MoIecdar I hermo-
Chemical Reactions
1
CHARLES A. ECKERT
The techhiques of molecular
to problems in chemical kinetics. The transition state model reduces the rote process to a much more . .
tractable equilibrium problem. Accurate predictions may be made for the
e k i of pressure and solvent on the rote of a chemical reaction an the
basis of thermodynamic data.
10
INDUSTRIAL AND ENGINEERING CHEMISTRY
any recent studies in chemical reaction kinetics M have been directed toward the physical aspects of the reactor problem-optimization, kactor dynamics, transport phenomena. However, less attention has been given to an equally important facet of the problem, the chemistry of the reaction itself. What can the chemical engineer do to iduence the speed of the chemical reaction or the relative speeds of competing reactions? An answer to this question would pmvide a powerful tool for the rational design of chemical reaction systems. The rate of a chemical reaction is largely a function of h e thermodynamic state of the system-the temperature, he pressure, the concentration of reactants and the lolvent, or inert species present. For an elemcntiw -action, A B + product h e dependence of rate on concentrations of reactants is
+
1
)@${$
Rate = k [ A l p ]
~~~~~~~~~~
The dependence of the rate constant on temperature is given by the Arrhenius law in terms of an activation en-, but also the effects of pressure and solvent can be important. We consider here how these e5ects on the rate of a reaction can be treated in terms of thermdynamics. Pressure effects on condensed-phase reactions are generally small, unless extremely large pressures are used. Although running reactions at these high pressures is usually not a feasible industrial practice, it is an excellent laboratory technique for the investigation of the molecular details of the reaction and, as such, is considered below. The effect of pressure on the rate of a gas-phase reaction is larger, and this effect can often be treated most adequately in t e r m of thermodymnics. Solvent effects on reaction rates can be enormous.
For example, Cram and coworkers (6) investigated the racemization of (+) 2-methyl-3-phenylpropionitrile in methanol and dimethylsulfoxide solutions, observing solvent effects on the rate constant as great as a factor of about 100. Although thii is an extreme case, solvent effects of the order of a factor of 100-1000 are common. In this paper we show how the engineer can influence the rate of chemical reactions or adjust the relative rates of competing reactions by a rational choice of solvent or solvent mixture, based on a judicious application of molecular thermodynamics. The approach here is the use of the absolute rate theory to reduce the rate problem to an equilibrium, or thermodynamic problem. The advantage of this is that thermodynamics is far better understood than rate processes. Furthermore, advances in molecular thermodynamics and applications of statistical mechanics provide much insight into the problems of phase equiVOL 5 9
NO. 9 S E P T E M B E R 1 9 6 7
21
libria and solution theory. Such techniques can be applied in conjunction with existing knowledge about intermolecular forces to characterize the large effects on chemical reaction rates of pressure and solvents. Theory of Absolute Reaction R&s
The link between the rate of reaction and thermodynamics is provided by the theory of absolute reaction rates (77) which postulates the existence of an intermediate species M in an elementary reaction, called the activated complex.
A
+B
-L
u
M + product
The complex, an association of A and B resulting from the collision, is a normal molecule except for its instability with respect to motion along the reaction coordinate. This activated complex is considered to be in strict thermodynamic equilibrium with the original reactants,
LzA= -'
A+B=M
0
and the reaction rate constant is assumed to be proportional to the concentration of complex M, found from the chemical equilibrium considerations,
[MI
= K[A][B]
-
I,& = - 1
! p c 7.
