Estimation of gas-phase hydroxyl radical rate constants of oxygenated compounds based on molecular orbital calculations

Pergamon

Chemosphere, Vol. 32, No. 4, pp. 717-726, 1996 Copyright 6 1996 Published by E~skier Science Ltd Printed in Great Britain. All rights reserved 0045-6535/96 $15.00+0.00

0045-6535(95)00352-5

-

ESTIMATION

OF GAS-PHASE HYDROXYL RADICAL RATE CONSTANTS OF OXYGENATED COMPOUNDS BASED ON MOLECULAR ORBITAL CALCULATIONS

Andreas Klamt Bayer AG, MD-IM-FA, 418 D-S 1368 Leverkusen, Germany e-mail: [email protected] (Received in USA 13 April 1995; accepted 7 November 1995)

ABSTRACT In the molecular orbital based estimation method for gas-phase hydroxyl radical rate constants MOOH so far most oxygenated organic compounds have been excluded due to indications for reaction mechanisms different from direct H-abstraction or OH-addition to double or aromatic bonds. In the present article we give an extension of the approach to ketones, alcohols, ethers, carbonic acids and aldehydes. The basic mechanistic assumption for these oxygenated compounds is an addition step of the hydroxyl radical to the oxygen lone-pairs prior to the H-abstraction. Finally an extension to alkynes is given.

1. INTRODUCTION

The gas-phase

hydroxyl

radical rate constant

compounds

in the environment.

complicated

and expensive,

The estimation introduced

by the MOOH method

more accurate

incremental

provide

system has to be extracted

descriptors

although

scheme

a great deal of systematic

the reactivity

strictly limited to a particular

more

reliable

for this physicochemical molecular

property.

orbital calculations compared

as

to the

being not significantly

for which increments estimates

is quite

have been

for compounds

being

validity. This is due to the fact that the molecular information

which in the case of an incremental

data.

parameters

derived

from molecular

the range of validity of an empirically

reaction

for the fate of organic

of these rate constants

1986 and 1988). Although

in the range of compounds

for considerably

from experimental

descriptor

1983) has proven to bear several advantages

scheme (Atkinson,

allows

for the rate constants,

measurement

on the basis of semi-empirical

a bit further away from this core range of

calculations

Nevertheless,

estimation

approach

an important

methods are of special importance

(Klamt,

than the incremental

the MOOH

structurally orbital

estimation

the experimental

of OH radical rate constants

more traditional

adapted

Because

has become

mechanism.

orbital calculations adapted

quantitative

are valuable equation

is

So far in the MOOH method the three most important 717

718 mechanisms olefinic

of hydroxyl

radicals with organic compounds

double bonds, the addition to aromatic bonds and the OH abstraction

While

being

rather

hydrocarbons

hydrogen

precise

underestimated,

abstraction.

by the presence

olefins,

dependence

appears to be plausible

influence

it is the closer sterical neighborhood

membered

ring OH-adduct

(Figure

for ketone OH radical rate constants

of the carbonyl

geometry

of the respective

of the carbonyl

enhances

the abstraction

la). This assumption

abstraction

observable

from the carbon

of hydrogen

section we therefore

at carbons in P-position

of hydrogen

equations

for the estimation

considered.

For these the abstaction

7

acids, esthers,

of these rate constants

of the aldehydic

hydrogen

which

a hypothetical

six-

observations

that the nature of the dipole moments

and no metastable

stabilized.

state is

Apparently,

such an

atoms which are reachable findings

start from the mechanistic

of H-atoms reachable for the radical oxygen considerably

quantitative

atoms in p-

adduct of the OH radical to one of

oxygen as given in Figure lb is considerably

carbonic

in the in-

the H-abstraction

is very unlikely under physical aspects: The considerable

of ketones,

for

factors f(-C(=O)-) = 0.76

it must be emphazised,

of the dipoles. Instead a hydrogen-bonded

In this way the rate constants

is very different

seems to explain most experimental

Nevertheless

atoms the initial formation

to be

which is induced

For these reasons Dagaut et. al. (1988) postulate

does. In the following

of oxygen

appears

would be very strange. Thus it is very likely, that

of the carbonyl group to the hydrogens

satisfactory.

