pKR+ values for pyridinium cations

Tetrahedron

Vol

43, No. 19. pp. 4277 to 4286. 1987

Printed in Grcal Britain.

6

PKR+Values

for

John

Department

Bunting

of Chemistry,

Toronto,

University

Ontario

S3.00+ .oO

Journals Ltd.

Cations

Pyridinium

W.

0040-4020/87 1987 Pergamon

M5S

of Toronto

Canada

lA1,

(Received in USA I Augur1 1987)

ABSTRACT Literature data for nucleophilic addition at C-4 of C-3 substituted I-methylpyridinium cations are evaluated. It is shown that sufficient data are available to allow the confident evaluation of a linear free energy relationship for the equilibration of these cations with their C-4 hydroxide adducts (pseudobaaea) in aqueous solution: This C-4 PKR+

The

allows

equilibration adducta)

heterocyclic an

(iO.4)

o-

= 20.5

of pKR+

of heteroaromatic in aqueous

chemistry.

equilibrium

equilibrium the

-8.5

=

+

:or

20.5

(+0.5)

pseudobase

formation

A less confident of the 1-methplpyridinium cation. = 18.2 at C-2 of this cation is also made.

(hydroxide

which

pKR+ prediction

the

lt2

solution

This

constant

constant

has

cation-pseudobase

the

cations is a well

process

KR+ form

can

with

can

be

of a Brensted

equilibration

is usually

their

phenomenon

by

equation

as

in equation

acid

ionization

described

at of

pseudobases

established

expressed

be defined

estimate

in

This

[2].

constant,

in terms

for

111,

and

of a pKR+

value.

R+

t

H20

-

ROH

[H+l K

Accurate

pKRt

cations.2'3

they

acid-base must

be

and are

for

not

readily

equilibrations sought

for

the

basicity

function

application competing

are many

121

available

simple

by

a large such

the

pKRt

the

direct

solutions. of pKRt

pKRt

aqueous to other reactions

for

species,

cations,

evaluation

method

irreversible

now

accessible

cation,

of this

[ROHI

in aqueous

in basic

t11

H+

[Rtl

isoquinolinium

N-methylisoquinolinium

by

=

values

However,

quinolinium that

Rt

t

values.

= 16.29 dimethyl

was

in strongly

the

values

of heterocyclic

N-methyl

are

In such For

large

of simple

cases,

indirect

methods

the

evaluated

495

by use

solutions.

cations, basic

pyridinium,

sufficiently

measurement

sulfoxide

heteroaromatic

4277

as

number

of a

However,

is severely

solutions.

Such

the

limited is the

J. W. BUNTISG

4278

case

for

the

N-methylquinolinium

disproportionation also

has

two

addition

and

values

pKR+ carbon

atoms

are

pseudobaee

at both

C-2

and

estimates

of pKK+

equilibration

is not

value

N-methylisoquinolinium

for

linear

the

free

dependence Linear C-4

energy

energy

in N-methyl

extrapolated at C-2

= 17.4

the

present

of PKK,

for

the

at both

for

reliable

the

C-2

of particular

there

or C-6. variety that

are

three

Furthermore, of

the

these

important

ring-opening

evaluation

values

The

present

formation

by

constants

for

also

linear

the

in the

free

ion

a result

of

estimate

can

these

energy

and

sites

the

of pKK+

for

C-3

and

data

the

and

of

are

that

addition

estimates

few

available

to pseudobase

pyridinium

substituted close

cations

pyridinium

relationship

very

susceptible

a simple

data

coupled

with

to pyridinium have

been

that

cations,

ion

addition

to C-4

are

and

to relate

cations

a reasonably

of

equilibrium

to

1-bensyl-1,4-dihydronicotinamide.

it is shown

so

paeudobase

developed

of heteroaromatic

2

matter.

that

for

C-4 to a

base,

assignment

constants

then

C-2,

in aqueous

sparse

to

in these

However,

is not

the

9

cation.

specific

C-6

nucleophiles

hydroxide

ion

to be

on equilibrium

by

hydroxide

and

been

nucleophilic.attack:

known

C-4

of classes

to reduction

very

a

8

of C-2 have

coenzymes.

evaluation

These

each

at C-3,

reactions

values

at C-2,

at

of their

for

well

relationships

deliberations,

he made

pKK+

of other

of a variety

attack

cation

are

literature

cations.