I
o. I
0.2
U.J
IIII-C of ionic strength on rates of reactions bttu .... ...u
spherical ion of diameter a in solution is expressed in terms of the charge number Z and the dielectric constant of the solvent, e, as
. YAYB YM
where K is the equilibrium constant, and the y's are activity coefficientsin the mixture. Equation 1 does not provide an absolute value of the chemical reaction rate constant, k, but yields relative values of k for differing conditions. This relationship was first applied by Bjerrum and Br$nsted (3,5)who showed that the rate of reaction in a thermodynamically nonideal system is given by
(4)
where
and I is the familiar ionic strength:
Rate = k, YAYB - [A][B] YM
where k, is the rate constant in the ideal reference system (where = yB = yM = 1). Then, if the activity coefficients are evaluated in terms of some theory of solutions, and the basic rate constant k. is a function of temperature only, then the apparent rate constant k in any real solution is given by
k = k,
re)
(3)
The original example that was used to demonstrate this approach was the study of reactions between ions in dilute aqueous solution, using the Debye-Hiickel treatment. According to this approach, discussed in many common texts (20, 28), the activity coefficient of a 22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
If the solution is dilute, as it must be for the theory to be valid, l > > K a
Using this result in conjunction with Equation 3, and noting that Z M = ZA ZB we find the variation of the rate constant of an ionic reaction with ionic strength to be
+
(7) where A' is a constant for a given solvent equal to 1.018 for water at 25" C. This result has been compared with experiment on many occasions and is in quantitative agreement. A typical comparison is shown in Figure 1 (.
We now proceed to consider a number of other cases where Equation 3 predicts the variation of the rate of a chemical reaction by accounting for the nonidealities through a suitable solution theory. 0.1
-8-Phaso
I
Reactions
For a homogeneous gas-phase reaction the rate constant is a function of temperature only for the ideal gas, but for a real gas it varies with density. For a bimolecn (8) the Br#nsted-Bjermm equa-
f
(8) 4.4
here the 4's are the fugacity coefficients of the reactants and activated complex and I is the compressibility factor >f the gas mixture. If one applies the virial equation of
1 I
-0.6
0
1
I 2
-A, = = = I + - + -B+ C
RT
I 3
I 4
I
I
5
6
mmda
v o '
where B and C are the second and third virial coefficients, respectively, the rate constant, relative to that in the ideal gas, is given by
where the summations are over-all components in the aixture, and
0,= B , Acjk
+ BBI - B M ~
= CAja
+
CBjk
- CMjk
Equation 10 is general, but limited to the region of conrergence of the virial equation, as well as to the number d coefficients that can be evaluated. Equation 10 was applied to the pyrolysis of hydrogen iodide:
+
2HI + Ht It ising a model of the complex based on that of Wheeler, ropley, and Eyring (43), and virial coefficients estinated by corresponding states theory. The results for the variation of the rate constant with density (l/o) are in excellent agreement with experimental results of Kistiakowsky (23),as shown in Figure 2. By contrast, an alternate prediction is also shown in this figure for the variation of rate based on the assumption that the rate is proportional to the activity of the reactants only, neglecting the activity of the complex, as suggested by Whalley (42). This assumption clearly predicts a variation in the wrong direction. Another interesting application is provided by the effect of a gas-phase solvent on such a reaction. Mills
and Eckert (30) used the modified Redlich-Kwong equation of state (34) to calculate the effects of various diluent g a m at elevated pressures on the same reaction. After verifying that thii equation of state also would give predictions in agreement with the data for decomposition of pure HI, they attempted to predict the rate of pyrolysis of HI very dilute in various other gases. If no competing reactions with the other species present are assumed, the type of results obtained are shown in Figure 3. In general, the effect was largest for a diluent gas near its critical point-in other words one which exhibits stronger intermolecular interactions with the activated complex to stabilize it relative to the reactants. Any valid equation of state could be used to study the variation with density and composition of any elementary gas-phase reaction. However, one munt have some basis for evaluating the constants in the equation pertaining to the structure and interactions of the activated complex. Solvmt h
h on Liquid-Phare Reactions
Effects of the solvent on rate of reactions in the liquid phase can be large. Recent discussions of many aspects of this problem include the work of Amis (7) and Baekelmans, Gielen, and Nasielski (2). Frisch, Bak, and Webster (9) have developed a moderately successful corresponding states treatment to correlate relative rates of certain types of reactions in various noninterVOL 59
NO. 9 S E P T E M B E R 1 9 6 7
23
I
F i w e 3. Solvent efecls on n gar-pharc reachn. The pyrolysis of hydrogen iodide at booo K.
