July 20, 1955
RATES OF SOLVOLYSIS WITH A FOUR-PARAMETER
mixed crystals. However, its identity was shown by a kinetic analysis. In 50% acetone-50% water solution i t hydrolyzed with a half-life of 980 sec. (as. 960 sec. fo: a n authentic sample of pure trityl acetate, m.p. 87-88'), in a strictly first-order fashion (followed to 88% reaction), and the endpoint ( a t ten half lives) indicated a quantitative yield of trityl acetate from the trityl fluoride present in the original starting material. Absence of trityl fluoride was shown by titration of the solution for fluoride ion using thorium nitrate and sodium alizarinsulfonate after completion of the solvolysis. As little as 5% trityl fluoride would have been detected by this test. Although a fourfold change in concentration of acetate ion had no observable effect on the rate when it was in escess over the trityl fluoride, omission of the acetate ion resulted in incomplete acetolysis. Evidently there is a considerable effect of the acetate ion on the position of equitibrium, similar to the effect of pyridine on the methanolysis of trityl chloride in benzene solution.4a Kinetic Measurements.-All of the runs except those in acetic acid were carried out with a cell and technique described previously.'*J6 Since the rate of solvolysis of trityl halides is independent of p H , i t was possible to follow these reactions by intermittent titration with base. I n the 100% ethanol and 80% ethan01-20% water runs, the base used was 0.05M sodium ethoxide prepared by dissolving sodium in absolute ethanol. I n the 97% methanol-3% water run 0.05 M sodium methoxide was used, prepared by dissolving (16) P. D. Bartlett and C. G. Swain, THISJOURNAL, 71, 1406 (1919).
[CONTRIBUTION FROM THE
EQUATION
3731
sodium in absolute methanol. I n the other runs, 0.05 .lI carbonate-free sodium hydroxide was used. The base solutions were standardized against potassium acid phthalate. Brom thymol blue was used as a n indicator in all runs. Identical results were always obtained whether the solution was kept basic or acidic between endpoints. The initial concentration of trityl fluoride was 0.02 M in acetic acid and 0.001 M in the other solvents. The procedure used may be illustrated by the 40% ethan01-60% water runs: 5 ml. of a fresh solution of trityl fluoride in ethanol (prepared gravimetrically) was added to the cell containing a mixture of 35 ml. of ethanol and 59 ml. of water; since 2 ml. of a n aqueous titrating medium was required for the complete solvolysis, the solution was 40% ethan01-60% water at 50% reaction. All percentage compositions reported are by volume before mixing, i . e . , deviations from additivity of volumes on mixing were ignored. It was not possible to use the intermittent titration method (or any other method based on titrating the acid liberated) in the acetic acid-acetic anhydride mixtures because of the very slight difference in acidity between acetic acid and hydrofluoric acid. These runs were followed by quenching 5-ml. aliquots in 50 ml. of benzene, extracting twice with 10-ml. portions of water, separating the aqueous layer, and titrating for fluoride ion with 0.02 N thorium nitrate in 0.01 M nitric acid using the yellow to orange color change of sodium alizarinsulfonate. No further fluoride ion was extractable by subsequent washing of the benzene layer with water.
CAMBRIDGE 39, MASSACHUSETTS
DEPARTMENT OF CHEMISTRY, MASSACHUSETTS INSTITUTE
OF TECHNOLOGY]
Correlation of Rates of Solvolysis with a Four-parameter Equation1 BY
c. GARDNERSWAIN, ROBERTB. MOSELYAND DELOSE. BOWN RECEIVED AUGUST30, 1954
+
A four-parameter equation, log ( k / k o ) = cldl c&, is tested, where k is the first-order rate constant for solvolysis of any compound in any solvent, ko is the corresponding rate constant in a standard solvent (80% ethanol) a t the same temperature, c, and c2 are constants depending on only the compound undergoing solvolysis, and dl and d2 are constants depending on only the solvent. All available data capable of serving as a test of the equation were used. Values of CI and c? are reported for 25 compounds ranging from p-nitrobenzoyl chloride t o triphenylmethyl fluoride, and values of d l and dz for 18 solvents ranging from methanol to formic acid. These values were determined from the above equation and 146 observed log ( k l k o ) data by the method of least squares. The mean and maximum ranges in observed rate for a fixed compound are factors of 1.4 X los and 7.6 X lo6, respectively. The mean and maximum errors in the calculated rate are factors of 1.33 and 4.4.
