Kin F. Wong Charles A. Eckert
Pseudo-Thermodynamic Techniques for the Estimation of Chemical Reaction Rates and Equilibria
chemical reaction represents an extremely complicated electronic process that is difficult to characterize theoretically, even for the simplest and most idealized cases. For any real situation, it is hopeless to try to make quantitative predictions about a reaction from first principles. Yet, in any chemical process analysis or design, a knowledge of chemical reaction rates and equilibria is essential to the chemist or to the engineer. Reliable experimental data are of course preferable, but not always available. However, over the years physical organic chemists have been rather successful in using a judicious combination of empiricism and theory to correlate rates and equilibria for many organic reactions in pseudo-thermodynamic terms. The application of such relationships should be of great use to design engineers in estimating the effects of substituents, reagents, and solvents on chemical reactions. This paper will deal primarily with a review of some of the simpler and better-developed correlations for reactions and equilibria in the liquid state. T h e majority of these treatments are referred to as “linear free energy relationships,” because they reduce the rate problem to a problem in equilibrium thermodynamics by correlating the free energy of activation, as demonstrated by the transition state approach. The overall goal of this review is to present to design engineers tested techniques for obtaining the maximum amount of information from a minimum of experimental data. Often the design engineer needs approximate data on chemical reaction rates or equilibria for process evaluation or screening. The effect of reaction conditions, especially the choice of solvent, can also be a vital factor in process
A
16
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
design. T h e use of thermodyiiarnic concepts to extend limited kinetic data to new situations often introduces no increased uncertainty in the reliability of the data, since thermodynamic data and techniques are in general many times more precise than rate measurements. T h e transition state theory views a general chemical reaction wherein the reactants are in thermodynamic equilibrium with the transition state M, which in turn can decompose to product.
A
+BS M
--t
Products
T h e overall rate of the reaction is given as the product of a frequency Y, independent of external paramet’ers, and the concentration of the transition state [MI, rate = v [ M ]
(1)
This concentration in turn is given by its assumed chemical equilibrium with the reactants, and the equilibrium constant K is related in the usual fashion to a free energy for the activation process, A@.
T h e earliest example of a correlation of the linear free energy type was the Br@nsted catalysis law (1924), relating catalytic activity to the strength of an acid or base in solution (72). In general, at constant temperature, the logarithm of the kinetic rate constant k is proportional to AG*. For chemical equilibria the corresponding quantities are the equilibrium constant K and the standard free energy of reaction AGO. The proportionality constant depends on the reaction series
as well as on the substrate. Many reviews have appeared in the chemical literature (78, 27, 27, 33, 38, 42, 49, 57, 59, 60), and the subject has continued to receive considerable attention not only in organic chemistry but also in other branches of chemistry. The free energy of activation (or the standard free energy of reaction for the case of equilibrium) is, of course, the sum of an enthalpy contribution and an entropy contribution,
AG* = AH* - TAS*
A‘+M‘+
AH* is unchanged AS* is unchanged Although both AS* and AH* change, the variathem is linearly related (57)
Most of the correlations proposed have been twoparameter relationships between the above quantities. Although attempts to account theoretically for many more types of specific effects have led to the introduction of additional parameters (78, 38, 49, 60),we shall not go into these here. Rather, we shall confine our attention to the simple relationships for organic reactions that should be of most use to the engineer. Effect of Substituents
O n e of the most frequent applications of linear free energy relationships is for the substituent effect. For these cases, the correlations are used to estimate the rate of a reaction using as data the rate of another reaction of similar niechanism, involving a related molecule with similar structure, but differing by a substituent group not a t the reaction site. For example, suppose one wanted to estimate the rate of esterification of p-chlorobenzoic acid by methanol. If the rate constant for the reaction of m-nitrobenzoic acid with methanol were known, this could then be used to make a very good quantitative estimate of the rate of the former reaction. To apply this type of correlation, organic reactions have to be categorized by their reaction mechanisms (24, 30,
AUTHORS K. F. Wong is a postdoctoral student, and C. A. Eckert is an Associate Professor in the Department of Chemical Engineering, School of Chemical Sciences, University of Illinois, Urbana, Ill. 67807. The authors gratefully acknowledge the jinancial support of the National Science Foundation.
