A New Method for Determining the Absolute Rate Constants of

26 A New Method for Determining

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the Absolute Rate Constants of Autoxidation of Some Hydrocarbons H . B E R G E R , A. M . W . B L A A U W , M . M . A L , and P. S M A E L Koninklijke/Shell-Laboratorium, Amsterdam, The Netherlands

When a slow steady-state autoxidation of a suitable hydro-carbonis disturbed by adding either a small amount of inhibitor or initiator, a new stationary state is established in a short time. The change in velocity during the non-steady state can be followed with sensitive manometric apparatus. With the aid of integrated equations describing the non-steady state the individual rate constants of the autoxidation reaction can be derived from the results. Scope and limitations of this method are discussed. Results obtained for cumene, cyclohexene, and Tetralin agree with literature data.

H p h e autoxidation at moderate temperatures of hydrocarbons, including inhibition by a hindered phenol, for example, is generally described by the following mechanism: Rate Initiation Propagation

—> RR- + 0 -> R O O 2

ROO- + R H - » R- + R O O H Termination Inhibition

2 R 0 · - » Non-radical products 2

R

{

Fast & [RH] [ R 0 ] = P

2

2k [ R 0 · ] 2

t

^ dt

2

R 0 - + A H —> R O O H + A- Î 2k [AH] [ R 0 - ] A- + R 0 · -> AOOR j Fast 2

A

2

2

(We adhere to the convention, also adopted by Walling (5) that rate constants of reactions which consume two radicals are written as 2k. To avoid confusion, the factor 2 is retained in Table I and in the equations. ) 346 Mayo; Oxidation of Organic Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

26.

BERGER E T AL.

Absolute Rate Constants

347

Only ratios of rate constants of autoxidation can be determined by ordinary steady-state velocity measurements. Non-steady state methods, notably the sector method, have been used to evaluate the individual or absolute rate constants. W e have devised a method for measuring the change i n rate of autoxidation under non-stationary conditions directly, by means of sensitive manometric apparatus, and for deriving individual rate constants from these measurements. In our method reproducible non-stationary states are effected as follows: the low stationary-state rate of an autoxidizing hydrocarbon is decreased by a factor of 2 to 5 by adding an appropriately small amount of inhibitor. Under the conditions outlined below, the time required to establish the new stationary state at the inhibited rate is not immeasurably small, as it would be i n conventional measurements, but of the order of 100-300 sec. W i t h sufficiently sensitive apparatus a number of determinations of the decreasing velocity can be made, which delineate the course of the non-steady state. Similarly a non-steady state with an increasing velocity can be realized by introducing a small amount of initiator. Experimental Apparatus and Procedure. The reaction vessel is a cylindrical borosilicate glass flask with ground glass joint, height 30 mm., diameter 75 mm. ( Figure 1 ). It is equipped with four side baffles projecting 15 mm. into the flask with a 45° slant upward, relative to the direction of stirring. Connections to the pressure transducer are made first b y means of a solid ground glass stopper with a wide (5 mm.) channel, so as to avoid blocking by liquid drops, and further with capillary tubing, submerged as far as possible i n the thermostat bath. Also provided are capillary side connections to a conventional gas buret (closed during non-steady state measurements), to a well-lagged Metrohm piston buret, equipped with a step-geared synchronous motor, and to a closely fitting rubber serum cap which allows insertion of the 250-mm. needle of a 50 /Jiter Hamilton syringe down to a few millimeters over the magnetic stirring bar. The magnetic stirring motor is fed from a constant-voltage transformer since varying the stirring speed varies pressure, presumably owing to changes i n centrifugal pressure on the gas bubbles. The thermostat is a lagged glass basin, the water temperature being regulated b y means of a Thermotrol controller and an efficient stirrer. Temperature constancy in our runs was 0.002°C. (range 20-50°C. ). The differential pressure transducer is a model PID-0.1PSID with model C D U carrier demodulator (Pace Engineering Co., North Hollywood, C a l i f . ) . It is operated at 0.4 maximum sensitivity and is connected to a standard potentiometric recorder using the 200-mv. range. The reference side of the transducer is connected to an empty flask submerged in the thermostat (Figure 2 ) . The reaction vessel is filled with hydrocarbon, leaving only enough gas volume to allow formation of a vortex down to the stirrer, so that gas

American Society

Chemical Library

1155 16th Compounds St. N. W. Mayo; Oxidation of Organic Advances in Chemistry; American Chemical Society: Washington, D. C. Washington, 20036 DC, 1968.