Many chemical reactions of uncharged substrates
or
(S) in solution appear to involve both a nucleophilic reagent (N) and an electrophilic reagent (E) attacking during, or prior to, the slowest step on the way to products. For these reactions we might expect log ( k / k o ) = sn
+ s'e
(1)
where k is the rate constant with N and E, ko is the rate constant with No and Eo (standard reagents), n measures the nucleophilic reactivity of N, e measures the electrophilic reactivity of E, and s and s' measure the discrimination of S among N and E reagents, respectively.2 The terms sn and s'e measure nucleophilic and electrophilic driving force. Equation 1 should apply not only to concerted mechanisms, which may be represented by k N f Sf E
--+
Products
(1) Abstracts of 13th A.C.S Organic Symposium, Ann Arbor, Michigan, June 17, 1953, pp. 63-69. This work was supported by the Office of Naval Research. Complete experimental data may be found in references 12 and 19. JOURNAL, 76, (2) C. Gardner Swain and Carleton B. Scott, THIS 1 4 1 (19B3)
N
+S +E
slow ~
I
fast
--+
Products
any speed
such as reaction of pyridine with methyl bromide catalyzed by phenol or mercuric ion in benzene s ~ l u t i o n reaction ,~ of quaternary ammonium azide with trityl chloride catalyzed by phenol in benzene solution,* mutarotation of tetramethylglucose by pyridine and phenol in benzene solution,j enolization of acetone by acetate ion and acetic acid in water solution (third-order term) reaction of iodide ion with epichlorohydrin catalyzed by acetic acid in water s ~ l u t i o n ,and ~ cleavage of organosilicon compounds in water solution,s but also to mechanisms involving successive attacks S
+E
K
I
N
-+Products slow
(3) C. G . Swain and R. W. Eddy, i b i d . , 70, 2989 (1948). (4) C.G . Swain and M. M. Kreevoy, i b i d . , 77, 1122 (1955). (5) C. G. Swain and J. F. Brown, Jr., i b i d . , 74, 2534, 2538, 2691 (1952). (6) H. M. Dawsan and E. Spivey, J . Chcm. Soc.. 2180 (1930). (7) C. G . Swain, THIS JOURNAL, 74, 4108 (1952) (8) F. P. Price, i b i d . , 69, 2600 (1917).
c. GARDNERS W A I N , ROBERTB
3732 or K
K
+S
I
E
-+Products slow
provided that the equilibrium constants (K)are small enough so t h a t [ I ] [SI, or when S is an anion or cation, or when the reaction is a four-membered cyclic process. When the electrophilic reagent is held constant, e . g . , when water is the solvent and acts as the only important E,we may set e = 0.00 for E = E" and log ( k / k o ) = s n
(2)
where both k and ko are second-order rate constants. One set of n values (and one set of s values) suffices to correlate displacements by water, acetate ion, chloride ion, aniline, hydroxide ion and other N reagents on carbon in esters, ethylene oxides, alkyl and acyl halides as S2. A diEerent set of n values, viz., n = log (KB,'ICBO)where K B and Kgo are the basic ionization constants of I\' and No in water a t 2 j o , is needed to correlate displacements on hydrogen or to reduce equation 2 to the Bronsted catalysis law for bases.* The differences between these two n scales seem qualitatively to correlate with A F s , the free energy of solvation of X . The main reason for the extraordinary nucleophilicity of the ions of heavy atoms (I-, O,SS-) may thus be the ease with which the loosely bound water molecules can be displaced when reaction occurs, relative to the tightly bound ones 011 small atoms (F-, CH3COO--). Unfortunately there are not yet enough data on free energies of solvation (gas to aqueous solution) to adequately test the equation TI
=
CY
+
log (KR/KHO) p (AF, - AF.0)
for displacements on carbon. Still different n values are needed to correlate displacements in phosphorus9 or tin (where n is abnormally high for F- due to the large per cent. ionic character and strength of the bond forming) or in other solvents than water. When rates of solvolysis are correlated by equation 1, the solvent is not constant. For this reason we prefer to change the symbols s, n, s' and e to cl, d,, r2 and (E2 to avoid the implication t h a t the solvent parameters are accurate measures of nucleophilic and electrophilic reactivity of f he solvent when equation 1 is applied in this manner. A possible approach t o obtaining true values of s, n , 5' and e would iiiv,)lvc c l i l u t i ~e:ich i ~ o f the solvents with a n (9) I I)ustrov\kv
aiid
\r
IT;ilniann, J . Chpm .COG., 5 0 8 (1953).
MOSELY A N D DELOSE. ROKN
ITc11.
7;
inert low-dielectric medium so that the experimental log ( k / k o ) values could be interpolated t o a ct,ii stant dielectric constant for use in equation 1. Ii-c have not done this, but would expect the paranieters to have much more obvious and simple physical significance if such a correction were made. e\' shall now describe a test of the equation log ( k / k O ) =
Cldl
+ (.d.
f3)
where k is the first-order rate constant for solvolysii of any compound in any solvent, k o is the corrcsponding rate constant in a standard solvent a t thca same temperature, c1 and cz are constants depentling on only the compound undergoing solvolysis, and d, and dz are constants depending on orily thc solvent. -4s the standard solvent we chose SOYc ethanol2OY0 water by volume because more data were available for it than for any other solvent. Table I lists log ko in 80% ethanol for 25 compounds. TABLE
1
RATESI N 80% ETHA\OI. Compounda
logla kO, sec. -- 1
TemJ,., O C
i o I Bere dropped in t h e course of t h e >lark IT calculation when t h r v ill)peared t o interfere with convergence of the s u c r e s l v e 'i1iproxiiiia11011 I,roct.dure employed. No d a t a in the presence of added salt' w r c available when this calculation was done 'fhe recent discovery thal lithium perchlorate in low concentrations can alter the rate of a f e u solvolyses and not t h a t of others" suggests t h a t all the parameter5 will shift and t h a t t h e correlation may be poorer u,hen such salt5 are added (11) S. Winstein, E Clippinger, A . H Fainberg and (>.C R u l t i i i w i i 'THIS J O U R N A L , 76, 2597 (1954) , (12) D. >E. 13own, P h 1) theiiu, hf I T , . A ) > r ~l'lJ;,R
July 20, 1935
RATESOF SOLVOLYSIS WITH
serving a check on the equation,1° i.e., for solvents studied with three or more compounds, or for cornpounds studied in four or more solvents either a t
A
FOUR-PARAMETER EQUATIOK
3iR3
the same temperature or a t enough temperatures to permit extrapolation to the common temperature for each compound listed in Table I.