addition

susceptibility

hydroxide

addresses

pyridinium the

attack

are

via

cations.

is susceptible

The

disproportionation

of accurate

work

available

species

and

to hydroxide

currently

with

these

there

because

non-equivalent

for

pKK+

substituent

quantitative

available.3

nicotinamide

the

formation

which

energy

quoted

confirmed

N-methylquinolinium

to be no

free

to provide

above

isoquinolinium

= 15.4

parent

interest,

the

for

ring

cation-pseudobase

substituents

of pKK+

linear

subsequently

N-methyl

on

relationships

example,

was

cation,

are

cation

nucleophilic

of

which

developed

In fact,

values

for

bearing

seem

C-4.

This

of substituents

of these

pseudobase

in the

there

and

pKR+

biologically

species

for

N-methylpyridinium

formation

(1) are

at C-4

since

development

For

was

evaluation

time,

The

cation

cations

the

nature

cations

substituted

quinolinium

pKK+

for

that

relationships

At

cations

C-4

to allow

and

which

for

3

accessible.

relationship

of pKK+

free

directly

formation,

extrapolation

values

base. 687

in aqueous

to the

cations.

the

rapid

C-4.

sensitive

allows

undergoes

reactions

for

quite

often

which

sites

of heteroaromatic

relationships accurate

condensation

potential

is possible

cation,

confident of the

As

pKR+ Valuesforpyridiniumcations

N-methylpyridinium is also

Martin

and

in aqueous

of

Cation

the

10

Hull

base.

The

an electronic possible

Martin

and

a proton

conjugate

absorption

from

the

of

addition

C-3

at C-4 isotope

cationa

by

This

pKH+

kinetica

has

k2

for

N-methyl

ll-fold

cation

cations,

and

The

to

one

can

in terms

regiochemistry

1.3.

then

id_& Hulf"

nm of of

absorption wavelength

are

more

estimate

pKK+

for

ion

of the

variable

of quinolinium derivative.

via

equation

[3],

degenerate derivative.

to nucleophilic

at C-4 for

the

effect

ammonium

ApKH+

pKK+ with

hydroxide

cations.

substituent

Using

the Thus

basis

of

propose

Conaiatent

for

susceptible

= 1.1)

N-methyl

13.2.

1,4-dihydro

addition

KCN)

cation.

predict,

N-methyl ion

controls

4,4-dideuterio

the

11

[31

5.1

A aimilar and

its

t

d(log

16

for

and

ala0

of a variety

accurately (k2)

12

Cuttman

13

ita

any

I 1.0

For

= 15.6

N-benzyl

is also

cation

seen

which

N-methyl

for

the

of nicotinamide

cations, for

instance,

than

in the

display

relative

to

the

cation.

8th

rnaIIu:ed

conjugate

332

15

0.7

and

1,4-

out

ionization

on the

reduction

and

cyanide

of N-benzyl

3-Acepl-l-methvlny

at

cation

rulea

not

repreaenta

this

deduced

does

maxima.

which

than

pKH+

(i.e.

and

cation

absorption

value

constant

corresponding for

pKa

be greater

to also

1,2-,

wavelength

been

clearly

since

of

the

cations

(KCN)

l-methylnicotinamide

longest

of this

equilibration

this

hydrolysis

cation

-0.56

the

cation.

range

Martin

=

and

Brooke

constant

this

greater

acidities

in the

result

at C-2

1-methplnicotinamide

in this

this

has

for

14

rate

than

constant

ia

maximum

= 14.6

heteroaromatic

cations

cation.

that

must

shown

between

reactions

association

N-benzyl

longer

equilibrium

observed

been

the

nm,

1-benzylnicotinamide

second-order

N-benzyl

relative

>300

of the

of pKH+

effects

log

the

produced

group.

cation

the

transhydrogenation

this

carboxamide

the

for

1-bensyl-1,4-dihydronicotinamide

observed

APK,

of pKH+

paeudobaaea,

argue

a similar

of

value

addition

base

display

for

a value

kinetic

the

all

I-methylnicotinamide result,

= 13.2

a

maximum

persuasively

ionization

this

pK

dihydronicotinamide

Hull

pH-dependence the

value

(1: W = CONliZ)

measured

1,6-dihydronicotinamidea

amide

reliable

deduced.

1-Methylnicotinamide

have

A less

cation.