acting solvents. In principle one could use the Br@sted-Bjerrum equation with appropriate experimental data on activities in solution to calculate solvent effects. However, because the required data are not generally available, we take here a few examples to show how solution theory can be used in conjunction with Equation 3 to predict solvent effects in certain cases. The regular solution theory of Scatchard and Hildebrand (78, 79) provides a good method for the prediction of mixture properties from the properties of the pure components for nonionic, nonpolar, or relatively nonpolar, liquids. Glasstone, Laidler, and Eyring (77) have discussed the idea 0f.applying the regular solution concept *"the -Br@nsted-Bjerrumequation for solvent effects, although the necessary relationships were not completely developed and no application was made. This application has been carried out by W o n g and Eckert (45). Hildebrand (79) shows that the activity coefficient of any species in a multicomponent mixture is given by
RT In y ,
= u,(&
- 8)'
I
2o
1.4 1.2
3 3 0.8 f 0.6 -
IB
-
0.4
O2
0
4.2 4.4
(11)
where u , = liquid molar volume of component i 8, = solubility parameter of component i 8 = average solubility parameter of solution =
CVf i
-0.6
4.8 -1.0
The solubility parameter is the square root of the cohesive energy density, which is usually taken to be the 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
0
I
M
I
40
I M)
I
80
I
100
I
l
120 140
l
I
1M)
180
a
tiW m q y d m IW ml/a.
a, = volume fraction of component j
24
-
1.0
soi&ts,
oft of isoprcncma[cicanhydridc reaction in umiour Comporisn with prediction of regular solution t+
t .d
I 1 .
are in reasonable agreement for nine of the 10 only for highly polar nitromethane is there
, even for nonionic reactions, the ch is often inadequate because of s of either reactants, complex, or An alternate approach is to characterize the thermodynamics in terms of the polar internd use this to predict solvent effects. ood (22) has shown that the free energy repolarizable molecule with a point the center of a sphere of radius r medium of dielectric constant e is
which is to take into account nteractions, such as London etc. The electrostatics here corrections can be made (4) herical molecules, real dipoles, a1 dipoles, and polarizable dipoles. Although change the value of the coefficient, they do not the functional dependence of the free energy on ielectric constant. Equation 13, applied to an ry bimolecular reaction, gives the result
Figure 5. SoImnt effecfs on reaction of pyridiw with methyl iodide at 50' C.,1 otm.
configurational energy divided by the .molar volume. Combining Equations 3 and 11, we obtain
and, if the reaction is being run at high dilution in some solvent, then to an excellent approximation 8 is the solubility parameter of the solvent. Equation 12 has been applied to Diels-Alder reactions (the simple, bimolecular ring condensation of a diene Nith an olefin). In these reactions, reactants and complex are not highly polar, and the thermodynamic propsties of the active state can be estimated from the known itructure by the recommended techniques of Reid and Sherwood ( 3 3 , and the solubility parameter of the :omplex is estimated (29). For example, calculations have been worked out for the Diels-Alder condensation 3f isoprene with maleic anhydride
usual procedure for applying Equation 14 inneglecting the last term. This is justified on the in some cases contributions from dispersion interactions are relatively small in relation to
If this assumption is made, the rate in various solvents should constant factor (e - 1)/ be proportional to the facto
to yield l-methyl-cyclohexene-4,5-dicarboxylic anhydride. Figure 4 shows a comparison of the predictions of regular solution theory with experimental data (7),
example, Equation 14 is applicable to the Menreaction between a tertiary amine and an alkyl give a quaternary ammonium salt. The recomplex are all polar but nonionic, whereas VOL 59
NO. 9 S E P T E M B E R 1 9 6 7
25
Table I. Parameters for Ccllculatlng Solvent Effects I1 at 25' Cl ~
x
7
- lcal./cc.)~~ - r./grmd. lll./4U'
the ionic product facilitates analysis. For instance, the results of Hartmann et al. (76) for the reaction of ppidine and methyl icdicje are given in Figure 5.