P’
M” +p”
A”
(3)
T h e usual form of the linear free energy relationship is to take some given reaction as a model and assume that in a similar reaction (with some relatively minor change in either the solvent, the substrate, or the substituent group) one of the following three conditions is satisfied :
(1) (2) (3) tion in
39), and treatments are available for various types of mechanisms. The Hammett Equation. T h e Hammett correlation for the effect of meta- and para-substituents on aromatic side-chain reactions provides a n excellent example of the development of the linear free energy relationship. Let us take as a model two similar reactions,
where A ’ and A ” differ only by a side group well removed from the reaction site. Then
AG*’ = AH*‘ - TAS*~ and
A@’’
=
AH*“ - TAS+‘
(4)
but since the substituent difference is far from the reaction site, the entropy difference is assumed negligible, AS*‘ = AS*” (condition 2), so that the free energy of activation, a n d therefore the logarithm of the equilibrium constant for activation is linear in a n enthalpy, AH*. The same argument can be formulated for an equilibrium constant in terms of AGO. The result may then be expressed as
(5) where k , represents the rate constant for some model reaction, u represents the relative energetic effect of a given substituent, and p is a constant for each type of reaction, representing the sensitivity of that class of reactions to substituent effects. Of course, Equation 5 could be used equally well to express either rate constants, as written, or equilibrium constants. As a matter of fact, the original development used the ionization of benzoic acid as the model reaction, for which p was defined as unity, and for which u was defined as zero for hydrogen substituents. The basic Hammett relationship has been applied successfully to a great many aromatic nucleophilic substitution reactions. Jaff6 (33)has listed 204 reactions, many under various conditions of temperature and solvent, yielding more than 350 reaction series. Schein and Kozorez (50) report on 58 reactions of mono-, di-, and trisubstituted aromatics with nucleophilic reagents. Examples of values of p and u are presented in Tables I and 11, and additional u values are given by Jaff6 (33) and by Pal’m (42). I n general, negative values of u indicate a n electron-donating group, and positive values a n electron-withdrawing group. T h e larger values of p indicate that a reaction is facilitated by a lower electron density in the aromatic system relative to that for the model reaction. VOL. 6 2
NO. 9
SEPTEMBER 1970
17
TABLE I .
H A M M E T T SUBSTITUENT C O N S T A N T 9 upara
urneta
-0.069 -0.043 -0.120 -0.027 0.415
-0.170 -0.151 -0.197 -0.066 0.551 -0.357 -0.268 -0,250 -0,660 -0,592 -0.611 0.265 0.522 0.628 0.227 0.276
-0.002 0.115 0.150 -0.161 -0,302 -0.211 0.355 0.398 0.678 0.373 0.352
TABLE I I .
H A M M E T T REACTION CONSTANTS FOR SOME REPRESENTATIVE SERIESe
Reaction Equilibria: Ionization of CBHICOOH Ionization of CeH5"3'
A4nilineswith formic acid to give formanilides Reaction Rates : Saponification of ethyl benzoates Ethanolysis of benzoyl chlorides Benzaldehydes plus HChT Hydrolysis of benzoic anhydrides a
Solvent
T , "C
H2O HzO 20y0Dioxane 70% Dioxane
25 25 25 25
677c CaHsN
100
75y0 MeOH
25
P
95% EtOH
20
2.329
75Tc Dioxane
58
1.568
:isH)
($)
= pu+
This form also applies to reactions such as solvolyses and nucleophilic substitutions in which the transition state involves considerable carbonium ion character a t the a-carbon of the benzene ring. An example of the application of Equation 6 is shown in Figure 2 (36). The model reaction in this case is the solvolysis of substituted 2-phenyl-2-propyl (t-cumyl) chlorides in 90% acetone at 25°C. Examples of u + and p (73, 47) are
-1.429
25
{%%
log
1.000 2.767 3.256 3.567
2.193 1.922 1.499
0
An example of the application of Equation 5 is the hydrolysis of meta- and para-substituted ethyl benzoates in 85y0ethanol (20, 32), shown in Figure 1. This relationship yields the best results when the reaction center is insulated from the aromatic ring by at least one methylene group; moreover, the polar and steric effects of ortho-substituents can invalidate the assumptions. Brown and coworkers (73, 47) have modified the Hammett equation to apply it i o electrophilic substitution on the benzene ring,
From the compilotion of J ~ f b (33).
-I6kJ -06
-08
I
j
-2.0
-3.4
9
-02
02
S"xt~t,enl
04
08
06
I O
Constant m
Figure 7 . g d r o l y s i s of para- and meta-substituted ethyl benzoates in 85% ethanol
Doto
P -%e
-C'6
O f Koch
-0'4
and HOmrnOnd
-0'2
!C
0!2
0'4
0'6
08
Substituent Constant m *
Figure 2. 18
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Sulvobsis of substituted benzyl tosylates in ethanol
TABLE 1 1 1 .
