Understanding Product Optimization Kinetic versus Thermodynamic Control King-Chuen Lin National Taiwan University and Institute of Atomic and Molecular Sciences. Academia Sinica, Taiwan, R. 0.C The concept of kinetic versus thermodynamic control of reactions has been frequently discussed in most organic textbooks (1,2)and has been demonstrated experimentally in this Journal (3-5). Its effect on product formation in such processes as sulfonation, the Diels-Alder reaction, isomerization, and addition is exemplified in the table. Klnetlc versus Thermodynamlc Control Reactlon Kinetic Control 1.
Reactants
Thermodynamic Control
Did-Alder Reaction'
Reaction Coordinate Figure 1. Reaction schematics for kinetic versus thermDdynernic contra.
'See ref 1. bSBe ref 5. -See ref 2: kinetic and themadynamic prodvcts are idemisal
I t is ordinarily expressed as a concurrent reaction scheme,
namic product C tends to dominate, which is energetically more stable than the kinetic product B. At high temperature, the increase of temperature raises the rate coefficients and facilitates the reacti"n equilibrium; this favors the thermodynamic product. Accordin&, thermodynamic control is also termed equilibrium control: In spite of the importance of the concept to general synthesis, the lack of detailed elucidation may obscure understanding. Snadden has clearly discussed reaction scheme 1 on the basis of the kinetic point of view (6). The objective of this paper is to help undergraduate students understand: (1) the role of kinetic and thermodynamic control reaction in terms of the kinetic equations, (2) the influence of concentration and temperature upon the reaction, and (3) to apply these concepts to synthetic chemistry. ~inetic'Equation
when the reactant A proceeds competitively toward products B and C. The time evolution of the product ratio is determined by two parameters: the rate coefficients and the thermodynamics. According to Figure 1( I ) , B is the kinetically controlled product, while C is thermodynamically stable. Under the condition of low temperature, the product B is favored, for its low activation energy speeds up the change of A to B. In contrast, for long reaction times, the thermody-
General Solution Based on reaction scheme 1 and Figure 1, the following kinetic equations are set up:
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857
0.1
0.2o , 3 0 0
p
y
,
,
2
4
,
a
,
a
e
,
10
TIYE(YIN)
1
b
C=A,
k&-I
[
+
-
A&
kdk-, A&
- Ax) e - As)
-.,t
+
kdk-1 - A,)
e
-
- A,)
]
(9)
The time evolution of A, B, and C is depicted in Figure 2 . Extreme Cases of Kinetic and Thermodynamic Control In the beginning stage of reaction scheme 1,the exponential function may he expanded in terms of the Taylor series in which the higher terms may be omitted. Replacement of exp (-A$) by 1 - Xit yields B = Aok,t C = A,k,t
(10) (11)
The products of B and C depend upon only the forward reaction rate. That is, in the early reaction, the reverse routes may he neglected, since the product of B and C are not significantly accumulated to compete with the forward rates. The case simplifies as, 8
ki
BCA+C
o
0.4
0.8
2
&*I nm(mln)
2.4
I
Figure 2. Tlme evolution of reactant A and product B and C: a, regime dominated by kinetic conwol; b, regime dominated by thermodynamic control: kq = 1, k-, = 0.01, k2 = 0.1, and k-* = 0.0005. The unit iss-'.
By uaing the matrix method, these coupled differential equations may be solved analytically (7).The following secular equation 0 k2 - A -k-, =0 k-, - A
The amounts of B and C are simply associated with the forward reaction rate constants, and the reaction is dominated by the kinetic control. As for longer times, the higher terms. X2t2 - X3t3 . . . become imnortant and the oroduct ratio is time dependent. -, eqs 8 and 9 are For long-term reaction times, i.e., t replaced by
+
t.6
(12)
-
The concentration of products appears invariant, and now the system reaches equilibrium, i.e.,
(5)
In this case the product ratio
yields the particular solutions, exp (-hit), Xi, eigenvalue of the equation, and i = 1 , 2 , and 3. Thus A,=O
depends only on the equilibrium constants K I and K2. The process is dominated by thermodynamics, and the products are termed thermodynamically or equilibrium controlled.
and
where b = k,
+ k, + k-, + k-,
and c = k,k_*
+ k.,k, + k_,k_,
Substituting the summation of particular solutions into eqs 2-4, and applying the initial condition: A = Ao, B = C = 0 a t t = 0, one obtains the general solution:
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Journal of Chemical Education
Deviation Estimate As pointed out earlier, the concept of kinetic versus thermodynamic control results from the sbort- and long-time limits of the kinetic equation. The forward rate dominates the process in the limit of kinetic control, whereas the equilibrium constant controls the limit at equilibrium. A deviation factor is defined as follows to take measure of temporal influence upon the validity of the approximation,
Figure 3. a. Comparison of vaiidny for kinetic controi, (BICk, = kdk2. b. Comparison of validity for thermodynamic control, (BIG)., = KllK*. C w e s obtainedare for the following conditions: (i) kt = 1, k-, = 0.01, kz = 0.1. and k-2 = 0.0005: (ii) kr = 1. k-I = 0.05, kg = 0.1. and k-. = 0.005: and(ili) k, = 100. k-i = 1, k2 = 10. and k-2 = 0.05. The unit is s-1.
