acetic acid at 50.0°,it was observed a
MACHWART, O. M.,
AND
Qnr~~cr, R. E., Ind. Eng. Chem.,
48,1194(1956),
' NORD,M., Chem. Ind.,63,666
(1948); 64,280,636 (1949). SCHWARZ-BERGKAMPF, E., A f t t . Chem. Forsch-Insl. Wirtschaft. Oslerr., 6,31(1952). 100 hrs
% REACTION
n
FIRST ORDER R A T E
70
C O N S T P N T (SEC,-')
3
--
2
--
60 rnin
-' 60
--
--
40 30
..20
Figure 1.
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Nomograph for flrst order reactions.
Journal of Chemical Educafion
that, after 15 minutes, 10.3% of the theoretical acid had been developed. By laying a straightedge across the nomograph joining these two points, one estimates a first order rate constant of 1.2 X lo-&sec-'. One is now in a position to program the pulling of samples to achieve a run of maximum utility. Thus for points a t 25, 35, 45, 55, 65, 75, and 85% reaction one can estimate times 40 min, 1 hr, 1 hr 25 min, 1 hr 50 min, 21/1 hr, 31/4 hr, and 4'/2 hr. Figure 2 is for use with second order kinetics and corresponds to the two special cases of equimolecular reactants and reactants in 2 to 1 stoichiometric ratio. The numbers on the left hand side of the per cent reaction line are for use in reactions using equal concentrations of reactants, and those on the right hand side are for reactions in which one reactant is present in twice the concentration of the other. This second order nomograph is a little more difficult to use than the first order one in that two lines, intersecting on the
center line, F(% Rx, Co), are required. For example, in the reaction of tetra-n-butylammonium bromide with 2-phenyl-l-propyl-p-toluenesulfonatein dry acetone a t 75.0°, it was observed that, for initial concentrations of 0.04 and 0.02 M respectively, 17.5% of the inorganic bromide had been consumed after 15 minutes. A straightedge is laid on the nomograph connecting 17.5% reaction (on the 2 : l scale) and 2 X on the initial concentration scale and intersecting the F (% Rx, Co) scale. A second straightedge is then used to connect this point to the observed time (15 minutes) and an estimate for kz of 6 X 10W3 liter/mole-sec is obtained. As pointed out above, the process can now be reversed so that aliquots may be taken a t the optimum times. For second order reactions with initial reactant ratios between 1 :1 and 2 :1, it is possible to interpolate between the two scales, though with larger errors (e.g., 2% reaction for 1.5:l falls roughly halfway between
2:1
% REACTION
INITIAL CONCENTRATION
SECOND O R D E R
RATIO OF
OF L I M I T I N G R E A G E N T
RATE CONSTANT
REACTANTS
MOLES / LITER
TIME
LITERS/MOLE-SEC
hrs
90
100
+ 4
3
2
..
.-
80
-- 6 0 -- 5 0
.-
-- 4 0
--
80 6 --
70
.. 2 0 -- 1 5
-3.. lo-! 2 -. 4
-- I 0
60 50
---
-- 4
: : lo-2 4 2
..
30
6
.-
4
--
3
..2
----mi0 n0
:: IO-3
50
-.4 0
2
..
-- 3 0
I 8 6
--::
---
20
15
8
-6
6-
40
30
:
4 2
--
15
.-I 0 -- 8 --- 5
IO-4
6
10 I Fig 2. Nomograph for second order reactions. One line connects Fl%Rx, Col,the second order rate constant,and time.
20
1 4
% reaction, initial concentration, m d FI%Rx,
Col.
A resond line conned*
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the correspondimg points for 1 :1 and 2: 1). For ratios much larger than 2:1, second order reactions follow first order kinetics to quite good approximations. A further useful feature of both nomographs depends on the logarithmic nature of the time and rate constant scales; they can be easily extended. Thus times shorter than 6 minutes or longer than 100 hours may be
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Journal of Chemical Education
employed by dividing or multiplying the numbers on the t i e scale by 10 (e.g., 10 hours becomes 1 hour or 100 hours) and correspondingly multiplying or dividing the numbers on the rate constant scale by 10 (e.g., 3 X lo-& becomes 3 X or 3 X 10-5). For the second order nomograph, a similar relation holds between kz and initial concentration.