Substituent
given in Tables I11 and IV. The reactions shown in Table I V include rearrangements, eliminations, addition of halogen to olefins, and Diels-Alder additions. I n some of these cases, for example in the Diels-Alder reaction of 1-aryl-l,3-butadienes with maleic anhydride, the original u values are quite sufficient if substituents such as methoxy- and nitro- are omitted (38,47). Conversely, for reactions involving strong electron withdrawal, another relationship is used
(7) Examples include aromatic nucleophilic substitution reactions involving phenol and aniline derivatives where unshared pairs of electrons are located on the atom next to the benzene ring. Typical u- values (33, 38) are listed in Table V. I n these reactions the stabilization of the negative charge is by resonance, which is important only in the case of para-substituents. Since the numerical values of u- are very close to those of u for meta-substituents, they are omitted from Table V. T h e standard process used for defining u- is the ionization of phenols. T h e u- parameters give better correlation than u for many reactions of phenols, phenolic esters, anilines, dimethylanilines, and anilides. An example of the use of Equation 7 is shown in Figure 3 ( 74)* The Taft Equation. Following a suggestion of Ingold (37), Taft (54, 55) developed a procedure to account for both the polar as well as the steric effects in aliphatic and ortho-benzene systems. A polar substituent constant is defined by 1 u* = __ 2.48 [log
(t), ($)*I - log
where k and k, are the rate constants for the hydrolyses of R C O O R ' and CHICOOR', B a n d A refer to basic and acid hydrolysis carried out for the same R ' under identical experimental conditions, and the arbitrary constant 2.48 adjusts u* to the same scale as that for u. The assumption here is that the mechanisms for acidand base-catalyzed hydrolyses are so similar that steric effects should cancel out, and thus k B / k A is a measure of the polar effect alone. A similar approach was applied to ortho-substituted benzenes, using the hydrolysis of o-CH&cH&OOR' as the standard reaction. Thus, the Taft equation is
log
u+mets
-0.16
2"
OH OCH3 CH3
0.047 -0,066 -0.064 -0.059
CzHs C(CH3)a
H
-0.311
-0.295 -0.256 0
0.399 0.359 0.562 0.674
I CN NOz a
=+para
-1.3 -0.92 -0.778
0
c1
0.114 0.135 0.659 0.790
From Brown and Okamoto (73).
TABLE IV.
REACTION CONSTANTS FOR u + CORRELATIONa
Reaction
P
Solvolysis of 2-phenyl-2-propyl chlorides in 90% acetone a t 25°C Chlorination of monosubstituted benzenes in acetic acid a t 25OC Nitration of monosubstituted benzenes by nitric acid in nitromethane or acetic anhydride a t 0' or 25 O C Solvolysis of benzhydryl chlorides i n ethanol a t 25 OC Reaction of l-aryl-l,3-butadienes with maleic anhydride in dioxane a t 45 O C Bromination of monosubstituted benzenes by hypobromous acid and perchloric acid i n 50% dioxane a t 25°C Solvolysis of t-cumyl chlorides in ethanol a t 25 "C Equilibrium constants for formation of ion-pairs from triphenylcarbinyl chlorides i n S O ? a t O°C
-4.54 -8.06 -6.22 -4.05 -0.62 -5.78 -4.67 -3.73
a From Okamoto and Brown (47) and Brown and Okamoto (73).
TABLE V.
(8)
ELECTROPHILIC SUB§TITUENT CONSTANTS u f a
SUBSTITUENT CONSTANTS FOR PHENOLS AND A N I L I N E S
Para-Substituent
u-
COOH COOCHI COOCzHs CONHz CHO
0,728 0.636 0,678 0.627 1.126
Para-Substituent COCH3 CN
NO 2 S02CHa CH=CHNOz
U-
0.874 1.ooo 1.270 1.049 0.88
a Taken f r o m Lejler and Grunwald (38).
($)
= p*u*
Typical values of u* (54, 55) a n d p* are given in Tables V I and VII. A series of reactions of diphenyldiazoVOL. 6 2
NO. 9 S E P T E M B E R 1 9 7 0
19
NOp5-
a Data
Of
methane with aliphatic carboxylic acids (56) is correlated by Equation 9 (Figure 4). Unfortunately, for systems to which Equation 9 applies, steric (entropic) effects are not always negligible; for ortho-benzene and aliphatic systems these can be important. Equation 9 has been modified t o allow for such effects by introducing a standard for the evaluation of steric effects-the acid-catalyzed hydrolysis of esters, defining the steric substituent constant ES as
B"n"ell
I Sub51\luenl Constan< v -
Figure 3. Reaction of para-substituted 2-nitrophenylchlorides with methoxide ion
TABLE V I .
Aliphatic Series Substituent 2.65 H 1.65 CHI 1.05 CZHS 1 .oo CH2CHsCH2CH3 0.85 CH(CH3)z 0.60 C(CH3)3
Substituent
U*
C13C CHaCO ClCH, BrCHz ICHz
C6H5 OCHI OCzHs
CH 3 H
POLAR SUBSTITUENT C O N S T A N T 9 U*
0.49 0.00 -0.10 -0.13 -0 1 9 -0.30
ortho-Substituted Benzene Derivatives -0.39 F -0.35 C1 -0.17 Br 0 .OO I
a From t h e compilation of
TABLE V I I .
where A denotes acid catalysis, and k , is the rate constant for the methyl ester. Then Equation 9 becomes a linear combination of two free energy effects, and for reactions where the steric effects resemble those in ester hydrolysis
0.24 0.20 0.21 0.21
T o f t (56, 5 5 ) .
