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62

RAPID DETERMINATION OF

THE ORDER OF CHEMICAL REACTIONS

FROM TIME-RATIO TABLES

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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technical ^Ttete

62

AUGUST 1960

RAPID DETERMINATION OF

THE ORDER OF CHEMICAL REACTIONS

FROM TIME-RATIO TABLES

Joseph H. Flynn

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RAPID DETERMINATION OF THE ORDER

OF CHEMICAL REACTIONS FROM TIME-RATIO TABLES

Joseph H. Flynn

A table of ratios of reaction times corresponding to 15%/5%, 20%/lOfo, ..,, 95%/$$%

reaction for orders of chemical reaction

from -10 to +10 is presented. Use of this

table allows the rapid calculation of the

order of chemical reactions that are kinetically simple and involves only one

subjective step. Methods are discussed for

interpreting deviations from constant order

resulting from errors in the initial times

and concentrations, from errors in stoichiometry, and from the reaction following

a more complicated rate expression.

INTRODUCTION

The various methods of fitting chemical kinetic

data to rate equations may be divided into two categories-

"differential" methods in which the data are fitted directly to a differential equation expressing the rate,

dx/dt, as a function of the degree of advancement or extent of reaction, x, and "integral" methods in which the

data are fitted to an integrated equation. (1* 2).

Integral methods are more commonly used than differential methods despite the fact that the latter permit a

better interpretation of deviations from constant order,

and can be used to fit data to equations of non-integral

order and to determine initial rates. (1, 2). This situation results to some extent from the tedious determinations of tangents to the curves, x(t), required in the

differential method.

Presented at the 133rd Meeting of the American Chemical

Society, San Francisco, California, April 14,' 1953.

- 1 -
This paper presents a rapid, simple, incremental

method for determining the order of chemical reactions

and may be utilized to detect erroneous assumptions concerning initial conditions and stoichioraetry. In many

cases it provides clues as to the form of complicated kinetic expressions that may be necessary to fit the data.

Also, it minimizes personal bias, as the only subjective

step involved in the method is the fitting of a smooth

curve to a plot of x with respect to time.

This time-ratio table should be a particularly

valuable tool for a chemist with a not too highly developed mathematical background, as its use, consisting of

the application of a set of simple, rapid, non-subjective

steps to a time as a function of concentration curve for

a single kinetic run, can result in considerable information regarding the form of the kinetic equations describing the reaction. Also, the detailed discussion

of the use of this table brings out several features concerning the treatment of kinetic data that are not apparent in the usual "differential" and "integral" methods

of analyses.

If the rate of a chemical reaction is proportional

to powers of the concentrations of a single reactant, or

several reactants present in stoichiometric concentrations, it may be described by the equation

ff

= k(l - x) n

(1)

where k is a constant that may include stoichiometric

coefficients and initial concentrations, and n is defined as the order. Many other reactions may be represented by equation (1) by means of Ostwald's "Isolation"

method. (3).

Equation (1) can be integrated to obtain

ln(l - x) = -kt (n = 1) 0< t < oo (2)

(1 - x)

~ n-l „ _kt ( n ^ i) 0<t<t (3)

1 - n ax

It can be seen from these equations that the ratio of

times for two different percentages of reaction will

have a constant numerical value for a particular value

of n. For example, for n = 3 and t^Q and t£Q, the

times for 60 and 20% completion, respectively,

- 2 -
- kt 60 = ^60 = -4

1" 3 -! -

" kt 20 t20 .g

1" 3

-!

9.333.

This method of time-ratios has been used to determine

integral orders (1) and recently several time-ratios

have been plotted as a function of order to allow an

interpolation of fractional orders (4). The method has

also been used to determine rate constants of simultaneous reactions (5).

Overlapping ratios of successive 10% reaction times,

t 15/'t 5> ^O'^IO' '••» ^Q^^rt^ ^or a number °^ orders are

tabulated in this paper and have been found to be useful,

not only for determining fractional orders, but also for

detecting and interpreting changes in apparent order.