T A B L EI1 RELATIVE RATESOF SOLVOLYSIS' Compound b
NOiPhCOCl NOZPhCOCl NOzPhCOCl NOzPhCOCl NOzPhCOCl NOzPhCOCl SOzPhCOCl N02PhCOF NOzPhCOF NOsPhCOF XOZPhCOF NOzPhCOF XOzPhCOF NOzPhCOF NOzPhCOF NOZPhCOF NOzPhCOF PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOCl PhCOF PhCOF PhCOF PhCOF PhCOF PhCOF PhCOF PhCOF PhCOF MePhCOCl MePhCOCl MePhCOCI LIePhCOCl MePhCOF MePhCOF MePhCOF MePhCOF MeBr MeBr MeBr
Solvent e
EtOH, 100 MeZCO, 90 MeZCO, 80 MeZCO, 70 MeZCO, 50 AcOH, 100 HCOOH, 100 MeOH, 100 MeOH, 96.7 MeOH, 6 9 . 5 EtOH, 100 EtOH, 40 MeZCO, 80 MeZCO, 70 Me2C0, 50 AcOH, 100d HCOOH, 100 MeOH, 100 MeOH, 9 6 . 7 MeOH, 6 9 . 5 EtOH, 100 EtOH, 60 EtOH, 50 EtOH, 40 MeKO, 90 Me2C0, 80 Me2C0, 70 MezCO, 50 AcOH, 100 MeOH, 9 6 , 7 MeOH, 6 9 . 5 EtOH, 100 EtOH, 40 MeZCO, 80 MeZCO, 70 MeZCO, 50 AcOH, 100d HCOOH, 100 EtOH, 100 MeXO, 80 Me2C0, 50 AcOH, 100 MeZCO, 80 MeZCO, 50 AcOH, 100d HCOOH, 100 EtOH, 100 EtOH, 50 MeeCO, 50
log ( k / k o ) a
Ref.
-0.68 12 - .91 12 - .52 12 - .35 12 - . I 2 12 12 -4.67 12 -3.37 12 -1.59 -0.51 12 12 .47 -1.77 12 f0.51 12 -1.07 12 -0.64 12 - .04 12 -6.37 12 -4.25 12 +o. 22 21 12 .33 .87 12 - .52 12 .48 12 .84 12 +1.30 12 -1.18 12 -0.72 12 - .37 12 f .50 12 -2.39 12 -0.23 12 .84 12 12 -1.58 +0.81 12 -1.19 12 -0.67 12 .ll 12 -4.47 12 -1.61 12 -0.79 21 - .90 12 .85 12 -1.94 12 -1.13 12 +O ,09 12 12 -3.79 12 -0.36 - .75 13 . 4 1 13,22 .22 37
+
+ + + +
+ +
+
+ +
Compound 6
MeBr MeBr EtBr EtBr EtBr EtBr EtOTs EtOTs EtOTs EtOTs n-BuBr n-BuBr n-BuBr n-BuBr n-BuBr n-BuBr n-BuBr PhCHzCl PhCHsCl PhCHzCl PhCHZOTs PhCHzOTs PhCHpOTs i-PrBr i-PrBr i-PrBr i-PrBr i-PrOBs i-PrOBs i-PrOBs i-Pr OB s PinOBs PinOBs PinOBs PinOBs PinOBs MeOCxOBs MeOCxOBs MeOCxOBs MeOCxOBs BrCxOBs BrCxOBs BrCxOBs BrCxOBs PhCHClMe PhCHClMe PhCIlClMe PhCHClMe (Ph )zCHCl
Solvente
log ( k / k o ) .s
Rei.