(94 where s is the sensitivity of the reaction to steric effects. Thus, Equation 9a is in reality a four-parameter expression; however, the saving grace is that the pairs of parameters can be evaluated independently. Typical values for s and Es are given in Table VI11 (38). Influence of reaction conditions. Reaction rates and equilibria vary with the environment, as, for example, with solvent and temperature. The usual convention is to assume that the substituent parameters u are constant, and all variation is in p . The temperature dependence of p can be derived (59, 60) as
POLAR REACTION C O N S T A N T 9
Reaction6 Aliphatic Compounds Ionization equilibrium of carboxylic acids in water a t 25°C Reaction of diphenyldiazomethane with carboxylic acids in ethanol a t 25 'C Sulfation of alcohols with equimolar H2S04, initial H20/H*S04, 1.290, a t 25OC Catalysis of dehydration of acetaldehyde hydrate by R C O O H in acetone a t 25OC Bromination of ketones, C ~ H ~ C O C H R I R ~ , base-catalyzed, in water at 25 "C
P*
1 . 7 2 1 i. 0 . 0 2 5 1.175 i 0.043 4.600 f 0.149 0 . 8 0 1 =t0 . 0 1 5 1.590 f 0.079
ortho-Substituted Benzene Derivatives Ionization equilibrium of benzoic acids in 1 . 7 8 7 zk 0 . 1 3 water a t 25 "C Benzaniiide formation from anilines with 2.660 i 0.22 benzoyl chloride in benzene a t 25'C T u t (55). * From Rates unlss otherwise noted.
Q
Subs.iluenl
C015tonl m *
Figure 4. Reaction of d+henyldiazornethane with aliphatic carboxjlic acids in ethanol 20
INDUSTRIAL A N D ENGINEERING CHEMISTRY
TABLE V I I I .
Aliphatic Substituent H CH3 CzHs BrCHz n-C4HB i-C4Hg
t-CdH, ClzCH BrzCH EtzCH
STERIC SUBSTITUENT AND REACTION CONSTANTS A T 2 5 ' 0
ES 1.24 0.00 -0.07 -0.27 -0.39 -0.93 -1.54 -1.54 -1.86 -1.98
Reaction Constants
orthoSubstituent CH30 CzHrO F c1 Br CH3 I NO2 Ce"
Es 0.99 0.90 0.49 0.18 0.00 0.00 -0.20 -0.75 -0.90 s
Acid-catalyzed hydrolysis of ortho-substituted benzamides in water a t 100°C 0.812 f 0.032 Acid-catalyzed methanolysis of R C O O - / ~ - C I ~ H ~ a t 25°C 1.376 0.057 Reaction of methyl iodide with 2-alkylpyridines 2.065 =I=0.096 in nitrobenzene a t 3OoC A H o for reaction of 2-alkylpyridines with diborane 3.322 =t0 . 3 6 5 a
From Lefler and Grunwald (38).
nitro- groups, especially a t the para-position where u + values may be needed. However, electrophilic substitutions, and some solvolyses and nucleophilic substitutions in which the transition state involves considerable carbonium ion character a t the a-carbon to the benzene ring, require the use of the electrophilic substituent constants u + given in Table 111. Derivatives of phenols and anilines with pairs of electrons on the atom next to the benzene ring require the use of ufrom Table V. For aliphatic and ortho-substituted aromatic systems the Taft equation (Equation 9) should be used with u* as given in Table VI. Steric effects of bulky groups can be incorporated via Equation 9a. As far as extensions to other more complicated systems, the best procedure is probably to choose new reaction series as models. A note of caution is needed to ensure that the reaction conditions are consistent with those in the model series. Effect of Reagents
where A and B denote two reaction series, and the constant is a function of the reference reaction series only. If (dAGB/du) is temperature-invariant, then p should be inversely proportional to temperature. However, in general (bAGB/bu) will not be temperature-invariant except when the reaction series is isentropic, which has been shown to be true only for a few cases, such as, for example, ester hydrolysis (59, 60). O n the other hand, the inverse temperature dependence of p observed in many other reaction series is related to "isokinetic" behavior, which arises when changes in AH and A S are linearly related. The temperature dependence of p cannot be predicted a przori. I n fact, p has been found to change sign with a moderate change of temperature (38). At present, not enough reaction series exist with data over a sufficiently wide temperature range to investigate this behavior. Despite the importance of the solvent in liquid-phase reactions and equilibria, relatively little is mentioned in discussions about substituent effects. Since solvents chosen for the model series are water or aqueous mixtures, the solvent effect on p is probably not often important for these cases. The general topic of solvent effects is discussed in greater detail below. I n summary, the linear free energy relationships provide an excellent correlation for estimating rate constants from data available for related reactions, and these are applicable to various classes of reaction mechanisms. For aromatic systems the Hammett equation (Equation 5) should be used except for ortho-substituents. For aromatic nucleophilic substitutions and for many side-chain reactions the use of u values from Table I is sufficient. Notable exceptions are the methoxy- and