4279

base

the

M-'cm

addition addition

is two-

pKa

formed

(L = 14400 the

(1: W = COCH3) = 13.3 in this -1

product

maximum

the

Although

was

3-acetyl-1-methylpyridinium

equilibration

of hydroxide

to three-fold

absorption

).

for

these

ion not

greater in the

displays workera

to the

normally

spectrum

interpreted

pyridinium

specified. than

an absorption

The

ring,

intensity

observed

this the of

l7 for

of a dihydropyridine

the

4280

J.W. BUN~NG

derivative. pseudobase cation.

This observation suggests that this conjugate base may not be a produced by hydroxide ion addition to a ring carbon atom of this

Johnson and Morrison

this 332 nm maximum is

a

18

have suggested that the species that generates

ring-opened derivative.

Such species have been n

identified for many other pyridinium cations in aqueous base.‘ Studies of the related 1-methyl-3-phenylacetylpyridinium COCIf2C6H6) have identified two acid-base equilibrations formation in basic aqueous solutions of this cation.

19

cation (1:W =

other than pseudobase The kinetically

controlled conjugate base from this. species is the hydroxide ion adduct at the carbonyl group (pK, = 12.31), while thermodynamic control favours the enolate ion formed by deprotonation

of the benzylic methylene unit (pK, = 10.30).

Analogoua equilibria are possible for the 3-acetyl-1-methylpyridinium which should display a similar susceptibility

to hydroxide ion addition to its

carbonyl group to that observed for the 3-phenylaoetyl cation. for enolate ion formation upon deprotonation

cation,

The pKa value

of the acetyl group can also be

predicted to be in the vicinity of 12 to 13 by analogy to ApKa m 2.0 between acetophenone

(pK, = 18.24)

20

and benzyl phenyl ketone (pK, * 16.1)

21

.

However,

one would not expect either the gem diol anion or the enolate ion to have such an intense absorption maximum in the vicinity of 332 nm. If the interpretation ring-opened derivative

of the conjugate base of ~~~~ = 332 nm as a

is correct, then the pKa = 13.3 reported by Martin and

Hull is actually a composite equilibrium constant composed of pKB+ for pseudobase

formation and a ring-opening

and probably also contains contibutions equilibration

tautomerization

equilibrium constant,

form gem diol anion and enolate ion

with thia pyridinium cation.

Equilibrium constants have been reported for additions to 3-acetylpyridinium

a

variety of nucleophilic

and nicotinamide cations.

Equilibrium data

which allow direct comparisons between these two classes of cations are given in Table 1 for addition of cyanide ion at C-4.

As indicated in Table 1, there

is general agreement from a number of laboratories, and for a number of different N-substituents approximately

in these pairs of cations, that log KCN is

2.2 greater for 3-acetylpyridinium

correspondingly

substituted nicotinamide

constants for the equilibration

cations than for the

cations.

The relative equilibrium

of NAD+ or its 3-acetyl analog with NADH and

its 3-acetyl derivative also fit this same pattern.

The lone exception in

Table 1 is for sulfite ion addition to pairs of pyridinium cations.

This

difference may be related to the different charge present on the nucleophile and 1,4_dihydropyridine

adduct in the case of sulfite ion addition.

This observed d(log KCN) - 2.2 in Table 1, can be translated into pKH+ being 2.2 lower when a 3-CONH2 substituent is replaced by a 3-COCH3 unit.

Thus

PK,+ Values for pyridinium cations pKR+

for

the

15.6

- 2.2

3-acetyl-1-methylpyridiniua

cation

Equilibrium

constants

for

of nicotinamide

Reference

No.

and

ion

addition

to matched

3-acetylpyridinium

of cation

pairs

Average

24

1

2.1

25

7

2.2

26

1

2.3

21

1

2.3b

28

1

3.2'

cations

differing

equilibration

(or

its

3-acetyl C-4

= [Py+l[-CNl/[PyCNl in only

of NAD'

analog)

addition

and of

for

(or

C-4

COCH3

its

addition

or CONH2

3-acetyl

a

,l(log KCN)

2.2

in KCN

pairs

cations

1

for

of cyanide

as

analog)

the

C-3

and

ethanol

ion

to

substituent. with

NADH

acetaldehyde.

sulfite

ion.

of

for

3

A further [31 and

an

reduction

= 12.9,

PKR+ given

the

of

this

by

in water

at

(ionic

which

in reasonable

is

of pKa

fortuitous.

estimate

this

25O

in each

= 13.3,

for

probably

the We

to be

have

reliable

of

are

complex.

hydration

with cation

quite

the

x 10

can

1.0).

agreement

predictive

from

upon

our

within

method.

above,

pKR+

laboratory

log

These

it seeme for

deduced

values and

that

this

for

[3] predicts

value

by Martin

= 13.2

+ 0.3

the

equation

in 20%

Equation

with

reported

given

taken

are

Hull*'

this

cation

above, both

in

for

this

agreement in Table

is 2, and

units.