.68 -1.78 -0.77 .58 +1.12 -1.15 -0.28 - .57 .37 -2.38 -0.36 - .21 .46 .70 -0.19 .27 -1.14 -0.26 - .85 .74 - .29 - .78 -2.09 -1.02 +O. 86 f1.99 -0.14 .37 - .81 -1.39 -3.08 -0.63 -1.29
13 13,24 13,25 13,22 13 13,25 13 13 13 13 15 15 15
Hz0, 100 HCOOH, 100e EtOH, 100 EtOH, 50 Hz0, 100 HCOOH, 100'' MeOH, 100 EtOH, 100 EtOH, 50 AcOH, 100 MeOH, loop MeOH, 96. 7gVh MeOH, 69.5"** EtOH, 100 EtOH, Soh EtOH, 60* HCOOH, IO@' MeOH, 100 EtOH, 100 EtOH, 50 MeOH, 100 EtOH, 100 AcOH, 100 EtOH, 100 EtOH, 50 HzO, 100 HCOOH, 100 MeOH, 100 EtOH, 100 AcOH, 100 Ac~O,9 7 , 5 MeOH, 100 EtOH, 100 AcOH, 100 AQO, 9 7 . 5 HCOOH' MeOH, 100 EtOH, 100 EtOH, 50 AcOH, 100 MeOH, 100 EtOH, 100 EtOH, 50 AcOH, 100 MeOH, 10OivL EtOH, loo'*" Me2C0, 80is" AcOH, 100 MeOH, 96.7
+ + +
+ -
+
+
15 15 15
26 13 13 13 13 13 13 13
13
13 13 16 16 16 16 16 16 -0.70 16 -2.08 16 +2.39 16 -0.46 13 -1.02 13 13 +0.70 - .97 13,27 .74 13 -1.42 13 +1.11 13 - 1 . 1 2 13,27 -0.43 28 28 -1.50 -1.12 28 -1.62 29 -0.07 19
-
-
(13) S. Winstein, E. Grunwald and H. W. Jones, THIS JOURNAL (23) C. G. Swain, C. R. Scott and K. Lohmann, THISJOURNAL, 76, 73.2700 (1951). 136 (1953). , , (14) L. C. Bateman, K. Cooper, E. D. Hughes and C. K. Ingold, (24) I. Dostrovsky and E. D. Hughes, 1. Chcm. S o c , 166, 171 J . Chem. SOL.,925 (1940). (1946). (15) hl. L. Bird, E. D. Hughes and C. K . Ingold, i b i d . , 255 (1943). 69, 2051 (1947). (25) E. Grunwald and S. Winstein, THISJOURNAL, 70, 846 (1948). (16) E. Grunwald and S. Winstein, THISJOURNAL, ( 2 6 ) L. C. Bateman and E. D. Hughes, J . Ckcm. Soc., 935, 940, 946 (1940). (17) A. M . Ward, J . Chcm. Soc., 445 (1927). (18) A. M. Ward, i b i d . , 2285 (1927). (27) S. Winstein, E. Grunwald and L. Inqraham, THISJOURNAL, 70, 821 (1948). 77, 3727 (1955): (19) C. G. Swain and R . B. Mosely, THISJOURNAL, R . B. Mosely, P h . D . thesis, M . I . T . , July, 1952. (28) E. D . Hughes, C. K Ingold and A. D . Scott, J . Chcm. Soc., 1201 (1937). (20) E. D. Hughes, J , Chem. Soc., 255 (1935). 8fi, 2536 (29) J. Steigman and L. P. Hammett, THIS JOURNAL, (21) J. F. Norris and H. H . Young, THISJOURNAL, 67, 1420 (1935). f,.V.,,, ioan (22) I Dostrovsky and I? D. Hughes. J. Chcm S o c , 164 (1946).
3731
C. GARDNER SWAIN,ROBERT B. MOSELY AND DELOSE. BOWN
1701.
77
TABLE I1 (Continued) Compoundb
SolventC
log ( k / k n ) a
Ref.
Compound b
Solventc
log ( k / k o j (I
Kef
( P h j2CHCl EtOH, 100 -1.51 30 t-BuC1 19 HCOOH, 83.3 + l . 5 0 (PhjzCHCl EtOH, 90 18 -0.5.5 f2.08 26 t-BuC1 HCOOH, 100 MeJCO, 90 ( PhjZCHC1 - 2 57 31 (Ph)aCSCK -0.30 19 MeOH, 96.7 (Ph)zCHCl MeZCO, 80 -1.38 31 (Ph)$CSCK MeOH, 69 5 -I- .40 19 -0.73 MelCO, 70 31 (Ph)zCHCI EtOH, 40 .5G 19 (Ph)aCSCN .98 23 (Ph)zCHCl IlezCO, 50 - .28 19 Me2C0, 80 (Ph)aCSCN AcOH, l O O P -2.36 (Ph),CHCI 19 Me2C0, 70 - .05 19 (Ph)iCSCN HCOOH, 83 3 + 2 . 6 1 (Ph)zCHCl 19 hleL!O, 50 .2G 19 ( P h ),CSCN 19 EtOH, 50 + I . 61 (Ph)zCHF MeOH, 96 7 .03 19 (Ph)tCOAc AcOH, 100" +2.11 19 (Ph)nCHF MeOH, 69 5 .90 19 (PhjaCOAc 19 HCOOH, 83 3 +5,9O (Ph)zCHF (Ph)sCOAc EtOH, 60 4- .56 19 32 t-BuC1 MeOH, 100 - 1. 0.5 MezCO, 80 -1 56 I9 (Ph)sCOAc -0.i2 :3 2 1-BuC1 hleOH, 96 7 Me?CO, 50 -to 11 I O (Ph)sCOAc MeOH, 69 5 32 t-BuC1 1 , 02 .83 19 (Ph)ZCOPhhTOz MeOH, 69 3 t-BuCl EtOH, 100 -1.98 20 t1.25 19 (Ph)zCOPhA-Oz EtOH, 40 -0.73 20 t-BuC1 EtOH, 90 Me2C0, 50 (Ph)&OPhXOz +0.37 19 +1.14 20 t-BuC1 EtOH, 60 (Ph)nCOPhKOz AcOH, 100" .97 19 20 fl.60 EtOH, 50 f .11 19 t-BuC1 MeOH, 96 7 (Ph)zCF 20 4-2.15 EtOH, 40 t-BuC1 MeOH, 69 5 +1.50 19 (PhhCF 20 hIezCO, 80 EtOH, 100 -1.73 19 -0.68 t-BuC1 (PhhCF 33 f1.29 Me2C0, Z0 (Ph)>CF EtOH, 40 I9 f2.02 t-BuC1 16,34 MeZCO, 70 -1.21 10 f3.55 HzO,100 t-BuC1 (PhLCF 16 -1.64 AcOH, 100 MeZCO, 50 fO.58 I9 t-BuC1 ( P h )zCF AcOH, 100' 16 -3.29 t-BuC1 AQO, 97 5 (PhjoCF i-1 76 19 These are decimal logarithms. For all compounds, log k / k o = 0,000 in 80% ethanol by definition. For each comKumpound, the temperature is that given in Table I. b Symbols for compounds are explained in footnote a of Table 1. ber after solvent is 7 0 by volume based on volumes before mixing; the residue is water except for 97.5% AclO which is Extrapolated from a higher Calculated from data a t 80 and 100". 