Linear free energy relationships have also been developed to correlate rates and equilibria of chemical reactions where the changes occur at the reaction site itself, due to the effect of reagents. The most common examples of this type are the rates of acid-catalyzed or base-catalyzed reactions, such as dehydrations or hydrolyses. Were the rates will depend on the strength of the catalyst and thus on the ionization equilibrium in solution. This, of course, is a thermodynamic quantity and lends itself readily to correlations of the type discussed here. As an example of the application of this type of technique, if one sought the rate of the acidcatalyzed hydrolysis of methyl benzoate in the presence of formic acid, but knew the rates in the same solvent using acetic acid and trichloroacetic acid as sources of H30+ ions, then the desired result could be predicted quite well from only a knowledge of the pKA's for the three acids. The Brpinsted Relation. Brpinsted and Pedersen (12) found that a simple equation related the rate constant for the acid-catalyzed decomposition of nitramide to the dissociation constant of the acid, KA
(12 )
k = uKA"
where a and CK are constants that depend upon the nature of the substrate, the class of acids, the solvent, and the temperature. The exponent a is known as the Brpinsted slope or Brpinsted coefficient. I n basecatalyzed reactions /3 is generally used in the place of a. By selecting an acid (or base) as a standard, Equation 1 2 can be expressed as a Hammett-type equation log
($)
=
a log
(5) KAO
VOL. 6 2 NO. 9 S E P T E M B E R 1 9 7 0
21
The extent of permissible variation in acid and base structure is much greater here than was possible for the Hammett or Taft equations. The Brgnsted relation has been applied to numerous reactions, including the decomposition of nitramide, dehydration of acetaldehyde hydrate, mutarotation of glucose, and the halogenation of acetone (5, 6, 38). I n Figure 5, the Brgnsted correlation for the acid-catalyzed dehydration of acetaldehyde hydrate (7) is shown as an example for seventeen different acids studied. By assuming a particular reaction mechanism, it is also possible to show that a Rrgnsted relation implies the existence of a rate-equilibrium relationship for one of the individual steps in the reaction (6, 3 8 ) . T h e Brphsted coefficient a or p, which can be related to the slope of the rate-equilibrium relationship, is always positive and never greater than unity. As the acidity difference between the catalyst and the conjugate acid of the substrate becomes larger, the rate of proton transfer becomes diffusion-controlled. All catalysts then become equally effective, regardless of their acid strength (79). At high acid concentrations, an acidity function developed by Hammett must be used to account for the activity effect (70, 27). Of course, the acidity scale is solvent-dependent, as are both a and a in Equation 12. If k and K A are limited to similar solvents or solvent types, the precision of the correlation is improved. For some reactions, aqueous dissociation constants give a set of parallel Brgnsted lines for primary, secondary, and tertiary amines, whereas nonaqueous dissociation constants give a single line. I n the case of aromatic acids two lines corresponding to ortho-, or to meta- and parasubstituted acids are found, as would be expected from the Hammett equation (38). The Swain-Scott Equation. Another correlation is available for base-catalyzed reactions without proton
transfer (53). In a solvent that is stroiigly electropllilic, such as water, the rate of nucleophilic displacement reactions is related by
where k , and k , are the rate constants for the nucleophile in question a n d for water, respectively. The nucleophilicity n is defined relative to k,, the rate constant for nucleophilic displacement on methyl bromide in water a t 25"C, in which the sensitivity s is defined as unity. Typical values of n and s are given in Table IX (59, SO). Note that in general the nucleophilicity is not parallel to the group basicity, but rather seems more dependent on the group polarizability. A n exainple of a correlation similar to Equation 14 is presented in Figure 6 (3). Similar expressions have also been suggested for other such reactions (30, 67). Suhstiturioiis a t the saturated carbon have been successfully correlated by Equation 14. However, the n values obtained are not always applicable to substitutions a t other atoms. Other alternatives are to develop other sets of reagent parameters for reactions involving other atoms (such as sulfur), or to use the Edwards equation (38,59, SO), which atternpts to .take into account both basicity a n d polarizability. However, these developments are of more theoretical rather than practical interest a t present.
TABLE IX.
PARAMETERS FOR THE SWAIN-SCOTT EQUATIONa
Nucleophile Nos'-
71
1.0 1.9 2.5 2.7
Picrate so4--
c1CEI&OOBr-
/1
4 &;id
5
Dirsociol~on Coniton! Loq KA x I O 6
Figure 5. E f e c t of the catalyst on the acid-catalyzed dehydration of acetaldehyde hydrate in water 22
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
5
2.7
3.5
Nucleophile €1POn---HO SCN-
n 3.8 4.2 4.4
I-
s.O
CN SZO 3- -
5.1 5.4
Substrate
S
Methyl bromide ,Methyl iodide Methyl p-toluenesulphonate Ethyl p-toluenesulphonate Benzyl chloride Benzoyl chloride Benzenesulphonyl chloride Chloroacetate Bromoacetate Iodoacetate p-Propionolactone
1 .00 1.15 0.69 0.66 0.87
From Wdis (59, 60).