(1: W = CN)

3-cyano-1-nethylpyridinium Products

from

to give

predominant

product

being

concentration.

There

do

be baaed

-3 M-ls-1

strength

nitrile

the

of

the

= 7.7

cation

1-bensyl-1,4-dihydronicotinamide

CetioR

reactions

this

originally

reasons

3-Cyano-l-methylwyridinium

solution

of k2

cation

although

The

pKR+

value

uncertainties

vicinity

cation,

prediction

unpublished

acetonitrile

the

cyanide

23

for

to be

1

2.4

KH

the

predicted

3

pyridinium

the

be

22

aDifference

=KSO

can

= 13.4

TABLE

b

4281

not

the

very

in basic

disproportionation, amide

have

dependent

appear

cation

all

upon

to be any

ring-opening

been the

aqueous

reported

solution

estimates

29-32

and ,

pH

and

the

of pKR+

for

this

4282

BUNTING

J. W.

species. Second-order cation

and

its

corresponding

these

constants

1-benzyl

analogue

are

with agent,

equation

by

nicotinonitrile

pKH+

values

cation

the

N-methyl

expected

dpKH+

(1 1 for

In principle

one

method

However,

there

nucleophilic

might

to that appear

Diekmann'I

reported

have

which

to C-4 log

KCN

for

11.0

and

~~~~

smaller 2.038

= 328

than log

the

for

larger

and

Diekmann

~~~~

and

log

3-cyanopyridinium fact

Amax

nm

for

show

KCN

is at

the

a wide

range

deviates

from

somewhat

shorter

derivative

equilibrium

a simple

of

this

the

cation.

is not

A similar

I-benzyl-3-cyanopyridinium

cation

relative

to

its

it is noted

In passing, quantitatively

describe

aqueous

We

base.

this

study

formed

from

related the

is

PKH+

far

this

to the value

the

have

that

observations,

3-CONH2

from

cation

spectrum

suggested

of

a

Wallenfels

and

Atlog

KCN

adduct

slightly

= 3.362),

and

exists this

dramatically.

expected

likely

for

that

KCN

In

this

this

ion addition for

only

cation.

quite

be held

an

to the

cations,

of cyanide

also

at C-4 33 for

reported

) = 2.6 derived

reactions

above

is in progress 29-32

reactions

from

do

base

value for

of

this

for

this

occur

has

~~~~

12.1.

laboratory

one

These

nm,

data

Although

initial

and are

in

spectral

solutions. of the

= 346

to

1: W = CN undergoes

electronic

in these that

in our

species

appears

to be

consistent

with

cation.

(1: W = NO2)

controlled cation

that

time-dependent

it is clear

Catioa

thermodynamically

typical

complex

by a pKa

1-methyl-3-nitropyridinium

a study

in aqueous

1-Methgl-3-nitrouyridinium The

that

complete,

cation

for

correlation

, and it seems

and

available

KCN

than

by

cation.

a 1,4-dihydro

correlation

must

cation

derivative.

confirmed

several

3-acetyl

linear

expression

cation,

the

wavelength

11

reservation

1 for

of pyridinium

this

display

this

nicotinamide

a close

the

for

(log

of

above).

is unexpectedly

corresponding

while

(see

in

this

C-4

analogue,

addition

cation

by

at

for

the

be used

pKH+

to give

constant

3-acetyl

3-cyano-1,4_dihydropyridine constant

ion

may

= 12.2

cations.

this to

addition

constants

cation

that

for

cation

328

q

than

of

cations

1-benzyl

Table

cyanide

association

ion

analogues

of nicotinonitrile

corresponding

units

Wallenfels between

for

its

equilibrium

= 3.256

This

nm.

reduction

constants

of pKH+

for

upon

3-cyano-1-(2,6_dichlorobenzyl)pyridinium havin$

hydroxide

to predict

based

few

rate

N-beneyl

expected

was

the

of pyridinium

estimates

and

to be very

addition

reduction

These

3-cyano-l-methylpyridinium

analogous

These

the

cations.

for

I-benzyl-1,4_dihydronicotinamide

[31 for

to estimate

33

available

1,4-dihydronicotinonitriles.

conjunction reducing

rate

pseudobase

(pKH+

= 9.42)

1,4-dihydro-3-nitropyridine

formed has

an

from

the

electronic

derivative.

14

absorption Thus

this

pKH+

pK,+ Valuesforpyridiniumcations

value

can

be

confidently

assigned

to

the

C-4

4283

hydroxide

ion

adduct

of

this

cation.

c3

5-Di

a ion

Wallenfels

and

Hanstein

35 have

3,5-dicyano-1-methylpyridinium the

formation

c-2. A

= 382

max

more

adduct.

in view

nm,

likely As

Linear

Free

fact

observed

C-4

values

pKR+

species

assigned

C-4

above

1-methylpyridinium

are

TABLE pKR+

values

for

C-3