TIME-RATIO TABLES

Values of tx were calculated for x = 0.05, 0,10,

0.15, ..., 0„90, 0.95 and for values of n = -10.0, -5.0,

-4.0, -3.0, -2.0, -1.5, -lo4, ..., -0.1, 0, +0.1, ..., +3.1, 3.3, 3.5, 3.7, 4.0, 4.5, ..., 9.5, 10.0. Timeratios were then calculated. Tables of Natural

Logarithms, Tables of Fractional Powers, and Tables of

First Ten Powers (6) were used in these calculations.

The values were checked to five significant figures by

differencing tests, though only four significant figures

are used in Table I. This is probably sufficient for the

most accurate kinetic data.

USE OF THE TIME-RATIO TABLES

The degree of advancement, x, of a chemical reaction is obtained from the quantitative measurement of

some physical or chemical property which is a known function of the concentration of a reactant or product. The

times for various degrees of advancement can be obtained

from a smooth curve drawn through the points of a plot of

x as a function of t. (If the initial time is unknown,

x may be calculated from any arbitrary zero time on a

curve representing the concentration as a function of

time.) The ratios of the ty values can be compared with

those in Table I and the order thus obtained. If the

reaction follows equation (1), and accurate initial concentrations and time are used, the values for the order,

n, determined from successive time-ratios should remain

constant.

- 3 -
Table I is represented in graphical form in

Figure 1.

The sensitivity of the time-ratios to the order

may be defined by

1 _____ A (time-ratio value)

(time ratio value

)

ave An

where A indicates an incremental value. It can be seen

from Figure 1 that if all time-ratios are determined with

equal relative precision, the sensitivity of these ratios

increases with increasing order and degree of advancement.

DISCUSSION

The greatest value of the time-ratio table lies in

its applicability to kinetic expressions more complex

than equation (1), and the discussion of this table will

involve the analysis of some of the more easily interpretable causes of deviation from constant order. The

effect on the order of errors in the times, concentrations, and, stoichiometric relations will be discussed,

and some consideration will be given to the deviations

caused by the reaction rate following a more complicated

kinetic law than equation (1).

A stoichiometrically pure chemical reaction may be

represented by the equation

aA + bB + cC + ,. r uU + vV + wW + ... (4)

where small letters represent stoichiometric coefficients

and capital letters represent concentrations of molecular

species. For such a system the stoichiometric coefficients are independent of the time. The combination of

the time derivatives of the relationships expressing the

conservation of each elementary species gives

(5)

1 dA = 1 dB 1 dC _ . . . _ 1 dU _ 1 dV _ 1 dW = a dt b dt

' c dt u dt v dt w dt

whenever the concentrations of all intermediates are

negligible or their concentrations change slowly with

time (steady state).

- 4

-
Here we are interested only in the case where the

reactants that contribute to the order of the reaction

are present in stoichiometric amounts, the coefficients satisfy the relation

1a-$b-1c«... (6)

Then, if the rate of disappearance of a reactant, A,

is proportional only to powers of the concentrations

of one or more reactants,

-ii = K®

q

(f)

r- AP + q

+ r + - <?>

In terms of the concentration of a product, U,

equation (7) will have the form

where the subscript, o, indicates an initial concentration.

If at t - , A = A , U = ; and at t - oo , A^ = ,

Uqo = ii A , then n, k, and x of equation (1), in terms

a ° of the parameters of eqs. (7) and (#), will be given by

n=p+q+r+... (9)

*--ffl

, ft)

r

...-r

,

* r+ "'- 1

<">

x = 1 - x (reaction followed by change in con- (11a)

o centration of reactant, A, that contributes to the order)

x = tt- ( E - E) (reaction followed by change in

o concentration of a reactant, (lib)

E, that does not contribute to

the order)

- 5 -
a 5L , x ~ u A (reaction followed by change in concentration of a product, U) (lie)

