2372
RUTHKORENAND BERTAPERLMUTTER-HAYMAN ~
~~~
Table VI : The Entropies and Microscopic Structure of Some Sulfate Ion Pairs
,320
l o n pair
uozs04 Cas04 CdSOa VOSOa cusoc ZnSOc MgSOa NiS04 18.
[ M - ( a q ) I, gibbs/mol
Predominant cation hydra(ion pair), tion gibbs/mol number
-23.1 -13.2a -17.4c - 32d -23.6c -26.gC -28.2' -31.6'
10.8 3 and 4
7.2b 5.26 -2 -2.2 -6.9" -8.7b -11.1''
4 4 5 5 6 6 6
Predominant type of ion pair
Mixed Contact Contact Solvent separated Contacte Solvent' separated Solvent srparated Solvent separated
Reference 24. Reference 2. Reference 23. e SO4 ion acting as a monodentate ligand.
Reference
UOzS04in solution would exist as a contact ion pair. More recently it has been shown that the UOz2+ ion is hydrated by a maximum of four waters in water-acetone mixtures.26 The partial molal entropy of the VOSOa ion pair should be greater than or equal to the entropy of the CuS04 ion pair that had previously been assumed to
have a primary hydration number of 5. Because these two ion pairs have almost identical entropies, we would conclude from this work that the VOSO4 ion pair predominantly exists in solution as a solvent-separated ion pair. This conclusion is not in agreement with a pressure-jump kinetic study of VOSO, solutions, from which it was concluded that approximately 30% of the VOSO4 ion pairs exist in aqueous solutions as solventseparated ion pairs and 70% as contact ion pairs.'2 It is also not in agreement with the results one would expect from the crystal structure of VOS04.5Hz0in which the sulfate is in contact with the VOZ+ion and the cation has a hydration number of 4.27 Although the partial molal entropies of these ion pairs appear to be predominantly determined by differences in the primary hydration of the cation, contributions from long-range electrostatic interactions may also be significant.
Acknowledgment. The authors wish to thank the Marshall University Research Board for financial support'. (26) A. Fratiello, V. Kubo, R. E. Lee, and R. E. Schuster, J . P h y s . Chem., 74, 3726 (1970). (27) C. J. Ballhausen, €3. F. Djurinskij, and K. J. Watson, J . A m e r Chem. Soc., 90, 3305 (1968).
Rapid Preequilibria as a Cause for Change of Activation Energy with Temperature1 by Ruth Koren and Berta Perlmutter-Hayman* Department of Physical Chemistry, Hebrew University, Jerusalem, Israel
(Received December 17, 1970)
Publication costs assisted by the Department of Physical Chemistry, Hebrew University, Jerusalem
A reaction scheme is considered where the substance taking part in the rate-determining step is present in two forms which are in mobile equilibrium (isomers; or differently solvated or protonated species). Equations are developed which show that if only one of the two forms is kinetically active, while both are present a t comparable concentrations, the apparent energy of activation decreases with temperature. On the other hand, if the two forms are present a t different concentrations, but nevertheless contribute about equally t o the observed reaction rate, having different, independent, rate constants, the apparent energy of activation is shown t o increase with increasing temperature. The meaning of the results, and their possible applicability, are discussed.
Introduction The apparent energy of activation of a chemical reaction, defined by the expression
E,
=
-Rdlnk/d(l/T)
ACp*
= dE,/dT
(2)
(1)
is frequently found to change with temperature; in The Journal of Physical Chemistry, Vol. Y6, N o . 16,1971
other words, what we may formally define as the specific heat of activation, vix.
(1) This paper forms part of a thesis to be submitted by R. Koren t o the Senate of the Hebrew University.
CHANGEOF ACTIVATION ENERGY WITH TEMPERATURE
2373
is found t o be nonzero. Kegative values of AC’, have been reported for many hydrolysis reaction^^-^ and have recently been d i s c ~ s s e d . ~ -Positive ~ values are less frequent and are usually ascribed to the existence of two or more different mechanism^.^^^ At sufficiently low temperatures, they may occur as a consequence of proton tunneling.’ I n the present paper we shall show how the existence of a rapid preequilibrium can lead to both negative and positive values of AC, Our calculations predict that these values may be strongly temperature dependent.
-- 90 .-- 00
-- 70 A H / R T = 6.0
*.
