THOMAS I. CROWELL
902
Vol. 64
OSCILIATING TEMPERATURES IN REACTIOK KIXETICS. I. A CTIYATIOK EXERGY FROM STEADY STATE COXCENTRATION BY THOMAS I. CROWELL Cobb Chrmical I,aborcttory, University of Virginia, Churlotfesville, Vu. Keceiued Februnry 4 , 1960
A simple, reversible reaction with forward and reverse rate constants kl and k - , is maintained alternately at temperatures 2“ and 7‘”. If the alternation frequency is rapid, a steady state is reached. The apparent equilibrium constant for these conditions, K,, is then equal to (?&’ kl”)/(nk...l’ k - I ” ) where n is the ratio of the times spent at T’ and T”. By observing K , as a function of n, the activation energies of the forward and reverse reactions can be evaluated.
+
+
s. Equating coefficients gives q = kI’,;k-~’ = K’, r = k-,”,/k-,’ and s = (kl”l:-l’ - 1 ~ ~ ’ k - l ‘ ’’ ) (k-1’)‘ = (K” - K’). Thus Y E X-l”/k-1‘ = aA + bB + . . . . Z e C + dD + (1) k- 1 (k-l”/k-l’)(Kr’ - K’)/(K, - K’), and a plot of n us. 1/(Ks - K’) has an intercept on the l/(KB with forward and reverse rate constants kl’ and k-l’ K’) axis of k-1”/7c-l’, the temperature coefficient of a t the temperature T’, and k,” and k-,“ a t T”. reaction rate for the reverse reaction. The ratio The equilibrium constants at T‘ and T” are given kl“,’kl’ is then evaluated from IC~~’:’I;-~’, K’ and by Intrations [.I],[R], . . . [C], butylamine to form piperonylidene-n-hut.ylamine [D], . . . vi11 change so that the system will be at a,nd water2 equilibrium practically all the time. ki If, howerer, the fluctuation in temperature is ArCHO + n-C4HsSHa1_ .ArCH==NC4Hs+ H20 more rapid, the system will be more sluggish in rek- 1 sponding to the change. A very rapid fluctuation Experimental will lead to constant concentrations of reactants Apparatus.-The reaction vessel was a 10” X l/,” o.d. and products (Fig. 1-4). These steady-state consteel tube fitted with a screw cap and Teflon gasket. centrations are easily calculated without consider- stainless It was situated inside a larger glass tube through which ing the course of the reaction. Suppose for simpli- water was circulated from either of two large thermostat city that the rate equations correspond to the terms baths a t 25 and 50’. The circulating pumps in the rein the stoichiometric equation 1, so that dx/dt = spective baths were alternately actuated by a simple tinier and two-way relay, to give the desired temperature cycle. li1’[,l]“[B]* . . . --Ic-l’[C]c[D]d . . . at T’ and The response inside the reaction vessel was checked xvith a similarly a t T”. Here the reaction variable r has t,hermocouple attached to a Sargent recorder, and is shown the usual meaning.’ Let the system be at T’ for (for n = 5 and At = 1 hour) in Fig. 1C and 1D. A special value ensured the return of the water to the proper the time interval nAt/(l n) and at T” for the in- plunger terval At/(l n ) ,where At is the time of one com- bath. Procedure.-A solution of n-butylamine ( -10 - 3 V), plete heating and cooling cycle and n is the ratio of piperonal (-10-4 M ) and water (5.0 M ) was prepared in a 100-ml. volumet,ric flask at 2 5 O , using reagent, grade meththe times spent at T’ and T” (Fig. 1B). For a steady State, Ir,the change in .r during At, anol as the solvent. About 3 nil. of this solution was placed in each of four reaction tuhes, which were capped and placed must tie zero in the oscillating temperature apparatus for two days. A Consider a chemical equilibrium
+
h.1
+
(n
+
+ 1)Az = nki’At[A]“[B]’ ..
2-ml. sample was then withdrawn from each tube at the - ~~_I’A~[C]‘[[D]~ . mid-point of a 25’ interval, diluted to 10 ml. with meth- k-i”A.t[C]c[D]d. . . = 0 ( 2 ) anolic hydrochloric acid, and analyzed spectrophotometrically a t 348 mp for Schiff b a ~ e . 3 ’Because ~ 1 M HZ0 was present in the diluted sample, conversion of piperonal to the acetal was incomplete and necessitated a small corrwtion equilibrium constant for the for absorption of piperonal a t this wave length.
K , is an apparent steady state. It should be noted that the system is never at or near equilibrium except momentarily during the rise and fall of temperature. Equation 3 shows that a determination of K,for a certain n, together with K’ and K”, is sufficient for the calculation of kl”/kl‘. Paradoxically, the activation energy, a quantity pertaining to rates, can thus he obtained from data which are not a function of time, though time is involved in fixing the value of n. More experimental data ran be utilized if K , is measured as a function of n. The expected form of the function is seen from 3 to he ( K 8- q)(n r ) =
+
(1) A. -4. Frost and R. G . Pearson, “Kinetics and Mechanism,” John Wiley and Sons, Inc., New York, N. Y.,1953,p. 9.
Results Equilibrium constants for Schiff lime formation were det,ermined a t the two temperatures used. Although the water concentration was practically constant at 5 M , it was included in t,he espression K = [ArCH=IVBu] [H,O],;[ArCHO][RuSH2] to conform with our preceding ~ v o r k . ~The . ~ value obtained at 25”, K’ = 3.10 x lo5,is in fair agrwment, wit,h that found by titratioii (3.18 X 10”). At 50°, K = 0.95 x IO3, though the Tdue used i n the calculations mas obtained diffcreiitly as esplained below. (2) R. L. Hill and T. I. Crowell, J. Am. C”hem. Soc., 7 8 , 2284 i1 %%I). (3) T. I. Crowell and D. W. Peck, ibid., 35, 1075 (1953). (4) C . E. Bell and T. I. Crowell, J. Ory. Cliem., 24, 1159 (1959).
July, 1960
OSCILLATIXG
TEMPERITURES I N R E A C T I O N KINETICS
The reaction was then studied with oscillating temperatures as described in the experimental section. The reaction and the oscillation rate were both quite slow; n ranged from 0 (temperature constant a t 50') to 7.09 (7.42 minutes at 50', 52.58 minutes a t 25"). The time for one complete temperature cycle, At, was one hour in every case. The maximum permissible At for a given variation from steady-state concentrations can be calculated from equation 2 if the rate constants are known. A plot of l / ( K ' - K,) is shown in Fig. 2. The straight line through the experimental points follows the equation l/(K' - K,) = 0.478 0.0078n. The n-intercept of -6.14 is multiplied by 1.026 to correct for the change in concentrations when the methanol expands a t SO", so that l ~ - ~ " / k -=~ '6.30. The ratio of the forward rate constants, kl"/kl', is then 6.30 K " / K ' ; the value 1.15 for I