1.43
1.25 1.00
1.10 1.33 0.77
”
3--
CH3COO-
1.1 3
4
NuCleophiIie Porarneter n
Figure 6. Displacement reaction of p-propiolactone with various nucleophiles in water
Effect of Solvents T h e influence of the solvent medium on chemical reaction rates and equilibria has been approached from many viewpoints. First, a number of semi-empirical treatments have been developed and widely applied, in terms of solvent polarity (47, 48, 52), solute-solvent interactions ( 7 , 2, 77, 23, 34), or corresponding states theory (22). However, we shall discuss here two types of approaches-first, two linear free energy relationships for solvent effects, and second, the more rigorous Brgnsted-Bjerrum expression. Solvent effects on reaction rates often amount to many orders of magnitude, so that methods for predicting kinetic solvent effects can be a powerful tool for the design engineer. This can involve the rational selection of an optimum solvent medium for a given reaction (often this may involve design of a solvent mixture), or the optimization of yields or selectivity for reactions in series or for competing reactions. Solvent selection can be as vital a factor in reactor design for liquid phase reactions as the more commonly recognized effects of flow profile and heat and mass transport. Grunwald-Winstein Equation. T o correlate the solvent effect on solvolysis reactions, Grunwald and Winstein (25) proposed a parameter Y as a measure of the “solvating power” of the solvent, based on the solvolysis of t-butyl chloride in various solvents relative to that in 80% aqueous ethanol as the model reaction. Then the relative solvent effect for a reaction is given by log
(e)
=
smaller for sN2 displacements. For a given substrate no unique value of m is found which could be used for all the solvents. Instead the correlation of log k with Y generally yields a series of almost parallel lines, one for each binary solvent pair. For example, Figure 7 shows two such curves for the solvolysis of benzhydryl chloride ( 6 4* Incomplete solubility of the substrate is a frequent limitation in using solvolysis as a model series in Equation 15 for application to other reaction series, since many additional solvent parameters have to be obtained indirectly. Other frequent complications are the effect of the leaving group, the nucleophilicity of the solvent, and the problem o f “internal return’’ (7, 59, SO). Drougard-Decroocq Equation. Recently a surprisingly simple and seemingly successful relationship has been suggested ( I @ , relating the rates of many reactions to the standard reaction of the Menschutkin addition of methyl iodide to tri-n-propylamine in a series of organic solvents. This can be expressed in the form
where T is a solvent parameter, and X a reaction parameter. Though the authors do not express their results in this form, we have evaluated X and T by assuming X equal to unity for the model reaction in benzene solution. This permits wide application of Equation 16. There are several advantages to this choice. First, rate data (37) are available for a large variety of solvents78 pure solvents and many binary mixtures. Second, benzene is a common solvent used in many kinetic
2 4
I
my
Wells (59, SO) has compiled values of m and Y for pure solvents and mixtures. Note in general that m is approximately unity for sN1 reactions, but is appreciably
-0 8 -3
-2
-1 S o l v e n l Parameter Y
Figure 7. Solvent effect on the solvolysis of benzhydrjl chloride VOL. 6 2
NO. 9
SEPTEMBER
1970
23
studies. A few values of X and r are presented in Table X . The correlation is quite good, and an example is given in Figure 8 (45). Certainly, additional work is necessary to delineate the usefulness and liniitations of this approach. I n the meantime it appears to be of more general applicability than the Grunwald-Winstein equation. The Brgnsted-Bjerrum Equation. Although this is not a linear free energy relationship, a rigorous therrriodynamic expression for solvent (and pressure) effects on the rate of a reaction in solution is provided by the Brgnsted-Bjerrum equation (9, 7 7). This comes directly from Equation 2, and relates the rate constant k to that in an ideal (Raoult’s law) solution k , in terms of activity coefficients y (for a bimolecular reaction)
0
&
The applicability of Equation 17 was first demonstrated by using the Debye-Hueckel theory for ionic reactions in dilute aqueous solution (7, 77, 23), and this very early constituted strong evidence for the transition state approach. The drawback of this approach is the necessity for u przorz information about activities in solution ; although such data might be available for reactants, it cannot bc determined directly for the transition state. However, in sorne cases solution theory ran provide sufficiently good estimates of the activity coefficients without the introduction of any adjustable parameters. For instance, regular solution theory (29) has proven quite successful for many cycloaddition reactions (63). As an example, the Diels-Alder addition of 1,3-butadiene to maleic anhydride is shown in Figure 9 (64,where agreement is good for all solvents except highly polar nitromethane, for which the regular solution theory is inadequate. Other expressions, such as those of Kirkwood and Born, have been applied to dipolar and ionic reactions in terms of dielectric properties of the solvent media (7, 17, 23). Conclusions