9.88 7.61 9.4
Pyridine Methyl iodide Activated complex Carbon tetrachloride
The data of Heydtmann e6 al. (77) on the Menschutkin reaction of a-picoline with o-bromoacetophenone are given in Figure 6. These plots are typical of the degree of agreement between the data for Menschutkin reactions and Equation 14, neglecting the final term. There are two major reasons for the discrepancies that do occur. First, the electric field in the vicinity of a dipole is enorm o u e a t a distance of 2 A. from a 1-Debye point dipole the field would be close to 108 volts/cm. A polar dielectric undoubtedly becomes saturated in such a field, and no account is taken of such an effect. Second, the term dropped from Equation 14, primarily because of the difficulty in evaluating it, may often be important. Certainly the large deviation for methanol in Figure 5 must be, at least in part, due to the neglecting of the strong association of that solvent. In the case of methanol, one may cite a strong, specific effect, but in all cases there exist variations of the dispersion forces which are not accounted for by the theory. One possible way to account for both the polar and nonpolar types of intermolecular interactions would be to adopt a modification of regular solution theory for polar materials. Van Arkel (37,38) suggested an approach for the interaction of two polar species where the configurational energy density of a polar molecule is broken into a Dolar contribution r and a nonpolar
i-Propyl ether
8.60 7.00
Mesitylene
8.88
Toluene
8.95 9.20 7.81 9.27 8.77 9.43 9.22 9.59 7.66 9.32 9.70,
Benzene Chloroform Chlorobenzene Dioxane Bromobenzene Anisole lodobenzene Acetone Benzonitrile Nitrobenzene
Table 11.
Sol
Carbon tetrochloride" i-Propyl ether Mesitylene
+
xput)*i*t
X
[(Xi
- ASa +
- 2 h t l (16)
where $la is the induction contribution found empirically for a class of molecules. 26
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.554 0.41 9.2 9.4 9.9
17 19
Anisole lodobenzene Acetone Benzonitrile Nitrobenzene
TI*
-
cold.
Chloroform
Bnxnobenzene
(Xi01
5.60 2.06 4.80 2.65 3.05 2.45 6.14 4.50 4.89
97 141 140 107 89 81 102 86 105 109 111 74 102 103
mnt M a c k
Chlorobenzene Dioxane
AsE
0 0 0 0
k.10"O Sdvenl
Benzene
Weimer and Prausnitz (40) modified this approach to account for induction contributions and applied their result to get good estimates of activities of a nonpolar material (2) in an excess of polar solvent (1).
0
80.9 62.2 137.4
d i d o Roc
Toluene
Then, assuming zero excess entropy, the excess Gibbs enerm of a binary mixture of polar species is given by
3.71 6.10 12.0
18 23 25 22 23 10 19
0.354
16
0.90 0.88 3.9
32 32 32 32 16 32 32
5.7 7.21 7.4 21 21.4 22 24 29 43 64 123 180 197 210
16
32 32 32 32 32 16 32 16 32 __
Using a combination of Equations 15 and 16 with Equation 3, it is possible again to estimate the kinetic solvent effect. Calculations have been made for the Menschutkin reaction of pyridine and methyl iodide, using the parameters listed in Table I. These were determined primarily by the method of Weimer and Prausnitz (&), except for the volume of the activated, complex, which has been measured experimentally (72). The rates calculated in various solvents are compared with the data in Table I1 where the rate in the inert solvent CClr is taken as the reference. For nonpolar and moderately polar solvents the results are good, but for highly polar materials the rate measured is greater than that predicted, probably owing to specific solvation effects of solvents such as acetone, benzonitde, and nitrobenzene on the highly polar Menschutkin complex. A development similar to that for polar molecules may be carried through for the solvent effects on ionic re actions in solution (77). The electrostatic free energy change to take a spherical ion of radius r from a vacuum and put it into a dielectric medium is given by
0.7 0.5
I
I 0.3
I 0.35
0.40
0.