2.5% AcOH; Me, E t , Ac = CH3, CzHj, CHeCO. ~ from a higher temperature using A E = 19.8 k ~ a l .(I~Extrapolated ~ temperature using AE = 20.2 k ~ a 1 .f~Extrapolated from a higher temperature using A E = 22.0 k ~ a 1 . l ~Interpolated from a plot of log k ZJS. mole fraction of water. Extrapo~ t h e values used in these calculations were not lated from a lower temperature using A E = 27.1 k ~ a 1 . i~Unfortunately the correct ones recorded here but were -1.3i, -2.45 and -2.06 for methanol, ethanol and 80% acetone, respectively, which correspond to incorrectly calculated values tabulated by othersI6 in the literature. We discovered this error subsequent Extrapolated from Extrapolated from a higher temperature using E = 21.7 kca1.G to completion of the calculation. from a higher temperature using E = 21.8 kcaLas p The a higher temperature using E = 21.9 k ~ a l . ~Extrapolated ~ datum given was obtained in 99.3% AcOH-O.7% AczO, but is used for 100% AcOH since this compound is relatively little affected by traces of HzO or A c ~ O . ~ '
+
+
+ + +
+
+
+
The data represent a wide range of structural variation. The compounds range from $-nitrobenzoyl and methyl to triphenylmethyl and from fluorides t o arylsulfonates. Some have strong neighboring group participation and the pinacolyl compound even rearranges. The solvents range from anhydrous alcohols and water to glacial acetic acid and anhydrous formic acid. ltrishing t o weight all 146 log ( k / k o l values equally, we chose the condition [log ( k / k o ) o t s
- (cldl + cZd2)]* = minimum
t o define the best fit. Thus no compound is given more weight than any other. Setting the partial derivative with respect t o each of the 25 c1, 25 cp, 17 d l and 17 d z unknown parameters equal to zero gave 84 simultaneous equations. These were solved by an iterative procedure (see method of calculation below) on the Mark IV digital computer of the Harvard Computation Laboratory. (30) E. D. Hughes, C K . Ingold and S. A . T a h e r , J . C h e m . .Sot., 949 (1940). (31) L. C. B a t e m a n , M . G. C h u r c h , E . D. Hughes, C K. Ingold a n d S. A. T a h e r , i b d , 979 (1940). ( 3 2 ) A. R. Olson a n d R . S. Halford, THISJ O U R N A L , 6 9 , 2644 (1937). (33) M. S. Swain, P h . D . thesis, Radcliffe College, 1948. (34) C. G. Swain a n d S. D. Ross, THISJ O U R N A L , 68, 658 (1946). (35) E. D. Hughes, C. K. Ingold, S. Masterman and B. J. M c N u l t y , .I Chem. Soc., 899 (1940) (313) S.Winstein and H . hlarshall. T H I S J O U R N A L 74, , 1120 (1952). (:37r C G Swain and C. B. Scott, ibid , 7 5 , 111 (19.53).
The solution obtained was not unique. To make it unique it was necessary subsequently t u impose four conditions (in addition to d1 = dz = 0.00 for SOToethanol). These are in the nature of scale factors or normalization conditions for the calculated parameters and were chosen arbitrarily as follows. = 3.00 c2 for MeBr = cz = 1.00 for t-BuC1 3.00 c1 = cz for (Phj3CF 61
c1
Qualitatively these are in the right order because sensitivity t o nucleophilic reagents decreases and sensitivity t o electrophilic reagents increases in the order methyl bromide, t-butyl chloride, trityl fluoride. This choice also makes all the cl and c2 values positive. However, new values (denoted by superscript stars) may be easily calculated for any other choice of the scale factors CY and /3 by use of the equations c1*
=
ac,
c2* =
PCl
d2* = (=I)
a-8
d,*
+ d2*
+ (1 - a ) ( ?