Negative Values of AC,
-- 60
K- 5
-- 50
*
Let is consider a reaction scheme
-
A
41)
AS”/R = 7.6
A(2)
-- 40
(mobile)
+ B -% C + D A(z) + B -% C + D A(l)
(1)
-- 30
(11) ( W
where A(l) and A(2) are isomers, or, for reactions in solution, substances which differ by their degree of solvation or protonation. The rate constant observed for the disappearance of B is kobsd
= [ k ( l ) A ~ ) k(z)A(z)I/[A(1) =
+
+
+ A(2)I
(hi) K k ( z ) ) / ( l K )
where K is the equilibrium constant of reaction I. From eq 3, together with the definition of E a (eq 1)) we get
where E ( l )and E(2)are the “true” activation energies of reactions I1 and 111, and AH is the heat of reaction of reaction I, all, for the sake of simplicity, assumed temperature independent. Consider the limiting case where only one of the two species makes an appreciable contribution t o the observed rate. When this species is A(*),ie., KJq2)>> k ( l ) ,eq 4 takes the form
Ea
E(z)
+ A H / ( K + 1)
(Sa) Conversely, when k(l) >> Kk(,) and thus only A(l) contributes, we have =
E a =
E(1) - K A H / ( K
+ 1)
(33)
I n both cases, Ea decreases with increasing temperature, and to the same extent. This becomes evident when we differentiate eq 5 t o get AC,’
=
AH2 --
E T Z( K
**
(3)
K
+ 1)2
For any given value of AH and T , the effect is at its
-I2
Figure 1. The dependence of ACp on 8E/RT, for various values of 8 A S */R, indicated by a number alongside the appropriate curve.
maximum when KI = 1. When K differs considerably from unity, ACPT tends to zero and E , approaches temperature-independent values. For instance, when the kinetically active species is A(z), we get from eq 5a that E , = E(2) AH when this species is present at very low concentration (small K ) and E, = E(2) Tyhen it is predominant. Similarly, when A(1) is kinetically active we get from eq 5b that E, = E(1) and Ea = E(l) - AH for small and large values of K1, respectively. I t should be noted that AC, is negative, irrespective of the algebraic sign of A H . This means that it does not matter whether the relative concentration of the kinetically active species increases or decreases with increasing temperature. This can be understood from
+
*
(2) E. A. Moelwyn-Hughes, “The Kinetics of Reactions in Solution,” Oxford University Press, London, 1947, Chapter 2. (3) R . E. Robertson, Progr. P h y s . Org. Chem., 4, 213 (1967). (4) (a) B. Perlmutter-Hayman and Y. Weissmann, J . Phys. Chem., 71, 1409 (1967); (b) B. Perlmutter-Hayman, Israel J. Chem., in press. ( 5 ) J. R . Hulett, Quart. Rev.,18, 227 (1964). (6) S.W. Benson, “The Foundations of Chemical Kinetics,” McGraw-Hill, New York, N. Y., 1960, p 27. (7) E. Caldin, Chem. Rev.,69, 130 (1969).
The Journal of Physical Chemistry, Vola7 6 , No. 16, 1971
2374
RUTHKOREN AKD BERTA PERLMUTTER-HAYMAN
the following considerations. If, for instance, A(2) is kinetically active (see eq 5a) and AH is positive, then the observed rate constant increases with temperature because both the rate constant and the relative concentration of the active species increase. Therefore, E, is higher than the activation energy of the rate-determining step. This effect decreases as the temperature increases and Acz) therefore becomes more predominant. Conversely, if AH is negative, a rise in temperature represses A(2). Therefore, the observed rate constant rises with increasing temperature less than does the rate constant of the rate-determining step; E, is lower than E(z),and may even be negative. The effect is more pronounced as the relative concentration of il(2)decreases, i.e., as the temperature increases. An analogous reasoning holds when A(l) is the kinetically active species.