O i r a k l l h kl/l.+ll
(17)
2r
If we combine Equations 17 and 3, neglect nonelectrostatic effects, and assume small concentrations to negate the effects of ionic strength (discussed above), we get
k k,
-+--("
ln-=-
2)
2dBT rA '* (18) Except at low values of the dielectric constant, Equation 18 is in good agreement with experiment, as shown by the reaction of bromoacetate and thiosulfate ions ( 2 6 , n ) in Figure 7.
I
FiEurc 6. SolOmr effech on rmction of a-picolim with o-bromoaceb-
pkMm at 300 c., 7 atm,
I
ERed of Pnrrun on Liquid-Phase Reactions
The effect of pressure on reactions in solution, though small at ordinary pressures, can be appreciable if pressures in the kilobar range are applied. This effect of pressure on the kinetic rate constant can give a great deal of information about the nature of the activated :omplex (72, 74, 75, 47). Because the chemical red o n can be characterized by the equilibrium be:ween reactants and the activated complex, the effect If pressure on the equilibrium gives, from strictly :hemnodynamicarguments,
- . -u--.
where Av* is the difference in volume, or more exactly .n partial molal volume, between the activated complex md the reactants. AD*
-
f l ~ gA
between ionr
- @B VOL 59
NO. 9
SEPTEMBER 1 9 6 7
27
10-1
0
I 1
I
I
I
I
2
3
4
5
I
6
7
muun IkiWnl Figure 8. Effectof pressure on th Dick-Aldar dimerization of isoprene
Figure 9. Effectof presmrc on thc p y r i a 7 m t h y I ioa
lclion in
um'our solwnfs
If the rate constant k is based not on concentrations, but on mole fractions, kz,then the isothermal compressibility of the solvent fl enters for a bimolecular reaction as
where Avo* is a constant, namely the volume change on activation in an ideal solution. Then a plot of the lefthand side of Equation 20 us. the pressure derivative of the dielectric constant factor (a 1)/(2a l), should yield a straight line of slope - p a - - - -rA'
However, this additional term is usually small-an order of magnitude or smaller than the Au* term. High pressure measurements give, in effect, the partial molar volume of the active complex. For example, Figure 8 shows results (39) for the Diels-Alder dimerization of isoprene. The slopes at zero pressure give volumes of activation of -24.3 and -25.6 cc./g.-mole at 60" and 75' C.,respectively. In this case, the experimental information was of value in helping to determine the structure of the reaction intermediate. The effect of high pressure on the pyridine-methyl iodide Menschutkin reaction has been measured (76) in several solvents, with the results shown in Figure 9. If nonelectrostatic interactions are neglected, differentiation of Equation 14 with respect to pressure yields
-RT
28
a In k (bp),= Av*
=
Avo*
-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
+
-
IAAa
'"> rBa
Because the same quantity is also available from the plot of the atmospheric pressure results, I n k us. (e - l ) / (2. l), an internal check is possible. Figure 10 shows such a plot of the data, and despite some scatter, the dope is in reasonable agreement with the results of Figure 4. An example of a complete study of a single reaction is provided by the work of Heydtmann, Schmidt, and Hartmann (77) on the Menschutkin reaction between a-picoline and w-bromoacetophenone. These authors measured the rate of this bimolecular addition of an amine and an alkyl halide over a temperature range of 30' to 50' C. and a pressure range of 1 to 500 atm. in nine different solvents. Their low pressure results were shown previously (Figure 6). The results for the effect of pressure are presented in the same type of plot as above, b* as a function of the pressure derivative of the dielectric constant factor (Figure 11). Furthermore, because the data were obtained over a range of
+
M
"
Figure 70. Effect of prcssure on pyridinsmelhyl iodide rcnction
01
50' C.
