Simultaneous determination of relative and absolute rate constants in ...

Simultaneous determination of relative and absolute rate constants in racemization studies; applications to kinetic isotope effects MICHALTENCER A N D ALLANR. STEIX' De~~cll.trnet~t of C h e m i s t n , Memoriul Unit,ersityof hTe\yfo~incilund, S t . Johtl's, AIffci., Canada A I B 3x7 Received May 25, 1978 ~ STEII. Can. J. Chem. 56,2994 (1978) MICHALTE\CERand A L L AR. The method reported allows the sinlultaneous determination of relative and absolute rate constants for two substances in a single kinetic run. The approach is of special interest for kinetic isotope effect determinations where rate ratios are often near one and failure to exactly duplicate reaction conditions can lead to uncertainties in the values calculated from independently measured rate constants. For raceniization studies, using two optically active materials with opposite signs of rotation, knowing their initial rotational contribution and having the faster racemizing one in higher concentration, the rates can be calculated if the times required to reach zero net rotation and the extrernuni value are measured. Other systems to which the method can be extended are suggested.

MICHALTE\CERet ALLA\ R. STEIN.Can. J. Chem. 56.2994 (1978) On dCcrit une nikthode qui permet de detel-miner simultanenient les constantes de vitesse relatives et absolues pour deux substances au cours d'un seul et mkme essai de cinetique. Cette methode presente un interct particulier pour la deterniination de I'effet isotopique cinetique, alors que les rapports entre les vitesses sont souvent voisins de un et que les valeurs calculi.es au nioyen de rnesures indkpendalites de constantes de vitesse peuvent ktre entachCes d'erreurs si I'on ne reussit pas a reproduire parfaitement les conditions de la reaction. Pour des etudes de racemisation, si I'on utilise d e ~ ~substances x optiquernent actives possedant des rotations de signes opposes, si I'on connait les contributions initiales respectives de ces substances a la rotation et que la substance qui se racemise le plus rapidement possede la concentration la plus elevee, on peut calculer les vitesses a partir de mesures du temps necessaire pour que la rotation nette devienne nulle, d'une part, et passe par sa valeur extremnni, d'autre part. On suggere d'autres systernes auxquels la methode est applicahle. [Traduit par le journal]

Introduction Frequently it is not possible to precisely duplicate reaction conditions. Consequently precision and. accuracy of rate and rate constant determinations suffer. When it is importailt to be able to compare rates or reactivities, one coillinonly used approach is t o use a competitive reaction with two or more substrates in the sarne reaction mixture and to follow the reactions by the appearance of products or the disappearance of starting materials. In this way more reliable relative rates can sometimes be obtained. It is perhaps in the area of kinetic isotope effects, where rate ratios are often very near unity, that the greatest problelns arise. If the isotope effect is to be deterinined by cornparing two indepe~ldently obtained rates, extreine precision is required. Temperature must be very carefully controlled, solvents and reagents must be equally pure (or impure), concentrations of reagents iilust be identical or precisely known and so on, conditions which are often difficult to meet and reproducibility becomes a problem (1). Unfortunately, it is in just such systems that the competitive approach is most difficult to ITo whom all correspondence should be addressed.

apply. Normally isotope-specific methods such as mass spectrometry or nuclear nlagnetlc resonance, methods which frequently are not preclse ellough for subtle kinet~cpurposes, must be used to follow the reactions. These methods usually require either very high coilceiltrations of the specles (2) or, inore commonly, concentration or ~ s o l a t ~ oof i l the species (3) (or rnater~als der~vedfrom them) which may ~ ~ l t i o d u cerrors e due to equilibration and, ~f recoveries are not quantitative, fractlonat~on. Even such methods are not useable for, for example, secondary deuterium k ~ l l e t ~ cisotope eff'ects in 'symmetrical' reactions of the type illustrated 111 reaction [ I ] where the product and the start~ng material are the same.

Here the only available methods for following the reaction are radio-tracer techniques (e.g., if the nucleophile-leaving group is a halide, a rrrdio-halide can be used) (4) and/or racemization kinetics (if optically active starting material is available) ( 5 ) . There are a number of cases where considerable kinetic data were extracted from single runs. Thus,

TENCER AND STEIN

for example Ahlberg ( 6 ) and others have calculated activation parameters from single variable temperature runs and Albery and Robinson (7) developed a differential method for enthalpies and heat capacities of activation. Such sequential o r simultaneous measurement eliminates many of the einpirical problems associated with independently determining a n d comparing the rates for two solutions. The method we wish to report is a 'differential' method but is quite different from the Albery and Robinson approach (7) since the two substrates are in a single solution. Using optically active substrates and following the optical rotation as a function of time it is possible to determine relative and absolute rates for the two substa~lcesin a single ru11.

from which eq. [4] arises.

1 IxOAl k , - k , = -Ill -2t.Z[X1q

luOBl

Later, at time t,, the observed rotation reaches a minimum (or maximum) value before again approaching zero, this time asymptotically. At this extremuill : da -= 0 dt Since [XIqis constant, differentiating eq. [3b]gives: -

2k,[Xlq1a,,l

exp ( - 2k,tE[XI4)

+ 2kB[Xlq1~0,1exp ( - 2kBtE[XI4)

Derivation F o r a racemizatioll reaction of the type in reaction [I], the rate expression below applies (8).