A pseudo-thermodynamic approach to the problem of chemical reaction rates and equilibrium does not in any way predict these quantities from first principles; however, it does provide a framework within which limited experimental data can be extended and exirapolated to new situations. Simple tN-o-paranieter linear free energy relationships have been shown to be useful in
&
c
c
c
V 0
m
c
B
S o l v e n t Parameter r
Figure 8. Solvent efect of the ionic decomposztzon of t-butjl-perouyformate catalyzed b) pjridzne
Prediction of Regular Solution Theory 15
TABLE X . PARAMETERS FOR DROUGARD-DECROOCQ EQUATION Solvent Hexane Ethyl ether Carbon tetrachloride Methanol Benzene Tetrahydrofuran Dioxane
r
-3.26 -1.18 -1.11 -0.14 0.00 0.21 0,31
Reaction Parameters (n-C3H?)3N 4- C H d (CZI15)3N BrCH2COOCZH5 Addition of CH3I to Iridium Tetracoordinated Complex Curtius Rearrangment of Benzazide Ionic Decomposition of t-butylperoxyformate Catalyzed by Pyridine
+
24
Solvent Chlorobenzene Chloroform Acetone Dichloromethane Acetonitrile Nitrobenzene Nitrornethane
y T
0.59 0.86 0.92 1.19 1.42 1.43 1.79
T , ‘C 20 20
X 1.00 0.84
Ref (37) (76)
25 65
0.62 0.12
(26) (40)
90
0.44
(45)
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
Nitromet hane
1.0
N
Dichloromefhane OCurbon Disulfide
r
.E
0
P
Tetrachloride 0
-2 0
0
40
80
120
160
200
Cohewve Energy Denslty~b‘)cal/cc
Figure 9. Prediction of solvent efects f i o m the Bi Znsted-Bierrum equation and regular solution theory for the Diels-Alder ieaction of 7,d-butadiene with maleic anhjdride
correlating and estimating the effects of substituents, reagents, or solvent media for many cases. Relations of this kind, though probably justified more on the basis of their utility than any rigorous theories, do not require a correct diagnosis of the actual operating mechanism. However, for exactly this same reason their applicability to any given reaction may not always be predicted a briori. When used with some caution and chemical intuition, these linear free energy relationships can go a long way. Clearly, we have devoted much of our attention to the effect of substituents since the framework is much better established. Comparatively less work has been done in studies of the effect of the reagent and solvent, at least on the basis of the number of types of reactions studied. Perhaps this will change in the future. Linear free energy relationships are not restricted to organic reactions in the liquid only. They have been recognized in many gas-phase reactions (58), inorganic reactions (4, 75, 46), reactions of organometallic compounds (35, 43), and biochemical reactions (28). However, in gas-phase reactions a better alternative (8) is to obtain the kinetic parameters from estimated thermodynamic functions based on bond and group contributions. Unfortunately, there is no comparable method available in the liquid-phase case, because of the complexity of the molecular interactions. The BrgnstedBjerrum equation provides a rigorous approach, but so far it has been applied only to calculating solvent effects in a very few systems, primarily because of the limitations of existing solution theory and of our knowledge of the properties of transition states. However, for a great many situations that are of interest to engineers, these linear free energy relationships can be of value, Nomen c la t ure a
= constant in the Brpinsted equation
steric reaction constant G = Gibbs free energy H = enthalpy k = rate constant K = equilibrium constant K A = acid dissociation constant m = reaction constant in the Grunwald-Winstein equation n = nucleophilicity R = gas constant s = reaction constant in the Swain-Scott equation S = entropy s = steric substituent constant ’ T = temperature Y = solvent parameter in the Grunwald-Winstein equation E8 =
Greek Letters = Brpinsted slope for acid catalysis p = Brpinsted slope for basic catalysis y z = activity coefficient X = reaction constant in the Drougard-Decroocq equation Y = frequency p = Hammett reaction constant p * = polar reaction constant u = Hammett substituent constant u + = electrophilic substituent constant U= substituent constant for phenols and anilines U* = polar substituent constant = solvent parameter in the Drougard-Decroocq equation T
a
Superscripts = activation ’ = standard state
*