348

OXIDATION

O F ORGANIC

COMPOUNDS

1

bubbles are swept through the liquid i n both horizontal and—because of the 45° baffles—vertical swirls, enabling a rapid exchange of gas between vortex and bubbles. The volume of the vortex and the required length of the flask's neck determine the volume of the gas space (ca. 20 m l . ) , i n cluding capillary lines and Metrohm buret. A rate of oxygen uptake of 0.002 ml./min. produces a 20-mv. change of output i n 20 sec. The "noise" from the stirred liquid determines the limit of sensitivity; the latter was influenced favorably also b y the baffles which contribute to a smooth pattern of flow. MICROLITER IJI SYRINGE ^

c

Figure 1.

I TO V E N T , METROHM B U R E T AND PRESSURE TRANSDUCER

Reaction vessel with 45° baffles

During measurement of the reaction rates the pressure drop is recorded in 20-mv. steps, the original pressure being rapidly re-established by the Metrohm buret. In this way one measures under essentially constant pressure; this is necessary since otherwise the change in equilibrium oxygen content of the solution would interfere with the measurement. The setup is calibrated by allowing the Metrohm buret (with geareddown motor) to introduce oxygen at a known rate of the same order of magnitude as that which is being measured. F r o m the apparent rate so measured one obtains a factor for the conversion from mv./sec. to m l .

Mayo; Oxidation of Organic Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

26.

BERGER ET

AL.


349

0 /sec. If rates do not come near the limit of sensitivity a narrow gas buret may also be used, but this is more time-consuming. To measure a reaction in its non-steady state, the injection needle is mounted through the serum cap, the syringe being filled with the required amount of inhibitor or 2,2'-azoisobutyronitrile, dissolved i n the hydrocar bon. The initial steady state velocity is measured, reference and reaction vessel are vented, the reference closed, the syringe emptied, and the reaction vessel closed. W i t h i n 5 sec. measurement can be resumed; three to ten 20-mv. steps of the non-steady state velocity can be obtained, depending on the duration of the non-steady state and the velocities involved. The final velocity is also measured. F r o m the results, graphs of velocity vs. time are drawn as well as graphs of the appropriate loga rithmic expression, whose slope is converted to the absolute rate constant required, 2

Figure 2.

Apparatus for non-steady state autoxidation measurements A: Reaction vessel with syringe B: Differential pressure transducer to recorder C: Piston buret (synchronous motor) D: Magnetic stirrer (constant-voltage transformer)

The rate constant for physical gas absorption γ! (see Appendix) was determined by measuring the logarithmic decrease i n rate of oxygen uptake by a hydrocarbon slightly unsaturated with respect to oxygen (not taking up oxygen chemically). It was found that γι values as high as 0.08 sec." could be obtained. Starting Materials. Cumene and Tetralin were washed with sulfuric acid, aqueous sodium hydroxide, and water, dried over magnesium sulfate and over sodium, and fractionated from sodium. Cyclohexene (Eastman Kodak grade) was distilled only. A l l hydrocarbons were stored under nitrogen at —15 °C. and percolated over alumina before use. Rate con stant ratios obtained on these hydrocarbons from conventional measure ments are listed i n Table I. 1


350

OXIDATION

Table I. ARi X 10-" Mole/Liter/ T, °C. Sec.

Hydrocarbon ([RH]) Cumene

(7.2)

40

O F ORGANIC

COMPOUNDS

1


[AH] X 10' Mole/Liter

7

4.7 4.7

5

10~* Mole/Liter/ Sec.

4 4 4

1.70 6.56 0.75 3.90 9.18 8.74

5.25 0.25 1.55 0.37 1.29 1.22

Cyclohexene (10)

30

2 4 2

2.70 3.34 2.34

1.05 1.15 0.97

Tetralin

40

5 5 5

11.7 11.5 5.90

4.97 0.02 1.10

a 6 0 d

(7.34)

Determined according to non-steady state Equations 1 or 2. From k and k /(2k )i/2. From k and 2k /k . Determined in conventional rate measurements. P

P

p

t

A

P

2,2'-Azoisobutyronitrile ( A I B N ) (purchased from Fluka) was recrystallized from ether, m.p., 103°C. 2,6-Di-tert-butyl-p-cresol (Ionol) was recrystallized from ethanol, m.p., 71 °C. Principles of Measuring The steady state condition for the concentration of R 0 * radical i n an autoxidation is 2

^

^

= 0= R - 2 * , [ R O p 4

(1)

2

W h e n a small amount of an inhibitor, A H , is added, [ R 0 * ] decreases according to 2

0*

d [ R

^ ' 2

]

= R, - 2ft,[R(VP - 2k [AH] A

[RQ -] 2


(2)

26.

