13.3.
7 Yukawa-Tsuno equation
There have been a number of attempts, by the introduction of a further parameter into the Hammett
equation, to quantify this graded response-via through-conjugation-on the part of a p- substituent.
Among the best known of these is the Yukawa-Tsuno equation, [7], which, in the form shown here,
is
applicable to electron-donating p-substituents; for electron- withdrawing p-substituents σx⁺ would, of
course, be replaced by σx‾ ;. The new parameter, r, intended as a measure of the through- conjugation
operating in a particular reaction, is given the value of 1.00 for solvolysis of the tertiary halides, 2-aryl-2-
chloropropanes (20). For this reaction [7] does, of course, then simplify to [8],
which is reasonable enough as it was this reaction that we used (p. 371) to define σx⁺ in the first place,
for electron-donating p- substituents capable of considerable through-conjugation! Similarly, for a
reaction in which no through-conjugation occurs r will be zero, and [7] will then, of course, simplify to
the original, simple Hammett equation [6]
To evaluate r for other reactions, we can obtain p for the reaction by measuring kx, values for m-substituted
compounds only, and then measure kx, for p-substituted compounds where the values of σp-x, σ⁺p-x, or σ‾p-x ,
are already known. Using [7], r can then be evaluated by calculation, or by graphical methods. Thus for the
base-catalysed hydrolysis of p-substituted phenoxytriethylsilanes
(22), the value of r is found to be 0.50.
13.4 USES OF HAMMETT PLOTS
13.4.2 Deviations from straight line plots
13.4.3 Concave upwards deviations
13.4.3.1 Acetolysis of 3-aryl-2-butyl brosylates
An interesting case in point is the acetolysis of 3-aryl-2-butyl p- bromobenzenesulphonates or brosylates (25),
for which the Hammett plot is shown in Fig. 13.6. The lower right-hand side of the
plot-where the substituents are powerfully electron-withdrawing- is a straight line whose slope indicates a ρ value
for the reaction of -1.46. On moving across to the left-as the substituents become less electron-withdrawing-the
plot now curves upwards, indicating that the rate of acetolysis of these species is faster than we would have
expected it to be on the basis of the σx values for their substituents.
What we might expect as a pathway for this reaction would be simple SN2 displacement (p. 98) of the good
leaving group- brosylate anion-by acetate anion:
The smallish -ve ρ value (-1.46) is compitable with such a pathway, given that in the transition state (27) breaking
of the C-OBs bond, is somewhat more fully advanced than formation of the AcO-C bond, resulting in the
transient development of a small amount of +ve charge at the reaction centre. This is in no sense unreasonable a
with (a) a secondary carbon atom as reaction centre (cf. p. 82), and (b) so good a leaving group (cf. p. 98); this
pathway would be increasingly aided, albeit weakly, as the substituent X becomes less electron-withdrawing, i.e.
the rate of acetolysis might be expected increase, gradually and linearly, from right to left across the plot in Fig.
13.6.
To account for the departure from linearity, as X becomes more electron-donating, it would seem that the
substituted benzene ring must gradually become capable of exerting some more direct effect on the reaction
centre in (25) than it does in the SN2 pathway. It is significant in this respect that increasing electron donation by
X will increase the nucleophilicity of the substituted benzene ring itself, thereby enabling it to function in
competition with ‾OAc- as a neighbouring group (p. 93) or 'internal' nucleophile, e.g. when X = MeO (28). This
alternative reaction pathway would then involve slow, rate-limiting formation of the cyclic phenonium ion inter-
mediate (29, cf. p. 105), followed by its rapid ring-opening by ‾OAc
Support for the suggestion that Fig. 13.6 involves a change in actual reaction pathway is provided by
acetolysis of the threo diastereoisomer (31) of the brosylate. Acetolysis leads to two different distinguishable,
diastereoisomers whose relative proportion will depend on how much of the total reaction proceeds by
external nucleophilic attack via the SN2 pathway (erythro product, 32), and how much by intern1 nucleophilic
attack via a cyclic phenonium ion intermediate (threo product, 33):
The two, alternative, acetolysis products (32 and 33), being diastereoisomers not mirror images, may then be
separated, or their relative yields estimated by spectroscopic methods. It is found that the yield of threo product
(33) varies considerably as the nature of X, the substituent in the benzene ring, is changed:
13.4.4 Concave downwards deviations
13.4.4.1 Cyclodehydration of 2-phenyltriarylmethanols
A good example is the cyclodehydration of some substituted 2- phenyltriarylmethanols (38), in 80%
aqueous ethanoic acid containing 4% H2SO, at 25ºC, to yield the corresponding tetraarylmethanes (39), as
shown in Fig. 13.8 (p. 381).