=

Substituting for k , tion of eq. [ 6 ] .

0r

a = a, exp ( - 2kt[XIq) [2bl 111 the equation, m, is the initial rotation, m that a t time t, [XI is the concentration of the nucleophilei leaving group and is., of course, constant throughout any given run, while the exponent q expresses the order of the reaction with respect to X , i.e. q = 0 for a reaction which is first order, that is, the rate depends only upon the concei:tration of the substrate, q = 1 for a second order reaction, and so on. The factor of '2' arises from the f'act that inversion of one molecule cailcels the rotation of another. If two compounds with opposite rotations, A and B, are placed in the same cell and A, which is the faster racemizing compound, has the higher absolute initial rotation, the observed rotation r a t any time will be the s ~ l ~ofn the contributions of each component. One of these rotational contributions will be positive, the other negative; but since it is not specified which is which, absolute values can be used (eq. [ 3 ] ) .Reaction of .4 and B must, of course, show the same dependence upon nucleophile concentraq, q. tion, i.e, q,

0

which, on rearranging, gives eq. [ 5 ] .

-

k , from eq. [4] allows deriva-

Thus determining the rate ratio requires that the ratio of the initial rotations due to A and B be measured and that t, and t, be determined. T o calculate the rate constants themselves using eqs. [4] and [6], the nucleophile concentration and the order of the reaction with respect to the nucleophile, q , lllust also be known. The values obtained for kt, and k , will be dependent upon the various uncertainties that affect a single run but at least any resultant errors should be the saine or similar for the two components in the si~nultaneousdetermination.

Results and Discussion Typical curves for several runs are shown in Fig. 1 and additional results are tabulated along with data obtained by conventional methods in Table 1. While it is probably in the area of kinetic isotope effects that the approach outlined is most useful, data for m -- Ia*I - ImBI other cases are included to illustrate its usefulness over a range of rate ratios. 111 all the cases reported, a = /u,,J exp ( 2/c,t [XIq) [3b] the reactions are first order in nucleophile, the - /aoBlexp (- 2kBt [XIq) bromide ion (i.2. q = I), and the reactants are At some time t,, the observed rotation will be zero variously para-substituted x-phenylethyl bromides with dezrteriurn-substitution,if any, 011 the a-carbon. and From eqs. [4]and [ 5 ] ,it is apparent that if the ratio la,,l exp (-2k,fz[Xlq) = IxoBl exp ( - 2 k B t z [ X l q ) of the initial rotations ~a,,~/la,B~is excessive or if

--

CAN. J . CHEM. VOL. 56. 1978

FIG.1. Rotation as a function of time. Curve A : p-br01110- VS.unsubstituted I-bromo-1-phenylethane: time intervals, 4000 s, r o , ~ : r O B =: 1.866, [LiBr] = 0.01 M in acetone, T = 313.15 K . Curve B: secondary c/e~rtei.iut?zkinetic isotope effect for 1-bromo-1-phenylethane; time intervals, 1000 s, :cr,,Jl,ao,! = 1.138, [LiBr] = 0.04 M i n acetone, T = 318.15 K. TABLE 1 Typlcal lesults for the s~ngle-rundeteln~lnatlonof relat~veand actual rate constants Independently determined values

I~OA~IUOB~

tz (nxn)

t~ (m111)

k, k,

Range of valuea

-

kAh

kBh

IcAIXB

k~~

k~~

3.538 3.767 2.077

y-Nitro- vs. ~lnsubstituted1-bro111o-1-phenylethanec:acetone, 40 C, [LiBr] = 0.0100 l\il 64.0 141 .O 4.57 3.75-5.57 21.06 4.61 70.3 149.3 4.43 3.77-5.14 20.30 4.58 4.52" 22.29" 43.0 131 . 0 3.89-5.03 4.46 18.81 4.22

8.761

p-Nitro- vs. p-methyl-s~~bstituted I -bromo-1 -phenylethane; acetone, 40TC, [LiBr] = 0.0100 I;\ 165.0 218.0 2.01 1.67-2.28 21.84 10.88 I .9Sd 22.29"

11.28"

1.866

p-Bromo- vs. unsubstituted 1-bromo-I-phenylethane: acetone, 407C, [LiBr] = 0.0100 .M 86.0 195.0 2.205 2.22-2.31 11.06 5.02 2.18" 10.75"

4.93"

1.135 1 ,094 I ,0595

48.0 79.0 65.0

I-Protiu- vs. I-deutero-I-bromo-1-phenylethanef; acetone, 45-C 80.7 1.092 1.082-1.101 6.647 6.087 164.0 1.107 1.066-1.150 7.747 7.037 1.08"3q 186.0 1.114 1.096-1.132 7.263 6.522

-

aEstirnated by taking the highest and lowest acceptable t L 1 t , ratlos fiom the recorder traces and calculatin,- k,,'A ratio 18 that for the 'best' times. bTlie nits are x lo-' !/mot s in every case.