351


BERGER ET AL.

of Hydrocarbon Autoxidation Rate Constant Ratios OL X 10-' =± X 1 (2k y* k d

Abs. Rate Constant Liter/Mole/Sec. k

2k X10~

a

p

0.42 0.44 0.50 0.40 0.37 0.33

0.02 0.03 0.03 0.02 0.02 0.01

6.65 5.95 7.6

f

6

0.024 (0.03 at 50°C.) > e

2.65

2.65 (5.6)' 4.0 14 13

av. 9.2 (8.5)'

9.5 (10.0)'

57.5

e

6.05 5.4 6.9

5.7 11.3 10.7

9

p

2.4

2.6 2.1 3.4

av. 6.7 (6.1)'

t

2.4 2.5 2.9 2.3 2.15 1.90

6

e

1

c

A

5

av. 0.41 (0.5 at 50 C.)*-

e

2k X10-*

6b

t

r.

1.30

9.10

3.0

6.0

6.1 (2.7)' 3.4 6.6 6.6 5.5 (5)'·'

From Melville and Richards (3). From Howard and Ingold (2). Not directly comparable.

Since at t = 0, R — 2 f c f [ R 0 - ] [ R 0 ] is 2

{

2

= 0, the initial relative decrease of

2

[ ° 'J° dt

rf R

[R0 ] 2

0

2

= -2fc [AH].

(3)

A

For experimental reasons (discussed below) the relative rate of decrease should be smaller than 0.01 sec." . Since k is of the order of 10 liter per mole per sec, [ A H ] must be of the order of 10~ mole per liter. This low concentration of inhibitor w i l l only have an appreciable effect on the rate of autoxidation if the rate of initiation—and consequently the rate of oxidation—is low. To measure the decrease i n rate during the short-lived non-steady state, one must be able to determine these low velocities within short periods of time. F r o m the usual inhibition formulas one can compute, for instance, that i n order to obtain a ratio of original to inhibited rate of about 5 with [ A H ] = 10" mole per 1

4

A

6

6


352

OXIDATION

OF

ORGANIC

COMPOUNDS

1

liter, one requires an initiation rate of ^ 1 0 " mole per liter per sec, resulting i n a rate of oxygen consumption of —0.01 m l . per min. for an 80-ml. reaction volume. n

B y restricting the gas space over the reaction mixture and by using a sensitive pressure transducer, it was possible to measure rates of,—2.10" mole per liter per sec. of 0 (i.e., 0.002 m l . 0 per min. in an 80-ml. reaction volume ) within 20-sec. intervals, and this is sufficient for measuring a number of velocities in non-steady states of 100-300-sec. duration. 8

2

2

A prerequisite for the measurement is that the rate of stirring should be sufficiently high to ensure that the rate constant of physical gas absorption does not become limiting; this problem is dealt with in the appendix. This condition is not to be confused with that applying at high rates—namely, that the physical rate of absorption should not become limiting; here we have the more stringent requirement that a rate of change in oxygen consumption of 0.01 sec." can be followed without physical limitations. The required stirring efficiency was obtained with a baffled reaction vessel described above. 1

W i t h the low rates of initiation and oxidation used, the difficulties usually encountered in attempts to obtain reproducible data from radical reactions are aggravated. The main problem is that with hydrocarbons showing the expected linear dependence of the oxidation rate on R in the usual range of (10~ ), deviations occurred at Ri 10" owing to either spontaneous initiation or spurious inhibition or both. The difficulties arising from these effects were circumvented by incorporating the latter into the integrated formulas describing the non-steady state. It is necessary to assume then that the inhibition can be described as a firstorder termination: — ^ ~ ^ ^ = — X [ R C V ] with X a constant during the 500-1000-sec. span of the measurement. 1/2

t

8

10

2

Adding inhibitor or initiator to create the non-steady state also introduces a physical disturbance of the system, owing to differences in temperature and/or in content of dissolved gas. These effects were minimized by adding microliter quantities of solutions only. Results Originally we attempted to determine initial slopes of

vs. time

graphically, but this procedure ( though feasible ) was abandoned i n favor of using integrated formulas which allow determination of the rate constants from the slope of a straight line. Since the low steady state velocities measured often deviated from the theoretically expected values, we finally used formulas which accounted for both spontaneous initiation and


26.