Positive Values of AC,
*
Let us now consider the case when A4(1) and A(2)make comparable contributions to the observed rate. From eq 3 we see that the condition for this to happen is k(1)
-
-
Kk(2) or Kk(z)/k(l) 1
(7)
For this case, eq 4 is valid (provided reactions I1 and I11 are independent, i.e., proceed via different transition states: otherwise [AH - (E(1)- E(2)) is obviously zero). Differentiating that equation with respect to temperature and introducing
SE = E(1) - E(2)
(8)
we get
AC,* =
AH2
K
+ 1)2+
_____
RT2 ( K
This expression is composed of two terms having opposite algebraic sign and may therefore be either negative or positive. Let us first consider the dependence of AC,* on 6E, while all the other parameters have some constant, arbitrary values. An arbitrary example is shown in Figure 1. Here K 5, with AH/RT = 6.0, and A S o / R = 7.6 (where AS" is the standard entropy change of reaction I). The quantity 6AS*is definedas AS*(l) - AS*(2),where A S is the entropy of activation. At very high values of /SEI the first, negative term is seen to predominate. I n a certain, fairly narrow range of 6E the second term predominates. and AC, becomes positive. Before determining the mathematical condition for (ACp*)maxas a function of 6E, we remember that
*
*
Kk(*)/k(l)=
AH - 6E - T ( A S o - 6AS*) RT T h e Journal of Physical Chemistry, Vol. 76, N o . 16, 1971
Differentiating eq 9 with respect to 6E and equating the result t o zero, we see that AC,* has a maximum when the condition
AH - 6E 2RT coth
[AH - 6E -2TR(TA S o - 6AS*) ]
(11)
is fulfilled. nlathematically, this can hold a t two values of the argument. At one of them, AH - 6E and T ( A S o - 6AS*) have oppositealgebraic sign, and a t the other, they have the same sign. Only the latter solution is of interest for our considerations. (Inspection of eq 10 shows that it is this solution which is compatible with condition 7 . ) From eq 9 i t is clear that the value of (ACp*)m,, increases as lAH - 6EI increases. At the same time, IT(AS" - 6AS*)j must also increase for the maximum condition 11 to remain valid. This is brought out in Figure 1, where A S o is positive; large negative values of 8AS* are seen to be accompanied by high values of (AC, *)max. At the same time, the position of the maximum i s shifted to more negative values of 6E. So far for the dependence of AC, on 6E, a t constant AH, and varying values of the parameter !(/A ' So 6AS*). If, on the other hand, we consider AC,* at a given value of (AH - 6E),we find i t to have a maximum when ( A H - SE) = T ( A S o - 6AS*), i.e. (see eq 9), when condition 7 becomes an equality and the two species make exactly equal contributions. Theoretically, there is thus no limit to increasing positive values (In fact, the maximum conditions for deof AC,*. pendence on 6E at constant AH and on T ( A S " AS*) a t constant (AH - 6 ~ are ) simultaneously fulfilled when ( A H - 6E) -+ .) To get a better understanding of the physical meaning of the mathematical condition for positive values of ACpi, let us again consider the example where k ( z )>> k(l); this usually means that 6E > 0. This must be accompanied by a particular, small value of K for condition 7 t o hold. We can then rewrite eq 4 to get
*
When (AH - 6E) < 0, E , is smaller than the activation energy of the slower reaction (and may even be negative), Homever, the negative term decreases in value with increasing temperature, because, under the conditions we have chosen, k ( ~ ) / l c (decreases ~, more strongly than K increases. Finally, as the contribution of A(z) to the observed rate becomes negligible, Ea --c E(1). Conversely, when ( A H - 6E) > 0, E, is higher than E ( l )for the reason discussed in the previous section. This effect increases with increasing temperature, as the relative contribution of A(z) t o the observed rate increases. When this becomes predominant, E , + E(2)
2375
CHANGE OF ACTIVATION ENERGY WITH TEMPERATURE
+ AH.
The argument for the case where k ( z ) >> k(l) is completely analogous. If,however, instead of k ( 2 )>> k(l) and K ) k(1) and K < 1 (or vice versa), the positive and the negative terms on the right-hand side of eq 9 will always be of comparable magnitude, and ACp* does not attain high values.
Applications An argument similar to that outlined above has been adduced by Liebhafskys to explain the negative values of AC, in the hydrolysis of halogens. Negative values of ACp* for the same reaction may be explained on a similar basis. The negative values of ACp* for the uncatalyzed hydrolysis of dichromate have been tentatively expIained on the assumption that a dichromate ion in the neighborhood of hydrogen-bonded or “icelike” water might be hydrolyzed more slowly than one situated in “free” water.4 Several years ago, in connection with a different topic, we measuredlo the esterification of acetic acid in methanol in the presence of -3.7 M chloroform, and in its absence; in both cases, varying small amounts of water were added to the reaction mixture. We found the observed rate constant kobsd to decrease, and E , to increase, with increasing water concentration. Furthermore, in the presence of chloroform, E , decreased with increasing temperature. The retarding influence of small quantities of water on acid-catalyzed ester reactions has been known for a long time. Goldschmidtll explained this effect on the assumption that the catalytic effect of the proton present as H 3 0 +is small in comparison with that of the proton present as ROHz+. Various have subsequently used the “Goldschmidt equation’’ to interpret their results on esterification reactions and to determine the equilibrium constant between the two forms of the proton from kinetic data. The Goldschmidt mechanism in fact implies that ROH2+ should be the kinetically active species. This, however, is a t variance with the A A Cmechanism ~ now universally accepted for the hydrolysis of simple esters like methyl acetate. 