REFER ENCES (1) Amis, E. S.,“Solvent Effects on Reaction Rates and Mechanisms,” Academic Press, New York, 1966. (2) Baekrlmans, P., Gielen, M., and Nasielski, .I., Ind. Chim. Belge, 29, 1265 (1964). (3) Bartlett, P.D., and Small, G., J . A m e r . Chem.Suc., 72, 4867 (1950). (4) Basolo, F., and Pearson, R . G.,“Mechanisms of Inorganic Reactions,” 2nd ed., Wiley, New York, 1967, pp 166, 397-400. (5) Bell, R . P., “ Acid-Base Catalysis,” Oxford University Press, Oxford, 1941, Chapter V. (6) Bell, R . P.,“The Proton in Chemistry,” Methuen, London, 1959, pp 155-182. (7) Bell,R. P., and Higginson, W . C. E., P7oc. RuyalSuc., Ser. A , , 197, 141 (1949). ( 8 ) Benson, S. W.,“Thermochemical Kinetics,” Wiley, New York, 1968. (9) Bjerrum, N., Z . Phys. Chem. (Leiprig),108, 82 (1924). ( 1 0 ) Boyd, R . H., in “Solute-Solvent Interactions” Coetzee, J. F., and Ritchie, C. D., Eds., Marcel Dekker, New York, 1969, Chipter 3. (11) Brgnsted, J. M., Z . Phys. Chem. (Leiprig), 102,169 (1922). (12) Brgnsted, J. M., and Pedersen, K . J., ibid., 108, 185 (1924). (13) Brown, H. C., and Okamoto, Y., J . Amer. Chem. Suc., 80, 4979 (1958). (14) Bunnett, J. F., e t a l . , ibid., 75, 642 (1953). (15) Diebler, H., and Sutin, N., J.Phys. Chem., 68, 174 (1964). (16) Drougard, Y., and Decroocq, D., Bull. Suc. Chim. Fr., 2972 (1969). 59 ( 9 ) , 2 0 (1967). (17) Eckert, C.A., INn. ENG.CHEM’., (18) Ehrenson, S., Pragr. Phys. Org. Chem., 2, 195 (1964). (19) Eigen, M., Angew. Chem., 75, 489 (1963). (20) Evans, D. P., Gordon, J. J., and Watson, H . B., J. Chem. Soc., London, 1937, 1430. (21) Farcasiu, D., Stud. Cercet. Chim., 14, 37 (1966). (22) Frisch, H. L., Bak, T. A , , and Webster, E. R., J . Phys. Chem., 66, 2101 (1962). (23) Glasstone, S., Laidler, K. J., and Eyring, H . , “ T h e Theory o f R a t e Processes,” McCraw-Hill, New York, 1941. (24) Gould, E. S., “Mechanism and Structure in Organic Chemistry,” Holt, Rinehart and Winston, New York, 1959. (25) Grunwald, E., and Winstein, S., J.Amer. Chem.Suc., 70, 846 (1948). (26) Halpern, J., and Chock, P.B., lOthInt. Conf. Coord. Chem., 1967. (27) Hammett, L. P., “Physical Organic Chemistry,” McGraw-Hill, New York, 1940. (28) Hansen, 0. R . , Acla Chem. Scand., 16, 1593 (1962). (29) Hildebrand, J. H., and Scott, R . L., “ T h e Solubility of Nonelectrolytes,” 3rd ed., Dover, New York, 1964. (30) Hine, J., “Physical Organic Chemistry,” 2nd ed., McGraw-Hill, New York, 1962. (31) Ingold, C. K . , J . Chem. Soc., London, 1930, 1032. (32) Ingold, C. K., and Nathan, W. S., ibid., 1936,222. (33) JaffC,H. H.,Chem.Reu.,53, 191 (1953). (34) Jungers, J. C., and Sajus, L., Rev. Znrt. Fr. Petrole Ann. Combust. Liguides, 21, 109 (1966). (35) Kabachnik, M. I., Z . Chem., 1,289 (1 961 ). (36) Kochi, J. K., and Hammond, G. S., J . Amer. Chem. Soc., 75, 3445 (1953). (37) Lassau, C., and Jungers, J. C., Bull. Soc. Chim. Fr., 2678 (1968). (38) Leffler, J. E., and Grunwald, E., “Rates and Equilibria of Organic’Reactions,” Wiley, New York, 1963. (39) March, J., “Advanced Organic Chemistry: Reactions, Mechanisms, and Structure,” McGraw-Hill, New York, 1968. (40) Newman, M. S., Lee, S. H., and Garrett, A. B., J. Amer. Chem. Suc., 69, 113 (1947). (41) Okamoto,Y., andBrown, H . C., J . Org. Chem., 22,485 (1957). (42) Pal’m, V.A., Usp.Khim., SO, 1069 (1961). (43) Perevalova, E. G., Gubin, S. P., and Nesmeyanov, A . N., Reakts. Sposobnust Org. Soedin., 3 ( l ) , 68 (1966). (44) Pickles, N. T. J., and Hinshelwood, C. N., J . Chem. Soc., London, 1936, 1353. (45) Pincock, R . E., J . Amer. Chem. Soc., 86,1820 (1964). (46) Prue, J. E., J . Chem. Sac., London, 1952, 2331. (47) Reichardt, C., Angew. Chem., 7 7 , 3 0 (1965). (48) Reichardt, C., and Dimroth, K., Fortschr. Chem. Forsch., 11, 1 (1968). (49) Ritchie, C. D., and Sager, W . F., Progr. Phys. Org. Chem., 2, 323 (1964). (50) Schein, S. M., and Kozorez, L. A , , Reakls. Sposobnort Org. Soedin., 3 (41, 45 (1966). (51) Shorter, J., Chem. Brit., 5,269 (1969). (52) Stefani, A. P., J . h e r . Chem. Soc., 90, 1694 (1968). (53) Swain, C. G., and Scott, C. B., ibid., 75, 141 (1953). (54) Taft, R. W., ibid., 74, 2729 (1952). (55) Taft, R. W., in“Steric Effects in Organic Chemistry,” M . S. Newman, Ed., Wiley, New York, 1956, Chapter 13. (56) Taft, R. W., and Smith, D. J., J . Amer. Chem. Soc., 76, 305 (1954). (57) Thorn, R . J., J . Chem. Phys., 5 1 , 3582 (1969). (58) Trotman-Dickenson, A. F., Chem. Ind. (London), 1243 (1957). (59) Wells, P. R . , Chem. Rev., 63, 171 (1963). (60) Wells, P. R., ‘LLinear Free Energy Relationships,” Academic Press, London, 1968. (61) Wilputte-Steinert, L., and Fierens, P . J., Bull, Soc. Chim. Belg., 6 5 , 719 (1956). (62) Winstein, S., Fainberg, A. H., and Grunwald, E., J. Amer. Chem. SUC.,79, 4146 (1957). (63) Wong, K. F., and Eckert, C. A,, Ind. Eng. Chem., Process Des. Develop., 8 , 568 (1969). (64) Wong, K. F., and Eckert, C. A,, Trans. Faraday Suc., in press, 1970.
VOL. 6 2
NO. 9
SEPTEMBER 1970
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