Thermodynamically and Kinetically Controlled Products M. E. Brown and K. J. Buchanan Rhodes University, Grahamstown. 6140 South Africa A. Goosen University of Port Elizabeth, Port Elizabeth, 6001 South Africa T o a physical chemist, the title ahove may cause some immediate alarm. A normal response is that thermodynamics and kinetics control all products. The title phrase is, however, commonly used in many textbooks of organic and inorganic chemistry as a shorthand way of discussing the relative yields of two different products, formed from the same starting material, by parallel and reversible reactions
under the rather restrictive conditions that and C is then said to he the "thermodynamically controlled" product, and B the "kinetically controlled" product (I). In using this shorthand description, it is important that the kinetics and thermodynamics of the ahove system should he clearly understood, and in this paper we provide some examples of how the relative concentration of products B and C vary with time for different combinations of rate coefficients. An often-quoted example of the ahove reaction system is the monosulfonation of napththalene ( 2 )leading to two isomers
If the sulfonating agent is present in large excess, the reactions are pseudo-first-order and the kinetic treatment is simplest. Parrallel Irreversible Reactions The simpler system of two parallel, hut irreversible, firstorder reactions (3),at constant temperature, will he reviewed first.
For this system, the rate of formation of product B is given by
and that for C by d[C]ldt = hz[A]
+ +
If [Bl0 = [CIo= 0, then [A] [B] [C] = [Ale. The ratioof the concentrations of products, [B]/[C] = kllkz and is thus independent of time. The rate of consumption of reactant A is -d[A]ldt = (hl+ hz)[A] = h[A]
he major product of such a pair of parallel reactions, a t constant T, is thus the one formed in the reaction with the larger rate coefficient, hut at any time t , the desired product has to he separated from a mixture containing hoth unreacted A and the other possible product. An explanation then has to be sought as to why one product should he formed more rapidly than the other, e.g., why k l > kz. If each rate coefficient is written in terms of the Arrhenius equation as h~= A I exp(-E1/RT) the fact that k l
and
hz = A2 exp(-EdRT)
> kz may arise from any of the following:
1) A1 = A s
and
2)A1>A2
and and but but
3) A, > A z 4) A, > A2
El
Figure 1. When k,lk. = 10, and K2/K, = 1 , equilibrium wncenlralions of all three species equal, [A], = [B],, [C]., = 0.333 mol I-'.
[B],, and C is then described as the "thermodynamically controlled" product. For kl > kz and (Kz = kzlk-2) > (KI = kllk-1) to apply simultaneously, k - ~must be >> k-2. Since kl > kz, and the concentration of A is common to hoth reactions (1)and (2), ratel > ratez and B, the product of reaction (I), builds up more rapidly than C, so that soon rate-1 becomes >> rate-z. The concentrations of A , B , and C a t equilibrium are calculated as follows: [Alo = [Ale, + PI, + ICI,
Figue 2. When * , / k g = loand K d K , = 2, equiliMum mcenwt.ansare [CI, = = 0 40 and [El.. = 0.20 and [BI > [CI for lint 550 s
Hence
and for [C]., to approach [Ale, i.e., complete conversion to C, kLlkz must be >> k - z ( k ~ +k-1). Some examples of curves of concentration against time for various values of kl, kz, k-1, and k-2, from the many combinations possible, are given in Figures 1to 7. All are based on initial concentrations, [Ale = 1.00 moll-' and [Blo = [Clo = 0. Times are in seconds and k l is arbitrarily fixed a t 0.010 s-'. The other rate coefficients are varied relative to k~ as summarized in the table. From the diagrams it can he seen that if k l > kz, about half the initial amount of A is rapidly converted into B before the concentration of C becomes appreciable. This initial conversion is decreased as the ratio KzIK1 increases (Figs. 1,2, and 3). The higher the ratio KzIK1, the greater is the eventual conversion of A to C (Figs. 4 to 7). The "crossover time," i.e., the time a t which [C] = [B], depends upon the ratio kllkz, if K2/K1 is fixed, becoming earlier as kllkz decreases. Under the conditions illustrated. the "time window" for isolating a significant proportion of "kinetic product,"B, is small. Even at half the crossover time, the concentration of B is only approximately twice that i f C. At shorter times, B will stillhave to be separated from A.
Figure 3. When kl/k2 = 10 and K21Kq= 4, equilibrium wncenbations are [C], = 0.57, 161- = 0.14, and [A].4 0.29. awl [B] > [C] for first 470 6.