+ (1 - B k ? di =
-1
+ (-Ea--13 dl
ds
+ dz
Table I11 lists the values of the constants ob-
RATESOF SOLVOLYSIS WITH
July 20, 1955
A
3735
FOUR-PARAMETER EQUATION
TABLE 111 VALUESOF COMPOUND A N D SOLVENT COSSTANTS Compound
Cl
NOtPhCOCl N02PhCOF PhCOCl PhCOF MePhCOCl MePhCOF MeBr EtBr EtOTs n-BuBr PhCHzCl Ph C H 2 0T i-PrBr
1.09 1.67 0.81 1.36 0.82 1.29 0.80 .80 .65 .77 .74"
Solvent
.69" .90 di
c2
0.21 .49 .52 .66 .65 .80 .27 ,36 .24 .34 .44" .39"
.58 dz
CdC2
5.2 3.4 1.6 2.1 1.3 1.6 (3.0) 2.2 2.7 2.2 1.7" 1.8" 1.5 dl
- d2
Compound
i-PrOBs MeOCxOBs BrCxOBs PinOBs PhCHClMe (Ph)zCHCl (Ph)*CHF I-BuC1 (Ph)aCSCK (Ph)3COAc (Ph)sCOPhSO* (PhhCF Solvent
c1
C?
ci/ca
0.63 .57 .80 76 1.47 1.24 0.32" (1.0Oi 0.19
0.45 .57 .87 .87 1.75 1.25 1.17" (1.00) 0.28 776 .59 1.12
1.33 1.00 0.92 0.86 .84 .99 .27" (1.00) 0.69
2.19h
0.18 .37
dz
di
... .31 ( ,331 dt
+ + +
- da
-0.05 -0.73 +0.7 MeOH, 100 -0.53c -1.52c MeZCO, 90 l.oc - .ll -0.1 MeOH, 96.7 - .05 - .45 MelCO, 80 -0.68 0.2 -1.4 - .06 4-1.32 MeOH, 6 9 . 5 - .age - .75e MezCO, 70 0.7' - .53 f0.5 - .25 -1.03 Me*CO, 50 .97 - 1.2 EtOH, 100 - .0lC - .54" .5" - .44d HzO, 100 +4. O l d - 4.5d EtOH, 90 -4.82 AcOH, 100 +3.12 EtOH, 80 - 7.9 ( .OO) ( .OO) ( .OO) - .22d +1.34$ -1.6$ Ac~O,9 7 . 5 -8.77' EtOH, 60 -114.1' +5.34" .12 -1.2 +1.33 EtOH, 50 -4.44" HCOOH, 8 3 . 3 +6.26' -10.7" - .26 +2.13 -2 4 HCOOH, 100 EtOH, 40 -4.40 +6,53 -10.9 Somewhat doubtful because based on only three log ( k / k o ) values. Based on only five log ( k / k o )values for very similar media; none was an absolute alcohol or contained over 50% water or any acetic acid or formic acid. Based on only three Based on only four compounds. compounds. Based on aromatic compounds only, hence may give poor predictions for aliphatic compounds.
+
+
+
tained. The values in parentheses are the ones arbitrarily assigned. Values based on very limited data are indicated by superscript letters referring to explanatory notes. The ratio cI/cZ is a convenient single number to characterize the reactivity of a compound. Compounds which discriminate relatively more highly among nucleophilic reagents than among electrophilic reagents tend to have high values for this ratio. As expected, this ratio decreases from p-nitro to p-methyl and in the order methyl, ethyl, isopropyl, t-butyl, benzhydryl, trityl. The difference d l - dz is a convenient single number to characterize the reactivity of a solvent. The most nucleophilic solvents have the highest values for this difference, with the difference decreasing in the order anhydrous alcohols, acetone-water and alcohol-water mixtures, water, glacial acetic acid, anhydrous formic acid. It must nevertheless be admitted t h a t some of the individual cl, CZ, dl and dz values are surprising. For example, i t seems odd that c1 increases in the order ethyl bromide (O.SO), isopropyl bromide (0.90), t-butyl bromide (1.00). The ratio cI/c2 decreases only because of an even larger increase in c2. Figure 1 shows the correlation for several typical compounds using these compound and solvent constants. Table IV lists some measures of fit.38 The compound with the largest E (mean error in log kcale.) is t-butyl chloride and for it our measure of fit,38a, is typical and excellent, viz., 85%. The largest individual error is for benzoyl chloride in methanol and corresponds to a factor of 4.4 in k. The mean E for all compounds is 0.124 (factor of (38) C. G . Swain, D. C. D i t t m e r a n d L. E. Kaiser, THISJ O U R N A L , 77, 3737 (1955).