Figure 7 7 . Effect of presswe on a Mawhutkin reaction: or-picoline with w-homoac~toplunoncat BoC.
temperatures, they were able to consider the temperature derivative
-g*) P
where AG*, the Gibhs energy of activation, is related to the rate constant through the equilibrium constant for complex formation according to
AG* = - R T l n K
(214
Then when we neglect nonelectrostatic effects,
(22) I'hus, the entropy of activation, obtainable from the temperature derivative of the rate constant, should be l)]. This linear in the factor (b/bT)[(e - 1)/(2s result is shown in Figure 12; the slope is in good agreement with that in Figure 11, and both give reasonable values of p1/r8for the complex. These authors attributed deviations from the lines for some of the solvents to nonelectrostatic interactions. They made separate measurements of the partial molal volumes of the reactants in each solvent, and these results substantiated the fact that the w-bromoacetophenone did undergo some specific
+
~i~~~~72. Effect of tampaorurc on Mmcchurkin reaction: picoline with w-bromoocctqhnonc ot I o h . ~
VOL 59
NO. 9 S E P T E M B E R 1 9 6 7
29
13
i 112
t
I
j!
11
i I
/
! '0 i
.
09
08
~
I
I
0
I
a6
I
+ HZ4 CsHlr CHaCOCHa + HZ4 (CHa)zCHOH CsHlo
Figurc 13. Comparirinc catalytic hydrogenation of cyclokxm and acttom in m i u s solwnts
interactions with just those solvents for which the largest deviations occurred. Hehmgeneour Readions
Because many important industrial reactions are heterogeneous, it would be desirable to devise some method for treating reactions of this type as well. Basically, the problem encountered is what assumptions to make to treat a heterogeneous rate problem in terms of equilibrium thermodynamics. Consider a bimolecular reaction occumng on a catalyst surface in solution. Steps involved include diffusion to and from the surface, adsorption and desorption, and the reaction iwlf. For our purpow here, we deal only with cases where it may be assumed that the reaction itself is the rate-limiting step. Then Equation 3 is still valid, with the proviso that the activity coefficients refer to molecules on the surface. T o carry this out, let us take the example of the competitive hydrogenation of cyclohexene and acetone on a nickel catalyst in various solvents, as studied by Jungers (27). The experimental data were taken at 25" C., a low temperature for a hydrogenation; therefore, it seems likely that the chemical reaction is truly the rate controlling step. There are two simultaneous reactions: SO
INDUSTRIAL A N D ENGINEERING CHEMISTRY
with rate constants kl and kz, respectively; it is the relative rate kI/kz which is the quantity of interest. The variation of the relative rate from solvent to solvent is given by
"p) .an kz
.%I
e
?MI
(23)
?A*
where AI and As refer to the cyclohexene and acetone, respectively, and MI and MI refer to the corresponding activated complexes. T o evaluate activity coefficients, it is convenient to use regular solution theory. Although it cannot be said to apply exactly to the polar species involved here, it will suffice for qualitative comparison. In so doing, there are two possible assumptions that may be made concerning the evaluation of the 7's: First, one could assume that all species are in equilibrium with the bulk solution, and that variations in activity on the surface may be found from the activity coefficients evaluated in the bulk. This assumption gives
The alternate possibility is to state that whereas the reactant on the surface is certainly in equilibrium with the bulk, because we have implicitly asoumed an adsorption equilibrium, the activated complex for a cata-
dace species, not in equilibrium with any complex in the bulk Thus, the aciivities of the complexes are independent of the solvent used and their activity coefficients should not vary and may be included in the constant term to give
These expressions are compared with Junger's data in Figure 13. The solid line is Equation 25 and it is in good qualitative agteement with all data except that for the alcohol, which would be expected to deviate most from regular solution theory. The dotted line corresponds to Equation 24, and was calculated assuming that the properties of the complexes are similar to those of the product of each reaction. This line indicates a trend in the opposite direction from the data. These results suggest that the second assumption is valid for this type of reaction-namely, that the activated complex on the surface is unaffected by the solvent.