Two of the benzene rings in (38) each carry a p-substituent (X and Z, respectively), and the value of σ
actually plotted is Ʃσ : the sum of the values for X and Z. The plot in Fig. 13.8 -of log kobs, for the reaction
against Ʃσ -is clearly a composite of two straight lines, one on the left with ρ = +2.67, and one on the right
with ρ = -2.51.
There seems little doubt that the overall reaction follows a four-step pathway, the first two steps
constituting an El (p. 247) elimination of water to yield a carbocationic intermediate (40), which then, in
the last two steps, effects internal electrophilic
substitution on the 2-phenyl nucleus to yield the product tetraarylmethane (39):
The question then arises-which step in the overall reaction is likely to be the slow, and hence rate-limiting, one?
It's unlikely to be step (1): initial protonation in acid-catalysed dehydration is generally rapid; or step 4: final
loss of proton in aromatic electrophilic substitution is also generally rapid. This leaves steps 2 and 3 as possible
candidates for the slow step overall, and fortunately a clear distinction can be made between them.
In step 2, +ve charge is increasing at the reaction centre (the carbon atom carrying the two substituted Ar groups),
while in step 3, +ve charge is decreasing at the reaction centre. How does this match up with the requirements of
Fig.13.8 (p. 381)?
The right-hand side of the plot in Fig. 13.8 has a -ve ρ value (-2.51) indicating the development of substantial +ve
charge at the reaction centre during the overall, rate-limiting step. This would, of course, be compatible with step
2 being rate-limiting, but not with step 3. For the left-hand side of the plot in Fig. 13.8, exactly the reverse is true;
here a +ve ρ value (+2.67) indicates a substantial decrease of +ve charge at the reaction centre, which would be
compatible with step 3 being rate-limiting, but not with step 2.
It is significant that the substituents involved at the far left-hand side of the plot (38; X, Z = MeO) are powerfully
electron-donating, and thus capable of stabilising the carbocation (41a ↔ 41b), developing in step 2, by
delocalisation of its +ve charge. It is indeed
found that the log k obs, values on the left-hand side of Fig. 13.8 give a better straight line when plotted against
Ʃσ⁺, rather than against Ʃσ , because of the through-conjugation (41a ↔ 41b) between these p-substituents and
the reaction centre.
In (38; X, Z= MeO) this conjugative stabilisation results in easy formation of the carbocation (41), i.e. to a
rapid step 2; but the consequent delocalisation of +ve charge, away from the reaction centre (41a ↔ 41b),
clearly makes (41) a less effective electrophile, i.e. step 3 -electrophile attack on the benzene nucleus-is there-
fore slow. It is thus step 3 that is slow, and hence rate-limiting overall, for compound (38; X, Z= MeO). On
moving across Fig. 13.8, from left to right, the substituents become less electron- donating, delocalisation of
+ve charge thereby becomes less pronounced, and the reaction centre progressively more electrophilic.
Rate-limiting step 3 is thus speeded-up, and the overall reaction rate therefore increases, i.e. the slope of the
plot is upwards from left to right (ρ is +ve). Also on moving from left to right, decreasing through-conjugation,
as the substituents become less electron- donating, makes carbocation formation more difficult; thus step 2 is
being slowed down as step 3 is being speeded-up.
There must, therefore, come a point at which speeding-up step 3 catches up with the slowing-down step 2; any
further decrease in electron- donation by the substituents must result in step 2 becoming slower than step 3,
thereby making it now rate-limiting for the overall reaction. This shift in rate-limiting step from step 3 → step
2 occurs, in Fig. 13.8, with the compound (38; X, Z= Me).
Still further decrease in electron-donation by the substituents, beyond this point, will result in still further
slowing-down of step 2 - now the rate-limiting step-and hence slowing-down of the overall reaction, i.e. the
slope on the right-hand side of the plot is now downwards from left to right (ρ is -ve). For a reaction in which
such a shift of rate-limiting step is observed (as the electron- donating/-withdrawing ability of the substituent is
changed) there will be one substituent, or narrow range of substituents, for which the balance between the rates
of step 2 and step 3 is such as to make the overall reaction rate a maximum.