Errors in Initial Time and Concentration . If the

reduced times, / . = kt. , are in error because the

initial / differ from the correct value by an amount

± € y , then instead of determining i^/^l' one W^-H

actually determine n from

r'2 x2 ±€

r

(1 - x

2

)1

"n

- 1 ± (i - n

^

y

r'

=

r. +- " Ihi

1 1 " € 7 (1 - Xl ) -1± (1 - n)€ v

If an error of ± € . is present in the value used

for the initial concentration, A

Q

, then instead of

A

x_ = 1 - t~> (equation 11a), one will have

a A

o

x' = 1 - _J± and (1 - x ') = t-t^tt-

(1 ~X

a>- {12)

*T

The apparent orders, n

!

, calculated from Table I

for errors in initial time for various n, were almost

identical to the n f € calculated from errors in initial

concentration if - A = €^«i # The difference between

Ao the apparent and true order, n' - n, is found to

depend only slightly on n. Deviations from the true

€ A

order for € - € y

= - j— = + 0.1, ± 0.01, ±0.001, are

plotted in Figure 2 as a function of m, the percentagereaction index in the denominator of the time-ratio

t m + lO/tm. The index, m, varies from 10 to 80 and

these curves represent average values for various n

calculated for this interval in m.

- 6 -
As is to be expected, the deviations from true

order resulting from errors in initial time or concentration are larger for early time-ratios. At later

ratios, the deviations are relatively small.

An induction period, or an initial period of

rapid reaction, which may be too short to be recognized by other methods of kinetic analysis, will appear

as an error in the initial time or concentration.

Therefore, an apparent error in the initial time or

concentration, as shown by abnormal n, that is too

large to be ascribed to experimental error, can be

interpreted as evidence for initial irregularities in

the rate. The calculation of a new degree of advancement from an arbitrary point on the curve representing

the concentration as a function of the time should

eliminate this effect, except for cases where the concentration of a product is obtained and the error in

initial concentration is due to the determination of

an erroneous value for the concentration of a product

at infinite time.

Errors in Stoichiometric Relationships . In cases

where more than one reactant contributes to the order

of the reaction, the reactants must be present in

stoichiometric concentrations if the kinetics are to

be described by equation (1). The case where one of

the contributing reactants is present in a nonstoichiometric concentration will be considered subsequently.

In cases where equation (1) is followed, errors in

the determination of the degree of advancement may result from incorrect assumptions about the sto'ichiometry

of the reaction. A similar situation could result from

a stoichiometric error in the function relating the

measured physical or chemical property with concentration.

If there is -a constant stoichiometric error, € a , in the stoichiometric coefficient, a, in a reaction

investigated by the change in concentration of a product,

then the degree of advancement as defined by eq. (lie)

( a + € \ U will have the value, x ' = §J. _a and

A

o

(1 - x ') - (1 - xj (! + -£&)- _£a (13)

- 7

-
The deviations from the true order, n' - n,

calculated by means of equation (13)> are approximately

inversely proportional to n. A plot of (n* - n)/n

as a function of the index, m, (m = 10, 20, ..., SO)

for stoichiometric errors of € = —fLa = + 0.1 and

a

± 0.01, results in families of points that may be

approximated by the curves in Figure 3*

The deviations from the true order for later timeratibs may be quite large for stoichiometric errors of

this type.

If there is an error, € g, in the concentration

of a reactant, B, which does not contribute to the

order, and B is experimentally measured, then the

degree of advancement as defined by equation (lib) will

have the value

*a' " ^f-

(B ±.«B " B

a>> and

(l-O-d-x,)-^* (H)

These deviations from true order are quite similar

in form to those obtained for errors in A (Figure 2),

except for increasing positive order, the deviations

change sign and become large in magnitude as m

approaches 90.