' I
b
Design of Solvents
One useful application of thermodynamic analysis of reacting systems lies in the design of solvents for optimization of reactions. The techniques described above usually permit qualitative design and in many cases quantitative results can be obtained. Thus, a valid thermodynamic treatment of solutions permits maximization (or minimization) of a given reaction rate by a proper choice of solvent or solvent mixture. Also, as in the heterogeneous reaction described above, one may work with the relative rates of competing reactions. A successful treatment requires a thermodynamic treatment of multicomponent mixtures, hut this should not be a barrier. Techniques for this were originally proposed by Wohl (44) and in recent years have been developed into workable schemes for the calculation of multicomponent properties from a minimum of experimental data (33). Analysis of a reacting system often shows that the best solvent for a given case is one which is, for example, as highly polar as possible, or perhaps one with the lowest possible cohesive energy density. On the other hand,
". -
i
FigUi and bviyl carbitol
Charles A . Eckert is Assistant Frofessor of Chemical Engineering at the Uniunsity sf Illinois, Urbana, Ill. The author acknowied8es the financia6 suppmt of the National Science Foundation for this work. AUTHOR
V O L 5 9 NO. 9 S E P T E M B E R 1 9 6 7
31
it is sometimes possible to design a solvent mixture which has better properties than either of the pure solvents in it. Grieger (73) has shown that it is possible to design a solvent mixture to minimize or maximize the activity of a given component. IYong and Eckert (45)have applied this concept to chemical reaction rates. As an example, suppose it were desired to minimize the rate of the DielsAlder dimerization of isoprene. Calculations based on regular solution theory show that a mixture of about 30 volume % of neopeiitane with carbon disulfide will give a minimum rate. The rate in pure carbon disulfide is about 6% faster than in the mixture, and in pure neopentane over 35% faster. These results are shown in Figure 14. Although the effect is not great for this case, it does demonstrate that solution theory does predict an extremum. For this case a more polar solvent mixture, with greater thermodynamic nonidealities, would show a much larger effect. Experimentally, a number of such phenomena have been observed. Kondo and Tokura (24) quote similar effects for other types of reactions. Another example is given by the Wolff-Kishner reaction of benzophenone hydrazone (36), which shows an extreme maximum in the rate in a mixed solvent of dimethyl sulfoxide and butyl carbitol (Figure 15). The use of the techniques presented here, along with sufficient thermodynamic data or an applicable solution theory, should permit the design of optimum solvent mixtures for other reactions. Conclusions
This paper has attempted to demonstrate that it is possible to solve certain problems in chemical kinetics using the absolute rate approach and the techniques of thermodynamics. Many of the methods of molecular thermodynamics and solution theory, yielding insight into molecular processes. are powerful tools in attacking kinetic problems. For some t>pes of reactions, adequate methods are now available for predicting how the rate of reaction is affected by the thermodynamic variablespressure, temperature, and solvent medium. But for many other classes of reactions a great deal remains to be done. and future advances will depend on developments in molecular physics and increased understanding of intermolecular forces. NOM ENC LATU RE = diameter of an ion A = Helmholtz energy A ’ = constant in Debye-Huckel equation B = second virial coefficient C = third virial coefficient CI
e
E g
G I k k,
= = = = = = =
ke = K = n
P 32
= =
unit charge internal energy molar Gibbs energy Gibbs energy ionic strength kinetic rate constant rate constant in reference system Boltzmann constant equilibrium constant number density pressure 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
1’
=
R
= = = = = = = =
S
T u x
y z
2
radius of a molecule gas constant entropy temperature molar volume liquid mole fraction vapor mole fraction compressibility factor charge number of an ion
Greek letters p
isothermal compressibility = activity coefficient = solubility parameter e = dielectric constant = defined by Equation 5 K X = nonpolar solubility parameter ,U = dipole moment 7 = polar solubility parameter 4 = fugacity coefficient = volume fraction @ ’ = term characterizing nonelectrostatic interactions ~ 1 = 2 constant for induction contribution to excess Gibbs energy =
y 6
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