Max S. Peters and Edward .I. Skorpinski' University of Colorado
Boulder
II
Definition of ~aritib~e-volumeand Con~tllnt-Volume Reaction Rates
W o r k e r s in the field of reaction kinetics seem to be divided when it comes to the fundamental definition of the reaction-rate expressions. There are those who believe that the rate of a chemical reaction should be expressed as number of moles of a reactant (or product) consumed (or produced) per unit time per unit volume, (-l/K)(dNA/dO). Others believe the rate should be expressed as the change in concentration per unit time, (-dCA/d8). Although the two expressions are identical under conditions of constant volume, they are different for variable-volume conditions. Part of this controversy can be traced hack to the interpretation of what is meant by the word rate in the classic Gnldberg and Waage's Law of Mass Action which appeared in 1867: "For a homogeneous system, the rate of a chemical reaction is proportional to the active masses of the reacting substances" (14). The 'I active mass" of a system is the amount of active material per unit volume of the system or the molecular concentration of the active substanre.
Here NA is the number of moles of component A. V, is the total volume (cc), so that CA, the concentration, will be in moles/cc. However, the rate of a chemical reaction is not necessarily the change in active mass, as defined above, per unit time. Two interpretations can be given as follows: (NAIVJ Reaction rate = ~ ' ( N A I V J = " - ~do 1 dN* Reaction rate = k1(Nn/Vt)" = - yt
(2b)
Article based on study carried out while authors were in the Chemical Engineering Division at the University of Illinois, Urbanrt, Illinois. The National Science Foundation supplied help for part of the experimental work in this study under Grant
- ....
' Present address is Rohm and Haas Company, Philadelphia,
Penn*+snis.
Obviously, (-l/Vt)(dNA/dO) can never equal - d C d do unless V, remains constant. A further mathematical indication of the inconsistency of the -dCA/dO expression relative to the (-l/VJ(dN~/d0) form is given in the following analysis: Consider the first-order gaseous reaction A uB which has a change in volume upon reaction where u represents the moles of B formed from one mole of A. Let the volume be changed at a constant rate, (V, = Vb hB), by the addition of a diluent. Assume that the rate of the reaction is zero so that there is no change in the number of moles of reactant A. The concentration of A at any instant will then he given by
-
+
(2a,
whcrc k' represents the reaction rate constant (sec-1) and n represents the reaction order. Among the many authors who apparently support equation (2a) as being universally correct are Glasstone (14), Benson (4), Daniels (8, O), Laidler (17, 18), Prutton and Maron (21), and Vvedenskii (30). Equation (2b) is supported as the correct form by men such as Denhigh (11), Danckwerts (7), Antonoff (1, 6,3), Hougen and Watson (167,Smith (SO), Levenspiel (19),
C. l 1 , l \ A 7, ,
Benton (5), and Harris (15). Frost and Pearson (13) have presented an interpretation based on a rate expression in the form of (-l/V,)(dNA/dO) while Nemeth (20) has proposed a theoretical derivation of the rate expression which supports equation (2h). Repeated misinterpretations in the literature, based simply on failures to use the correct rate expression, make it desirable to clarify the situation by a mathematical indication of the unacceptability of the concent,rat,ion-based rate expression. The first step will be to establish that the two rate expressions can never give t,he same result for any order of r e a h o n unless the volume of the reacting system remains constant. This can be done simply by expanding equation (2a).
where h is a constant (cc sec-') and 0 is time. Differentiating with respect to 8, gives the reaction rate, r(mole cc-'sec-I).