Figure 4. a. Product ratio BIC increases from k,/k2 to infinity as one reverse channel B- A is blocked, i.e.. k, = 1, k-, = 0, k2 = 0.1, and k-. = 0.0005: b. C is Product ratio decreases lo zero as the other reverse channel A blocked, i.e.. k7 = 1, k., = 0.01, k2 = 0.1, and k.2 = 0. Theunit isr-'.
where (B/C),,is the product ratio approximated by eqs 13 and 17, respectwely; (B/C).,is product ratio in terms of eqs8 and 9. The resulting plot of A against time is shown in Figure 3 under the condition of various sets of kl, kz, k-I, and k-%. As the reverse rate is decreased. the time for the reaction reigned hy kinetically controllid product increases, and therehv it takes a loneer time to reach eouilibrium. Eouilibrium control becomes more important as the reverse rate increases.
energy and A the preexponential factor. Both are considered independent of temperature for simplicity. Thus the rate coefficient a t high temperature Tz is expressed as follows in terms of that at low temperature TI,
-
-
Concentration Effect McNaught (5) has studied the reaction of Hg2+ with I-; the kinetically controlled product, Ha12 (a yellow rhomhic isomer) appears first when-dilute solutions are used; whereas, with concentrated reactants the solution appears orange quickly, yielding the tetragonal isomer. In fact, according to eqs 10 and 11, the increase of initial concentration speeds up both competitive routes to an equal extent. We should note, however, that when the orange product increases, the yellow isomer is masked. Temperature Effect Influence of temperature upon the rate coefficient is usually based on the Arrhenius equation, i.e. (8) k = Ae-EJk~T (19) In this expression E. denotes the experimental activation
As shown in Figure 3, the increase of rate coefficients as a consequence of temperature rise facilitates the equilibrium of reaction. That is, the thermodynamic control prevails sooner with an increase of tenmeratwe. Based on ea 20. the rate coefficient of the reverse route, which has higher activation energy toovercome, is enhanced more than that of the forward route. This factor, as important as rate increase, accelerates hackflow of the products and therehv hastens the achievement of equilibrium. Accordingly, from eqs 14 and 15 and a temperature-independent free energy, we clarify the fact that the reaction tends to favor the kinetic product a t low temperature, hut at high temperature the equilibrium product tends to prevail.
a
Appllcatlon to Synthetic Technique According to the kinetic equation, the chemical system will favor the kinetic products at the early stage of reaction and then will gradually shift to the thermodynamic products as the reaction time is increased. This requires the establishment of a reversible channel connecting the reactants and Volume 65 Number 10 October 1988
859
products. As the concentration of the kinetic product B in reaction scheme 1 increases, the reverse rate increases so that A will apparently produce the thermodynamic product C. Consequently, to increase B the reverse reaction must be decreased by either removing the product B or isolating it in terms of a protective reaction. Analogously, the B/C ratio decreases to zero as the reverse channel A + c is blocked, These results are shown in Figure 4. Thus either the kinetic or thermodynamic product may be controlled. Acknowledgment
The author wishes to thank Professors and Stwalley for critically reading the manuscript and Tze-Cheng Shu
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Journal of Chemical Education
for calculating and plotting the figures. This work is supported by the National Science Council, R. 0. C.
Lneralure I. Men~er,F.M.:Galdsmith,D.J.;Mandell,L.,OrgonicCh~mialry;Benjamin:California, 1972: pp 216-216 pp 245-246. 2. R O ~ C ~ J. ~ SD. , :csserio. M c., ~~~i~ fiineiples 01 organic chemistry,2nd ed.i ~ ~ ~ mi": 1977: pp 374-376. 3. McCrew.L. A,: Kruger,T. L. J. Chem.Educ. I971,46,4W401. 4. Yousaef, A. K.: Ogliaruso. M. A. J , Chem. Educ. 1975.52. 473474. 5. MeNaught.1. J. J . Chem.Edue. 1978,55,722-723. 6. Snadden,R. B. J. Chem.Educ. 1985.62.653-655. 7 Mataen,F.A.:Franklin, J. L J Am. Cham.Sac. 1950.72.3337-3341. 8. Laidler, K. J. J c h e m . ~ d u r 1984.61.494-496. .
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