Deviations from Constant Order Resulting from a

More Complicated Rate Expression . A large number of

chemical reactions that give a changing value for n

when represented by equation (1) may be better represented by

4^ -* (1 - x)M (l - ax)*/ (15)

dt

where k , GL , ll , and v may be considered in

general as arbitrary parameters.

A few examples of reactions that are well represented by equation (15) are:

- 8 -
a) first order reversible reactions ( JUL = 0,v = 1, 0C>1)

b) parallel reactions: one zero order and one first order

(/X= 0, V = 1, 1 > 0C > 0)

c) parallel reactions: one first order and one second order

(fJL= 1, v = 1, 1 > 0C > 0)

d) Hydrogen bromide formation {ll = 5/2, V = -1, 0C > 0)

e) many catalytic reactions {v > for positively catalyzed

reactions, and v < for negatively catalyzed

reactions). If a reaction is catalyzed by a reactant,

CC > 0: if catalyzed by an intermediate or product,

0C < 0).

f

)

reactions in which two or more reactant s contribute to

the order in proportion to powers of their respective

concentrations and one of these reactants is present in

a non-stoichiometric concentration.

Equation (15) was solved for values of t corresponding to x * 0.05, 0.10, ..., 0.90, 0.95;OC= 3, 2, 1.5, 1.2,

1.1, 1.0, 0.9, 0.5, 0, -0.5, -1.0, -2.0, -5.0, -10.0, -00;

and various combinations of V m ±3, ±2, ±1, and ll = ±3,

±2, ±1, 0. These computations were performed with the

Analog Computer Facility of the National Bureau of

Standards for the equations involving cubic s in ll and V . Values for n

f , the apparent order as defined by

equation (1), were obtained by means of Table I from the

time-ratios that were calculated from these data. The

function, n " «= was found to be relatively independent

of ll and V . Tables II and III demonstrate that even

for large absolute values of 0C(CC= +2.0 and -5.0),

n 1 - U~— as a function of m falls into two sets — one

for y > 0, the other for V ^ 0. The mean curves for

V > o are shown in Figure 4 and for v < in Figure 5.

In the common case I v I = 1, a curve n'(m) can

be matched with a curve in Figures 4 or 5 by a displacement

/ll units along the ordinate. Thus the parameter, c£ , can be

determined. por Jz/| ^ 1, curves 21* (m) may be tested

- 9

-
(1 - x) is given by

n

x

= P

+ V

OC (1

1 - oC

x)

X

for various values of V . The estimates of cC > ^ and

v obtained by these procedures may be used as guides for

more precise curve fitting.

Thus the time-ratio method allows for a treatment of

kinetic data similar to that which may be performed in the

differential method of analysis (7) where the slope, nx , dv determined from a plot of log ~ as a function of log

(16)

with the limiting characteristics

lim nx

= /J. + OC v (17)

x —*

lim n =f± ( OC < 1) (Id)

x -* 1

and the function x "" H— vs. x for various oC can be

V

used to estimate values for OC > M tV an(i\

LITERATURE CITED

(1) Letort, M., J. Chim. Phys. 2k» 206 (1937).

(2) Laidler, K. J., "Chemical Kinetics", McGraw-Hill

Book Co., Inc., New York, N. Y. , 1950, pp. 15-18.

(3) Ref. (2), page 16.

(4) Chang, W. , J. Phys. Chem. 61, Sl9 (1957).

(5) Frost, A. A., and Schwemer, W. C, J. Am. Chem. Soc.

2t, 1268 (1952).

(6) "Mathematical Tables Project", Lowan, A. N., Director,

1941-1946, New York, N. Y.

(7) Flynn, J. H. , J. Phys. Chem. 6l, 110 (1957).

- 10 -
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n - n

Table II

as a function of m for<=<= + 2.0 and various v and p.