Inspection of equation (5) shows that, even though the reaction rate constant has been defined as zero, there is a finite rate of reaction if the volume changes. This is physically inconsistent. This situation is not enrountered if the rate expression is defined as (-l/VJ(dNA/dB). Writing out this expression, the result is
Here it is seen that the rate of the reaction is zero as defined because dNJd0 is zero. On the basis of the preceding presentation and discussion, it appears that the correct and logical rate Volume 42, Number
6, June 1965
/
329
expression to use in reaction lcinetics is (-l/VJ(dNn/ dB) and not -dC JdO. The balance of this paper is devoted to the results of an experimental study which, also, indicates that (-l/VJ(dN,/dO) is the correct rate expression.
system. Express the decomposition reaction as given in equation (7) in the form .4-2B
fC
(8)
For a first-order reaction, therefore, the rate expression under constant-volume conditions can be written as
Experimental Analysis
A survey of the literature does not reveal any coinpletely satisfactory experimental data to compare rate expressions under coutrolled conditions of variable volume. Among the earliest experimental studies on variable-volume reactions was one by Wegscheider (52). Unfortunately, his analytical methods were poor and he was unable to draw any conclusions as to the carrect rate expression. More recent experimental studies by Delmon, et a1 ( l o ) ,Rudakov (M), and Nemeth (20) tend to support the (-l/VJ(dNJdB) rate form, but their results are not totally conclusive. For this study, the gas-phase decon~position of di-t-butyl peroxide was carried out under variablevolume conditions a t constant temperature (28). Di-t-butyl peroxide decomposes in the vapor phase to acetone and ethane according to the following overall reaction: The reaction has been studied extensively under varying conditions of concentration, surface/volume ratio, and pressure (22, 26, 26, 27, 51). Experiments have also been conducted to study the effect of diluents in both the liquid and gas phases (6, 12, 24). Previous experimenters have reported that the reaction follows a first-order rate expression up to at least 40y0 conversion of the peroxide. The apparatus for the investigation consisted of a 1000-ml glass flask with necessary attachments for carrying out the reaction a t constant volume and a t constant pressure with volume changes being measured by a series of calibrated floating syringes. The entire system was immersed in a constant-temperature bath, and necessary controls were included to permit evacuation and precise measurement of temperature, pressure, and volume during the reaction. The initial concentration of di-t-butyl peroxide for the various runs varied from 0.0151 to 0.0181 moles/liter, and constanttemperature runs were carried out a t temperatures from 137.7'C to 160.Q°C. All reactions were run until about 60% of the di-t-hutyl peroxide was converted. Under these conditions, the total pressure changed from an initial value of approximately 18 in. Hg to a final value of about 38 in. Hg for constant-volume runs, and a volume change of approximately 700 cc was ohserved for the constant-pressure runs. All variablevolume runs were carried out at atmospheric pressure. The diluent gas used was nitrogen. Details of the conventional experimental apparatus and procedure, along with comments relative to the need for purifying di-t-butyl peroxide to remove all traces of t-hutyl hydroperoxide (otherwise an explosion occurs) are given by Skorpinski (28). Constant-Volume Reactions
Under constant-volume conditions, the lcinetics of the gas-phase decomposition of di-t-butyl peroxide can he followed by measurement of pressure change in the 330 / Journal of Chemicol Education
Note that
Applying the perfect gas law, making the appropriate substitutions into equation (Q), and integrating gives
Thus, for the constant-volume runs, a plot of In (3 P4 P,, -PJ versus 9 should give a straight line with the slope equal to the reaction-rate constant k. Typical plots of this type for two different temperatures are shown in Figure 1. The constant-volume runs were made to determine the reliability of the experimental equipment and compare the results of this study with published results (22). The straight lines obtained, as illustrated in Figure 1, show that the reaction is firstorder, and the Arrhenius plot, as presented in Figure 3, shows good agreement with the results of Raley, Rust, and Vaughan (22) which are presented for comparison in Figure 3.
+
9- MINUTES Figure 1. Plots to determine rate conrtontr for the decomposition of di-tbutyl peroxide a t constant-volume conditions.