~~~~

well

in this

Paeudobaee

to C-4

cations

thia

hydroxide

ion

= 352

ia actually

fits

formation

for

the

assigned

display8

interpretation

Enerrty Relationship

from

for

value

to

addition

at

3,6-dicyano-1,2_dihydropyridine

isomer

pseudobase

= 3.0

workers

nn)

that

pseudobase

this

pKR+

Theae

(II350

35

1,4-dihydro

below,

for

~~~~~

of the

its

the

shown

assigned

substituted

while

that

values

The

cation.

of a paeudobaae

However,

measured

nm,

the

with

the

has it seema

C-4

hydroxide

other

pKR+

work.

Formation

pseudobase

formation

summarized

in Table

in these

C-3

2.

2

substituted

1-methylpyridinium

cations

Substituent

log

b

KS0 3

3-CONH2

15.6

3-C02CH3

0.62

15.2

-1.2

0.74

14.2

-0.3

3-COCH3

13.2

0.04

13.4

2.0

3-CN

12.2

1.00

12.0

3.2 4.0

3-NO2

9.42

1.25

9.9

3,5-(CN),

3.8

2.00

3.5

aCalculated bKSO

This

correlation

is the

aseociation

Equation

[4] shows

equation

constant

for

[41

, using

sulfite

o- parameters P

ion

addition

from

to C-4.

ref.

(36).

28

3

Hammett are

from

o- substituent P

those

defined

for

correlation

via pKR+

these

constants the

C-3

resonance =

-8.5

pKR+

for

ionization

is expected

electron-withdrawing pseudobases

that

in view

C-3

closely

aubstituenta,

of

the

resonance

provide

contributors o;

the

are

+

such 20.5

for ae

(~0.6)

with

the

the

anilinium stabilization

these

that

correlated where

of para-substituted

substituents

(kO.4)

values

o- values P cations. which

1,4-dihydropyridine

shown

by

2.

(corr

coeff

= 0.997)

r41

4284

J. W. BUXTING

Analogous

close

for

the

C-4

the

C-l

pseudobases

pseudobases

close

correlation

value

indicated

quinolinium 'KR+ does

correlations of C-3

of C-4

and

this

equation,

estimated

assume

parallel

linear

and

current free

cations,

it

equation

[41

which

should

be noted

experimentally Table

2 also

coworkers28

for

includes

sulfite data

(equation

[51),

although

fit

is probably

concentrations

The

only

equilibrium are

also

Kso3

constants

those

for

of Wallenfels

no

reliable

passing,

it should

relationship

log

fit

3-NO2

constants of

substituted

that

in the

equation

[31

the pyridinium

correlation

line

3,5-(CN)2

in

cations

of Pfleiderer

these

for

is obtained

and

1-methylpyridinium

a reasonable

-

7.0

correlation

than error

with

in equation from

on the

uP

[4].

working

for

with

This low

cyanide

relationship

Johnson

and

for

KCN

data can 37

Smith

substituents

addition

ion

show

nicotinamide

cations

so much

scatter,

although

shown

cations

upon

to

be deduced, have

[51

= 0.972)

of pyridinium

at C-4

These

coeff

of C-3

influence

addition 24

(corr

(21.2)

cations.

and

p

solutions.

a;

that

The

the

confidence

and

and

and

available.

to experimental

energy

KS0

are

to C-4

Diekmann

be noted

between

values

data

free

for

of the

by the

to show

pyridinium

linear

slope

between

it is. true

relationships

nucleophilic

and

N-(2,6-dichlorobenzyl) that

(~1.4)

extensive

While

(-p)

sulfite

= 9.8

considerable

9

(p = 8.8)8.

similarly

the

a poorer due

for

derived

(p = 8.4)

similarity

of quinolinium

addition

mainly

other

work.

association

appear

of unstable

log

provide

previously

cations