1 4 , 1 5 According to that mechanism, it is the protonated acid (and not the protonated alcohol) which is involved in the rate-determining step of the esterification reaction. We do not feel in a position to comment on this discrepancy, and we therefore derived expressions for E , for both the Goldschmidt and the A ~ c 2mechanisms. On the very reasonable assumption that only a small fraction of the catalyzing proton is present as CH,COOHz+, we found the two expressions to differ only by the heat of reaction of the protonation of acetic acid in ROH solution.16 They both contain the temperature-dependent term
AH
1
+ K[ROH]/[HzOl
(13)
where AH and K refer to the reaction 4 CH3OHz+. This may be compared with eq 5,; a t a given mater concentration, K[ROH]/ [HZO] is a constant, and we get the temperature dependence of E , as given by eq 6. On the other hand, if we increase [HzO]from zero to a sufficiently high value, expression 13 increases from zero to AH. The Goldschmidt mechanism would thus constitute a straightforward application of our formulas: the active ROH2+ and the inactive H30+ are present a t comparable concentrations, and the equilibrium between them changes with temperature. According to the A A Cmechanism, ~ the temperature dependence of E , arises in a more subtle way. The kinetically active substance is present a t very low concentration; this usually leads to a temperature-independent E,. In the present case, however, that substance is simultaneously in equilibrium x i t h two other substances, ROH2+ and H30+, whose equilibrium, in turn, changes with temperature. To test the agreement between our expression for E , and the experimental results, we determined’l K in the presence of chloroform a t different temperatures,” and hence AH. We found it to be fairly constant in the range between 0 and 45”, and equal to -4.1 kcal mol-l. Our results are presented in Table I. The condition that E, a t high water concentration should differ from (8) H. A. Liebhafsky, Chem. Rev.,17, 89 (1935). (9) A . Lifshitz, Ph.D. Thesis, Jerusalem, 1961; B. PerlmutterHayman, unpublished results. (10) B. Perlmutter-Hayman, unpublished results, 1952. (11) H. Goldschmidt and 0. Udby, 2. Phys. Chem., 60, 728 (1907); see also R. P. Bell, “The Proton in Chemistry,” Cornell University Press, Ithaca, N.Y . , 1959, p 21. (12) A. T. Williamson and C. N. Hinshelwood, Trans. Faraday Soc., 30, 1145 (1934). (13) H. Goldschmidt and R. S. Melbye, 2. Phys. Chem., 143, 139 (1929); C. N. Hinshelwood and A . R. Legard, J . Chem. Soc., 1586 (1935); R. J. Hartmann, H . M . Hoogsteen, and J. A . Moede, J . Amer. Chem. Soc., 66, 1714 (1944); H. A . Smith and J. Burn, ibid., 66, 1494 (1944), and several earlier papers by Hartmann and coworkers, by Goldschmidt and coworkers, and by Smith. (14) See, e.g., C. K. Ingold, “Structure and Mechanism in Organic Chemistry,” 2nd ed, Cornell University Press, Ithaca, K. Y . , 1969, Chapter 15, Section 58. (15) We are indebted t o one of the referees for drawing our attention t o this fact. (16) The dependence of the observed rate constant on [HzO]is, in principle, different for the two mechanisms, but the t w o expressions remain indistinguishable as long as the change in [CHaOH] is negligible, L e . , at low water concentration, where the Goldschmidt equation was applied.’*-14 (17) For this purpose we used the expression
and plotted l/kobsd vs. [HzO] for a series of experiments. At the lowest water concentration, where [HzO] cannot be considered constant during the experiment, a suitable correction was applied. The meaning of k is ~ R O H or~ K ( R O +H ~ H ~ - A) ~~ H ~+,Aaccording ~ to the mechanism we assume. +
-
The Journal of Physical Chemistry, Vol. 76, hro. 16, 1971
2376
RUTHKORENAND BERTAPERLYUTTER-HAYMAN
Table I : Apparent Energy of Activation in kcal mol-' for Esterification of Acetic Acid in Methanol Solution in the Presence of Chloroform; [CHaCOOHIa = 8 X 10+ M ; [HCl] = (1-3) X lo-' M ,------[H@], 21.7
1.1
E , (273 -+ 289°K)
M-----0.83
0.55
14.0 13.6 1 4 . 7 2 ~ 0 . 1 14.3
E, (298 4 318'K) a
0
10,2a
13.2 1 2 . 7
See ref 13.
that at zero water concentration by AH is fulfilled as accurately as can be expected from the quality of our experiments.
The Journal of Physical Chemistry, Vol. 76,No. 16,1971
It remains to explain why we did not find the expected temperature dependence of E, in the absence of chloroform. This may be ascribed to the fact that the corresponding A H j as calculated from data given in the literature, l s increases with increasing temperature, thus masking the effect. A more detailed investigation of the esterification reaction would be outside the scope of the present paper. Acknowledgment. The authors wish to thank Professor H. J. G. Hayman for helpful suggestions, and Professors S. Patai and Z. Rappoport of the Department of Organic Chemistry for discussions on the mechanism of esterification reactions.