BERGER

ET

353


AL.

10° 3 7 ^ * M O L E / L I T E R / S E C . ατt "LOG"

"Ο

100

200

300

400

J

500

L

600

700

J

800

L

900

1000

TIME, SEC.

Figure 3.

Autoxidation of cumene at40°C.

Non-steady state after adding 4 X I0" 7 M 2,6-di-tert-butyl-x?-cresol. Values of "Log" on ordinate defined according to Equation 6 in text

first-order termination (discussed above and in Appendix). The formulas are: Av = v -

_2* [AH] ~ fcJRH]

(4)

A

v,

t

0

Λ

(^[RH])*

C 2

For adding a small amount of initiator, effect ΔΚ : (

Δ

υ

(5)

ARj

+

log

ο (ν -υ ) 2

χ

0

=

0

4

3

4

3

( Δ Κ ^

+

c

J

^

k p [ R H ] i +

c

For adding a small amount of inhibitor A H (Δϋ < 0 ) :

log c

2

v 0

(6)

v

+ ΔΌ x

= 0.4343 < j - ^ -

- c (v - υ J ) k [RR]t 2

0

p


+ C

354

OXIDATION

OF

ORGANIC

COMPOUNDS

1

The values of absolute rate constants of propagation given i n Table I have been obtained by using these formulas. Discussion Except for cumene (discussed below), no great accuracy is obtained in the rapid individual measurements of the low velocities employed, but the deviations are smoothed out considerably in the logarithmic straight lines, so that the resultant slopes are reasonably accurate (Table I and Figure 3 ). A fundamental drawback of the method is the limited range of conditions under which measurements can be made ( except for cumene ) ; the deviations observed are therefore regarded as a measure of the reproducibility of the method rather than of the accuracy of values obtained. Since the duration of the non-steady state can be shown (4) to be inversely related to k and R , a high value of k is unfavorable because it reduces both the time available for measurement and the rate to be measured. For this reason cumene, having a low k , is the obvious choice for demonstrating the method. Cyclohexene and Tetralin, on the other hand, probably represent the limit of what can be measured; their relatively high k is partially offset by high values of k [ R H ] , which increase the velocities, hence also the accuracy obtainable at a given R{. The method proved ineffective with fluorene, which must be measured in solution (e.g., 1.8M in chlorobenzene which reduces its k [ R H ] by a factor of 4 relative to Tetralin), seobutylbenzene and cyclohexylbenzene, which probably combine a relatively low k with considerably higher k than for cumene (1). Concerning the results, the values obtained for cumene, Tetralin, and cyclohexene agree well with those reported i n the literature. 1/2

t

{

1/2

t

t

t

p

p

p

t

Appendix Non-Steady State Equations with Correction for Spontaneous Initiation and First-Order Termination. Thoroughly purified hydrocarbons should exhibit a square-root dependence of oxidation rate on initiation rate, R ; we found, however, that even if this behavior is obtained with Ri of the order of 10" mole per liter per sec, deviations may occur with the low rates of initiation used in the non-steady state measurements ( R — 1 0 " ) . Also, spontaneous initiation of the order of Ri ^ 10" may occur. If we assume that the deviations can be described as a constant first-order termination, we can derive corrected formulas for the nonsteady state behavior upon adding a small amount of inhibitor A H or initiator AR as follows. t

8

f

n

12

h


26.

355


BERGER E T AL.