1.33 error in k ) . For typical solvents E is 0.12 for methanol ( n = 12), 0.22 for acetone ( n = 13), 0.07 for acetic acid ( n = lS), and 0.04 for formic acid ( n = 10). Thus the fit is about as good for the extreme compounds and solvents studied as for the ones of intermediate reactivity. TABLE IV MEASURES OF FIT^^ FOR CERTAISCOMPOUXDS Compound n t % SOzPhCOCl 7 0.07 95 NOzPhCOF 10 .15 90 PhCOCl 12 .23 72 PhCOF 9 .11 91 MeBr > 06 93 n-BuBr , 0.i 89 (Ph)zCHCI 9 19 84 I-BuCI 15 25 85 7 25 79 ( Ph )ICF
-
Discussion.-The results are compared with those using other equations in the following paper.38 The correlation is particularly gratifying for p-nitrobenzoyl chloride and triphenylmethyl fluoride, which gave almost random scatter plotslg of log (k/k0),b, 8s. log ( k / k o ) c a ~when c using the equation log ( k ' K O ) = m 1'. Acknowledgment.-\Ye are greatly indebted to Professor Howard H. Aiken and the Air Force for making the Harvard Mark IV digital computer available, and especially to Mr. Peter F. Strong and Mr. J. Orten Gadd, Jr., of the Harvard Computation Laboratory for setting up and solving this mathematical problem on the computer. Their abbreviated descriDtion of the procedure is given below,
C.C;
x730
IRDNER
SWAIN, RORERT B . MOSELYAND D ~ r , o sE. R O W N
1-01. 77
2 p-N02C6H,COF
0
I
70%
e -I
.'O Y \
-2 -:
Y
EtW 0 -01
4 -3 HCOOH
,
-4 K O H ,
-5 -5
,
1
,
-2 -I C o l c . l o g k/k',
-4 -3
0"
0
-1
-2
0 -3
-4
I
-5
-5
-2
-4 -3
Colc.
-7
-6
-5
-3
-4
-2
0
I
3
log k/k'.
-2
3
-1
C g IC.
0 Ca IC. log k / k ' ,
- 4 V '
-4 -3
I
Fig. 1.-Correlation
'
-2
z;, & atcj + hid,, i
=
j
=
1,2, . , 1,2, . . . ., ,
'
"
3
Pi7 ?1
CC E;,(Zij - nit, - bid,)' j
a- ( 8 2 )
d a i ($) = dbi ~
a ar,
(81)
=
a - ($)
3
adj
+
a,Z,E,,c12 b,Z,E,,c,d, = a,Z,EL1c,dl b,z,E,,dJz = c,ZLE,,a,2 d,Z,E,,a,b, = c,S,E,,a,h, d,S,E,,hI2 =
96.7% M
- 2 - 1 0 I 2 Ca c log k / k ' ,
4
A
1 3
If approximations to the values of r j and d j be given, equations 1 and 2 can be solved simultaneously to give approximate values of ai and bi. Then equations 3 and 4 can be solved to give new approximations to the values of c j and d,. This is a simple process since, for a given value of i, equations l and 2 are just a pair of linear simultaneous equations involving ai and bi as unknowns, and ;i similar remark applies to equations 3 and 4. .llthough equations 1, 2 , 3 and 4 may be used as an iterative process to obtain a solution of the problem it was decided to employ an extrapolation process39in order to decrease the time required to obtain the answer. Let the sets of numbers ai, bi, c j and d j obtained after the n-th iteration form the vector V x . Let
=
Vn+1
(1
for all i and j. Carrying out the differentiations n ) non-linear simulone obtains a system of 2 ( m taneous equations as
+ + + +
2
70%Me,CO
'
be minimum, From this it follows that
a --
I
of rates of solvolysis for eight typical compounds
S o t all possible elements Zij are given; the elements Eij are introduced to form an existence matrix such that Ecj = l if the corresponding Zjj is given and Eij = 0 otherwise. To obtain a least squares fit, it is required that the quantity i
"
-1 0 I 2 C o l c . log k / k * .
Method of Calculation In this problem it is required to find the best values of ai, bi, cj, ( i j to represent a given matrix of the form E;,Zij by the scalar product
=
2
oq k/k',
p !.h ,,
e2
I
k/k'.
43%Me&O
t-C,HQCI
-3
-2
0
-1 og
1
4
0
I
Cdc.
-I
log k / k ' .
Calc.
-2
-2
-
= Dn
Vn
Now e? may be written as e* = f(
v,,+
1)
since E " is a function of the variables which form the components of 17%. One may calculate
so
= f(Vn+l)
+
f( Vn+ 1 hD9,) f.. = f ( Vn+ 1 + 2 m , ,1' fi
._______
(39) The principle on which this proces, is based way siiggeited Clapp
nr. R. E.