Vorioble-Volume Reodions
All variable-volume reactions were started initially as constant-volume runs and then switched to variablevolume a t constant pressure when the pressure within the reactor reached atmospheric. This procedure was followed so that initial fluctuations in temperature would not be present during the variable-volume portion of the run. To allow sufficienttime for the fluctuations to disappear, the runs were planned so that from 15% to 20y0 of the reaction was completed before the switch to variable-volume conditions was made. The experimental proof as to whether (-l/Vt) (dNa/dO) or (-dCA/dB) is the correct expression can be
+
shown as follows for the reaction A -+ 2B C. Each of the preceding rate expressions can be set equal to k ( N A / V J ,and the constant for each case can then he determined and compared to the correct rate constant as established under constant-volume conditions. For thevariable-volume experimental equipment, N, = N* + A's + A'o + NN,+ NN.* (13) Equation (13) differs from equation (10) by the small correction term NN,*which partially corrects for the fact that a small increase in the amount of nitrogen diluent in the system occurred at the start of the variable-volume runs due to nitrogen a t atmospheric pressure in the line connecting the reactor flask to the expansion system. Under these conditions and assuming ideal gases, for the variable-volume runs at constant P,, dNA = -
P
2RT
dVt
For the case of ( - l / V t ) ( d N n / d O ) tegration gives
Variable-Volume Experimental Values 1 dN* Basedon - Based on V. d8
Temperature ("C)
Volumes k X lo4 seccL
k X
109Deviation seccL from a
%
10' sec-'
0.78 0.95 1.44 2.07 2.72 3.30 4.12
-16.0 -7.8 -4.6 -3.3 -5.2 +4.8 +51
1.40 1.64 2.34 3.43 4.62 5.34 610
- dCn dR
k X
%.
Deviation from a
from solid line in Figure 3 or (15)
where the constant, F, is defined by =
Comparison of Rate Constants Obtained from at Variable Volume with Those Obtained at Constant Volume
I - l/Vt)ldNA/dO)and I-dCA/d8)
k N A / V a in-
=
F-V
F
Table 1
(14)
- In --= 2 kB F - Vt,
PA.
tween the con~tant~volume values of the rate constants and the variable-volume rate constants determined by use of -dCA/d8. A further comparison of the values of the rate constants as determined by the various methods is presented in Table 1.
Summary
+ P N ~ V+~Nw*RT ,
(16)
Pt
Similarly, for the case of ( - d C Jd8) integration gives
=
k(NA/V,),
Thus, a plot of 111 ( F - V J and of In[(V,/Vd (F - V*)] versus 8 should give an experimental comparison of the rate expressions (- 1 / V J (dNJd9) and -dC Jd8, and the one giving the correct value of k will be the correct rate expression. Typical plots based on equations (15) and (17) are shown in Figure 2. The best straight lines are obtained for the ( - l / V , ) ( d N Jd8) expression, but the real proof of the experimental correctness of the rate expression is shown in the Arrhenius plot, Figure 3. This plot shows that the constant-volume values of the rate constant agree almost perfectly with the variable-volume values determined by use of ( - 1 / V J (dNA/d8), while there is considerable difference be-
By mathematical analysis, it is evident that the rate expression ( - l / V J ( d N A / d O ) is self consistent for all reaction-rate conditions and can only equal -dCJd8 for the case of constant-volume reactions. An experimental determination and comparison of the reaction-rate constants for the first-order gas-phase decomposition of di-tbutyl peroxide under conditions of constant volume and variable volume further shows that the correct form of the general reaction-rate expression is ( - l / V , ) ( d N ~ / d 8 ) . Literature Cited (1) (2) (3) (4)
(5) (6) (7) (8)
ANTONOFF, G., J. CEEM.EDUC.,21, 420 (1944). ANTONOFF, G.,J. CHEM.EDUC.,22, 98 (1945). ANTONOFF, G., J. Phw. Colloid Chem., 51, 513 (1947). BENSON, S. W., "The Foundations of Chemioal Kinetics," McGraw-Hill Book Co., New York City 1960, p. 10. BENTON, A. F., J. Am. Chem. Sac., 53, 2984 (1931). BOSE,A. N., AND HINSHELWOOD, C., PTOC.Roy Soe., London,A249, 173 (1959). DANCKWERTS, P. V., Nature, London, 173, 222 (1954). DANIELS,F., "Chemical Kinetics," Cornell University Press 1938, p. 8.
TEMPERATURE = 155.7-C
0
B
16
P