Initial condition: spontaneous initiation Rf, first-order termination con stant Χ : Ri - 2fc [RO '] 2 - X [ R O ] i

2

2

0

= 0

0

(Al)

Non-steady state after adding AR* or A H :

l °^ *

d R

+

=Ri

A R

.-

2^[R0 ]2 2

- X[R0 -] Φ 0 2

or

(A2) d

l °^ R

R

=

. o - 2 * [ R 0 - p - X [ R 0 - ] - 2k [AH] i

2

2

[R0 '] * 0 2

A

F i n a l condition: R,° + AR, - 2 ^ ^ 0 · ] 2 _ χ [ κ ο 2 · ] α = 0 2

χ

or

(A3) Ri* ~ 2 f c [ R O ] 2 - X t R C V ] * - 2fc [AH] [ R O , ] , = 0 i

2

A

00

B y subtracting Equation 1 from Equations 2 and 3, one can eliminate R/. Subtracting Equation 1 from Equation 3 (Rf eliminated) gives an ex pression for X to be substituted in Equation 2 (R/ eliminated) which can then be rearranged to give a quadratic differential equation i n A R 0 · = [R0 ] - [RO ] : 2

2

2

0

dt -AR0 { 2

2

M

dt

[R0 ]„ 2

- [RO,;].) -

+ [RO 2

] a 0

A

- [ R 0 - ] j " *< (A4) Δ

2

or _ dAR0 ° _ 2k (AR0 ') dt 2

T

Κ

+

2

- A R 0 - \ 2k ( [R0 • ] . -

2

2

^

^

t

}

+

ι

α χ

^

[RO ']J

2

[ Α Η ] [ Κ

°

s

2 ] ο

( A 5 )

The square roots i n the expression

—

dx

1

ax + bx + c ~~ y/\p - 4ac

η

2

~ ~ +C ax + b + y/b - 4ac 4 a c

+

2

can be extracted. After converting to rates of oxygen consumption ac cording to υ = k [ R H ] [ R 0 ] and introducing the constants c = 2k [ A H ] /k [ R H ] and c = 2fc,/[fc RH] , whose values are available from steady state measurements, we obtain the following expressions for the non-steady state when a constant spontaneous initiation and/or 2

p

A

p

2

1

p

2


356

OXIDATION

O F ORGANIC

COMPOUNDS

1

first-order termination are involved ( Av = v — v ) : t

0

ARi ,

A

AR,

—

Av +

log A H ( Δ ϋ < 0)

«

C 2 ( V

C i / c

log-

v

0

2

— 3 —

(A6)

f

-

—

Ν

/

°> =1 0 . 4 3 4 3 \ - ^ - + c (v„ - v ) } ^ [ R H ] i + C

V

2

_L

+

,

Δυ

^ — ^

0

(A7)

ν

=: 0.4343 \-^—c (v 2

- t) )k[RH]i + C e

0

Oxygen Uptake by a Solution in which the Oxygen Consumption is Rapidly Changing According to a First-Order Law. D E F I N I T I O N S . The dC symbols C and

refer to oxygen concentration in the solution and C

œ

is the saturation value; ν refers to the chemical reaction which consumes oxygen;

is the rate of absorption of oxygen gas by the solution.

The rate of oxygen absorption by an unsaturated solution is given by: ^

= yi ( C . - C )

(A8)

If the chemical reaction consumes oxygen at a rate

τ ^

—

+ C 7l

=

*

*

- e-y,
A9

(A10)

C„

7l

This is a linear differential equation whose solution is: C =

t y r V + C + aervi' Ύί—Ύ*

(A l l )

X

V

The value of a follows from C =

-

0

h C + a, so that we x

y i - 7 2

obtain an expression giving -

y i 7

( C - CJ =

Λ

directly—viz., =

t; - i v r V ( ^ * -

γι-γ

\Ύι"Ύ2

2

- l) (A12) I

It is seen that, when y χ > > γ , the oxygen consumption follows ν 2

exactly; any discrepancies

are governed b y — — .

B y computing

curves for different values of γι and y , we find that there are large differences between observed and actual rate if y > 0.3 γι. However, 2

2


26.

BERGER

ET A L .

if γ = 0.05 and γ = 0.01, for example, we find that 2

1

357


and ν take the

form of two apparently congruent curves which are separated by 15 seconds. Literature Cited (1) (2) (3) (4)

Alagy, J., Clement, G., Balaceanu, C., Bull. Soc. Chim. France 1961, 1792. Howard, Α., Ingold, K. U . , Can. J. Chem. 44,1119 (1966). Melville, H . W . , Richards, S., J. Chem. Soc. 1954, 944. North, A. M., "Kinetics of Free Radical Polymerization," p. 25, Pergamon Press, Oxford, 1966. (5) Walling, C., "Free Radicals in Solution," p. 67, Wiley, New York, 1957.

RECEIVED

October 9, 1967.


OXIDATION

OF

ORGANIC

COMPOUNDS-


A New Method for Determining the Absolute Rate Constants of

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