RATESOF SOLVOLYSIS WITH
July 20, 1955
A SPECIAL
where h is a scalar. Now assume that f( V , + 1 4mD,) is approximately a quadratic in x (in practice the validity of this assumption depends mainly on the choice of h). Then f( V,,+I f xD,)
= fo
+ uAf + e) Azf 2
where u =
z,h ~f = j l
- jn, A Y
=
j 2 - 2f1 + f o
This expression for f may now be minimized with respect to x by setting the derivative equal to zero. sDn) is minimum for One finds that f(17, +
+
TWO-PARAMETER EQUATION
3737
+
and the vector V = V,, + 1 sD,may be used to start a new series of iterations. In the course of the problem values of s of 10-20 were usual, although much larger numbers were encountered in certain circumstances. The size of s is, of course, dependent to some extent on the number of iterations between extrapolations. This number was varied somewhat in the course of running the problem, though it was usually found that three to five iterations gave good results. Running time for this problem, starting either with given approximations or with all starting values equal to unity, was about one hour. CAMBRIDGE 39, MASSACHUSETTS
[CONTRIBUTION FROM
THE
DEPARTMENT OF CHEMISTRY, MASSACHUSETTS INSTITUTE OF TECHNOLOGY~
Correlation of Rates of Solvolysis with a Special Two-parameter Equation' BY C. GARDNER SWAIN, DONALD C. DITTMER~ AND LAURA E. KAISER RECEIVED AUGCST30, 1954
A special two-parameter equation, log ( k / k n ) , - log (k/ko)AO = ab, is tested, where k is the first-order rate constant for solvolysis of any organic chloride or bromide (A) or of the standard compound, methyl bromide (Ao), in any solvent, ko is the corresponding rate constant in a standard solvent (807, ethanol) a t the same temperature, a is a constant depending on only the chloride or bromide and b is a constant depending on only the solvent. Values of a are reported for 15 compounds ranging from picryl chloride t o t-butyl chloride, and values of b for 19 solvents ranging from triethylamine t o formic acid, These values were determined from the above equation and a total of 124 log ( k / k n ) data by the method of least squares. The minimum, mean and maximum ranges in observed rates for compounds are factors of 10, 2 X l o 3 and 3 X 106, respectively. Themean and maximumerrorsin the calculated rates are factors of 1.5and 7 . 6 . A measure of goodness of fit which is applicable t o any correlation of rate or equilibrium constants is proposed and is evaluated for typical applications of the Bronsted catalysis law, the Hammett equation. and various two- and four-parameter correlations of rates of solvolysis.
This paper describes a test of a special two-parameter equation log ( k / k o ) r
- log (k/ko)Ao = ab
(1)
where k is the first-order rate constant for solvolysis of any organic chloride or bromide (A) or of the standard compound, methyl bromide (Ao), in any solvent, ko is the corresponding rate constant in a standard solvent (80% ethan01-20% water) at the same temperature, a is a constant depending on only the chloride or bromide and b is a constant depending on only the solvent. As the standard solvent we chose ROYo ethanol2OVo water by volume because more data were available for it than for any other solvent. Table I J comlists log k o in XOyc ethanol for all of the 1TABLE I ADDITIONAL RATESIN 80% ETHANOL Compound a
1ogio kQ sec
-1
T e m p , 'C
Rei
-5 30 50 5 PicCl PhCOCH2Br -5 80 50 5 i-BuBr -7 61 50 6 t-BuBr -3.44 25 7 Pic = picryl (2,4,6-trinitrophenyl); Ph = C6Hj or p substituted C6H4: Me, E t , P r , Bu = CH3, CgHs, CIHT, CJIg(1) F u r t h e r details and discussion m a y be found in D . C. D i t t m e r , P h . D . thesis, M.I.T., September, 1953. Cf.also C.G. Swain a n d D. C. Dittmer, THIS JOURNAL, '76, 4627 (1953): Science. 118, 576 (1953). T h e work carried out b y Miss Kaiser (kinetics of picryl chloride and phenacyl bromide) was supported b y t h e Office of Naval Research. (2) National Science Foundation Fellow, 1952-1953.
pounds not previously given in Table I of the previous paper.3 Table I1 lists 42 of the 124 log ( k / k o ) values which were used. The other 82 log ( k / k o ) values are for the compounds listed in Table I11 and the solvents listed in Table 117, and may be found in Table I1 of the previous ~ a p e r . ~One , ~ half (62) of the 124 values were measured in this Laboratory. The compounds correlated are all chlorides or bromides, but include p-nitrobenzoxl, methyl, benzhydryl and t-butyl. The solvents are especially varied, including n-butylamine, triethylamine, alcohols, water and anhydrous formic acid. We used the method of least squares in a simple iterative procedure to obtain the best values of a and b (see method of calculation below). T o make the solution unique three conditions were imposed arbitrarily as follows: b = 0.00 for 80% EtOH; a = 0.00 for MeBr; a = 1.00 for t-BuC1. A renormalization for any other choice of scale factor (y) may be made easily using the equations a* = Y a b* = b/r
for new values (denoted by superscript stars). When data for a secondary standard AO'are used (3) C.G.Swain, R. B. Mosely and D. E. Bown, THISJ O U R N A L , 77, 3731 (1955). (4) T h e log (k/kO) values for a-phenylethyl chloride used in this t r e a t m e n t were t h e correctly calculated ones ( c f . f o o t n o t e i in Table I1 of t h e previous paper3). T h e omissions were log ( k / k 9 values for 40% ethanol, 83.3% formic acid and 97.5% acetic anhydride a n d t h e vnlur for p-nitrobenzoyl chloride in acetic acid.