Thermodynamic Aspects The requirement, Kz > K1, is often stated in the form A G $ < AGP
although conditions normally used are so far from standard conditions (all reactants and products a t unit activity or, for ideal hehavior, concentrations of 1mol I-') as to make any
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Journal of Chemical Education
Figure 4. When k,/k2 = 10 and K2/K, = 2.5. equillbirum concentrations are [c], = 0.625, [a]- = 0.250, and [A], = 0.125 and [B] > [C] for first 1850
s.
conclusions drawn about the spontaneity of reaction under standard conditions irrelevant if not misleading. (For example, if k z = k - z , then Kp = 1and K1 must then be [C]for first 900 s.
where J is the reaction quotient (of the same form as the equilihrium constant, but with general values of the concentrations) which will be very small a t low product concentrations, giving a large and negative, initial AG value, decreasing to zero as the ~ r o d u cconcentration t builds uo to its eauilihrium value, that is as the thermodynamic exient of reaction 1. a~vroaches -In discussing "thermodynamic and kinetic control," the situation has been made more confusine. ... throueh misuse of diagrams of potential energy against reaction coordinate ( f i , 7). Such diagrams do not zive information cuncernine AG". and hence for a reaction. What may be represente& with reservations, is the relationship between the experimental activation energy and the enthalpy of reaction, AH, (at constant pressure). Since AHwill not differ greatly from AHe, the equilibrium constants for the two parallel reactions, a t the same temperature T, may be written as:
AGT = -RT In K z = A H - TAS?
So K z will be > K t , or In Kp > In K I if -AH; Figure 6.When kdk2 = 10 and K2/K,= 9,equilibrium mncenbationsare [C], = 0.a26,[BIT = 0.092,and [A], = 0.082 and [B]> [C]for first 900 s.
+ TAS: > - AH? + TAS?
Of the many possible combinations which would make this relationship true, two extreme cases are: 1) AS;
2) AH?
-
E f f e c t of
AS?
and
-AH? > - AH?
AH?
and
AS?
01
AH?
< AH?
> AS?
Temperature
The discussion, so far, has been concerned with parallel reactions occurring a t the same constant temperature. The parameten governing the differences between the two reactions are A and E (or AS$ and AHt) and AHe and ASe. Although these quantities themselves are not strongly dependent upon temperature, the magnitudes of E and of AHe determine the effect which temperature will have on the rate coefficients, k , and the equilibrum constants, K , respectively. d In kldT = EIRT2
Figure 7. When kl/k2 = 5 and K2/K1 = 9. equilibrium concentrations are [C], = 0.826,[BIT = 0.092,and [A], = 0.082and [B]> [C]for first 450 5.
d In KIdT = AHeIRT2 (constant pressure)
Summary of Conditions Illustrated for the Systema
Figure
kt
1 2 3 4 5 8 7
0.01 0.01 0.01 0.01 0.01 0.01 0.01
.With h,
k-
?
0.01 0.02 0.02 0.005 0.005 0.009 0.009
> k ~ s n d ( K , = hdk-g)>(K,
k2
k-P
krlk2
0.001 0.001 0.001 0.001 0.002 0.001 0.002
0.001 0.001 0.0005 0.0002 0.0004 0.0001 0.0002
10 10 10 10 5 10 5
k-llk-2
10 20 40 25 12.5 90 45
KdK, 1 2
4 2.5 2.5 9 9
crossing point [Cl = [Bl m
-550 s -470 s -1850 r -900 5 11900 s -450 s
[cl~/[Alo 0.333 0.400 0.571 0.625 0.625 0.826 0.826
[Blmu/[Alo 0.5 0.5 0.5 4.6 4.6 0.5 0.5
= k,lk-,).
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Number 7
July 1985
577
If, for reactions (1) and (2). A1 c Az = A, hut El a t temperature T., say, k l . may he greater than
< Eq, then
where At temperature Tb(> TJ, let k l b = Bk%b
A t temperature T. k l ,
= Ae-ElIRT.
neeative. . so that K values mav either increase or decrease with increasing temperature. For the parallel reactions, an extreme, although unlikely, case would he where one K value increased as the other decreased. I t is thus impossible to make generalizations about the effect of temoerature on "kinetic and thermodynamic control.'' Conclusions Thus. in summarv. the relative amounts of oroducts B (reaction (1)) and C Tieaction (2)), formed by coicurrent reversible first-order reactions from reactant A. deoend uoon both the ratio of the forward rate constants, kllk2, a n d t h e ratio of the equilibrium constants, K 1 / K 2 .If kllk2 is large, about half of A is initially converted rapidly into B, hut if in addition K d K z- is small, this product B soon reverts to A and is eventually ccmverted toC (ihe..thermodynamic product"). At some time durinr! reaction, called here the "crussover time." [B] = [C]. Only a t time less than the crossover time can significant proportions of the "kinetic product" B he isolated. This "time window" becomes smaller, for a fixed value of KlIKq as the ratio kllk2 decreases. ~
Similarly, a t temperature Tb
If Tb> Ts then In a > In 8, i.e., a>B
but only a t T = a is = 1, that is the rate coefficient for reaction (1) is greater than that for reaction (2), at all temperatures, provided that the A values are equal. Except for some rare instances, activation energies are always positive (or zero) so that k values will increase with increasing temperature. AHe, however, may he positive or
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Journal of Chemical Education
Acknowledgment The diagrams were drawn using a program DEQN, written in Fortran by P. D. Terry, for numerical simulation of the simultaneous solution of first-order differential equations, on an ICL 1900 computer. Literature Cited (1) Hine. J., "Physical Olganic Chemistry: McCrsw-Hill, New York, 2nd ed., 1962, p. C4
(2) Lantz. R.,BuII. Soc. Chim.2.2092 (1935). (3) Smbo, Z G.. '"CompmhensiveChemical Kinetia,"Elaevier, Amsterdam. 1569, Vol. 2,