10 20 30 40 50

-2 + 2 2.28 2.59 3.33

1 2.28 2.60 3.28

2 2.28 2.63 3.42

3 2.29 2.66 3.42

+ 1 1 2.28 2.61 3.32

1 2 2.28 2.64 3.46

2 1 2.28 2.62 3.36

Avg ( v > 0) 2.28 2.62 3.37

-1 -1 2.28 2.56 3.12 4.71

-1 2.27 2.56 3.07 4.49

-2 2.26 2.54 3.31 4.58

-3 2.30 2.66 3.12 >3.33

+1 -1 2.28 2.56 3.10 4.60

+2 -1 2.27 2.56 3.12 4.71

+2 -2 2.27 2.55 3.04 4.73

Avg ( v < 0) 2.28 2.57 3.12 4.64
Table III

n " ^ as a function of m for °< = -*~~2t0" and various v and |i. V n vx 10 20 30 40 50 60 70 80

+ 1 (3.5) (2.5) 1.90 1.54 1.24 .97 .76 .55

2 (3.3) 2.45 1.96 1.60 1.32 1.06 .83 .61

3 (3.3) 2.66 2.25 1.65 1.36 1.13 .91 .67

+1 1 (3.3) (2.4) 1.87 1.48 1.18 .92 .69 .47

1 2 3.15 2.57 1.91 1.59 1.25 .99 .76 .54

2 1 3.06 2.35 1.84 1.44 1.12 .85 .61 .38

+ 3 + 3 (3.5) (2.5) (2.0) (1.6) (1.29) (1.03) (.81) (.70)

Avg (v> 0) (3.3) 2.49 1.96 1.55 1.25 .99 .76 .54

-3 -3 (3.2) (1.8) (1.6) (1.3) (1.08) (.83) (.68) -

-2 -1 3.04 2.32 1.82 1.46 1.17 .94 .73 .55

-1 -2 2.99 2.25 1.73 1.33 1.08 .83 .63 .45

-1 -1 3.02 2.30 1.79 1.42 1.12 .89 .68 .50

-3 2.94 2.17 1.64 1.25 .95 .72 .52 .36

-2 2.98 2.23 1.70 1.31 1.01 .77 .57 .39

-1 3.01 2.28 1.76 1.38 1.08 .83 .62 .43

+1 -1 3.00 2.26 1.73 1.34 1.03 .77 .56 .36

+2 -2 2.95 2.18 1.64 1.24 .92 .67 .46 .27

+ 2 -1 2.99 2.23 1.70 1.30 .98 .72 .50 .23

Avg (v<0) 2.99 2.25 1.72 1.34 1.04 .79 .59 .39
ORDER (n)

Fig. 1. A family of curves representing the dependence of

time-ratios on the order and extent of reaction.
+3

tf+\ -

I

c

I

-I

-2 -

n-,

—

1 11 1 1

\€ = +0.l

1 1 1 1

—

€= + 0.01 \

—

—

€ = + 0.001

^^

€=-0.001 ^^-"~~""

—

€ = -0.01 / —

/€ = -0.l

1 / 1 1 1 .1 1 1 «L

o —

30 50 70 9.0

m

Fig. 2. Effects of errors in initial time or concentration

on the apparent order of reaction. Deviations from

true order, n' - n, are plotted as a function of

time-ratio index, m. The error in reduced time

or concentration is € = k € =

t

- €A

A,
+0.6

0.4

0.2 -

c

J

-

-0.2 -

-0.4

30 50 70 90

m

Fig. 3. Effects of constant stoichiometric errors (product

measured) for an nth order reaction. The function

(n* - n)/n is plotted as a function of the timeratio index, m. The error in the stoichiometric

coefficient, a, is € = .la

a
I

m

Fig. 4. A family of curves showing the dependence of the

function

Curves f

coincide.

on the time-ratio index, m. Curves for < v < 3, I /^.l < 3 almost
m

Fig. 5. A family of curves showing the dependence of the

function -

" c~ on the time-ratio index, m. Curves for -3 < ^ < 0, |/l|<3 almost

coincide.

USCOMM-DC-60792

U.S. DEPARTMENT OF COMMERCE

Frederick H. Mueller, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

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