the

found

been cations

reduction

pKR+

the

ion

These

those

predominantly

measured

cations.

poor

that

is determined

[41 and

energy

I-bensyl-1,4_dihydronicotinaride

have

ieoquinolinium

cations

in the

op-

quinolinium

by equation

isoquinolinium

values

with

substituted

substituted

indicated by

of pKR+

in

a linear which

differ

3 only

in the

nature

of the

substituent

on N-l.

7

Gf d

6)-

2

I

CH3

H3

I-MetblDvri&&um

the

than

cation 141 predicts

I-methylpyridinium

estimate pKRt

of

this

= 17.4

pKR+

recently

estimation

= 20.5 This

cation.

equilibrium

I-methylquinolinium The

9

t

N

Equation

7

/N-+.ro-

,

/=-CH3

cation of pERt

appears

constant.

deduced'

for

in aqueous for

2 0.5

As

C-4

for

pseudobase

to be

the

expected, pseudobase

first

it

formation

at C-4

quantitative

is significantly

formation

by

larger

the

solution.

pseudobase

formation

of

by hydroxide

ion

addition

4285

PK,+ Valuesforpyridiniumcations at C-2 the

of the

present

to be

system,

the

product

upon

11.6

C-2

only

for

pseudobase

= 9.42).

ha8 of

An

= 6.2

units

Applying

since

at C-2 the

resonance of

contributor

cations

the

pseudobase

of

l-methylpyridinium PKR+ the

= 20.5 beat

present

- 2.0

at C-2

cation,

quantitative

pseudobase

= 6.82

at C-4

at C-2

the

estimation

cation.

This

estimate

2 is expected

This

C-4

hydroxide

the

and pKR+

lower

of the

) allows

in the

cation,

= 9.16

pKR+

by

of pKR+

to be

far

more

system

consideration be

and

C-6

= 11.6

- 9.16

and

atoms

t 6.2

somewhat

important

than

the

suggests

somewhat

16.4

cations,

ia probably

pyridine

probably

C-2

at

cation.

in ppridiniua

the equivalence

cation

controlled

1-methylquinolinium

formation

at

In this

of this

for

is to

14

kinetically

is the

exists

unaubstituted

for

the

solution

substituent

should

greater

in

- 0.3 low,

in the

analogous

that for

ApERt

pyridiniuo

cations.

may be made of the

cation.

- 0.3

(pKR+

system.

estimate

this

stable

pseudobaae

quinolinium

formation

at C-4

C-2

substituent

for

than

latter

quinoline

An alternative

PKRt

factor

contributor

in the

3-nitro

C-3

aqueous

attempted

of a pyridiniua cation.

aa

situation

3-nitro

cation

of this

stabilization

the

a neutral

to pseudobase

1-methylpyridinium

= 17.5

for

the

a statistical

aa confidently

at C-2

establiehed

analogous

to the

ApKR+

been

cation

of

relative

this

including the

effect

be

value

pKR+

more

I-methyl-3-nitroquinolinium The

cannot

l-methyl-3-nitropyridinium

thermodynamically

(pKR+

C-2).38

cation

reliable

the

baaification

the

adduct

The

time.

appeara

although

I-methylpyridinium

= 18.2

estimate

the

observation'

1-methylquinolinium

Applying and

from

this

including

differential

the

This

latter

that

be made

for

value this

pKR+

is 2.0

for

leas

than

to the

statistical

at C-2. can

cation

that

factor,

leads

ia considered parameter

at

to to be the

time.

Acknowledgements. I appreciate the continued the Natural Sciences and Engineering Research coworkers, Ms. J. Bolton and Ms. Y, Goda, for are relevant to this work,

support of work in this area by Council of Canada, and thank my as yet unpublished studies that

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