groups are

1989). Especially

‘dangling’ oxygen of the radical and thus it explains the experimental

the former proposal in the presence

species

rate constants

by the enhancement

halogenated

functional

in the reaction mechanism

adduct is very well suited to act as an inital step for abstraction the quite flexible

for these

group and the OH radical would give rise to a strong repulsion

with this alignment

the lonepairs

having oxygenated

and

method the carbonyl group deactivates

such an alternation

for the higher reactivity.

adduct in the proposed

hydrocarbons,

mechanism

as expressed

atoms while it considerably

For an electronic

from aliphatic carbon atoms.

also from the strange systematics

= 4.4 in Atkinson’s incremental

from direct neighbor

i.e. the OH addition to

reactions and indicates an initial addition step (Atkinson,

fluence of the C=O group on the rate constants:

is responsible

reaction

Most probably this is due to a difference

such an addition

position.

aliphatic

hydrocarbons

the overall

of the oxygens. The temperature

and f(-CH,C(=O)-)

aromatics,

of most aliphatic

although

from that of pure abstraction ketones

for most

the rate constants

significantly

possible

have been parametrized,

bonded

adducts

contributes

as well as

assumption

that

and the consecutive

to the overall rate constant.

and alcohols

will be analyzed

will be given. Finally, is the dominating

for

aldehydes

mechanism

and

and

will be this will

H-O

a

‘H

Jl./k a)

Figure I: a) adduct complex according to Dagaut et al. (1988)

b\

b) hydrogen bonded adduct as proposed in this work

719 be treated as a separate reaction mechanism a preliminary

adduct is of importance

data, but fortunately

in close analogy to the H-abstraction

in this mechanism

this question is not of importance

TREATMENT

for the estimation

equation.

A simple extension

of

for the first MOOH paper are given as Addenda.

OF THE ADDUCT COMPLEXES

Starting point for the formal treatment of the H-abstraction lone-pairs

Whether

can not be decided on the basis of the available

the MOOH scheme to alkynes as well as two corrigenda

2. FORMAL

from sp3-carbons.

out of hydrogen-bonded

OH-adducts

is the intrinsic reactivity kabsH of a particular H-atom. This is calculated

at oxygen

for each hydrogen atom

at an sp3-carbon within the MOOH algorithm according to eq. 3 in the original MOOH paper (Klamt, 1993). In order to indicate that this is the basic reactivity we denote it as kcH , furtheron.

Now we assume that the

indirect reactivity of a hydrogen atom out of an adduct state formed at a particular lonepair i is given as the product of three factors. The first of these is some lonepair specific factor wlpi which takes into account the relative probability

of the lonepair to build a hydrogen-bonded

out of this adduct. A second factor fstericHFiresembles adduct state at lonepair i. The third factor represents we make the reasonable

assumption

OH-adduct

the steric availability the abstractability

as well as the tendency to react of a hydrogen

atom H from an

of the hydrogen atom H and for this

that it is just given by the intrinsic reactivity kabsH . Altogether

we then

have

(1) where the summation

is over all oxygen lonepairs.

Let us now first consider the steric factor. This factor should become unity, if the H-atom is ideally available for the hydrogen bonded OH radical, and it should strongly decrease for hydrogens further away or too close to the oxygen lonepair. Following

the results of several semi-empirical

al., 1985) on such OH radical adducts at oxygenated radical hydrogen denote

at a distance

these positions

lonepair positions a hydrogen

compounds

(AMl)

(Dewar et

we define the ideal H-bond position of the

of 2.16A straight in front of the particular

as lone pair positions,

MO-calculations

below. The ideal distance

turns out to be about 2.5A, but due to the flexibility

bonded adduct we have to take into account a considerable

oxygen

lonepair and we shall

of target hydrogens

in the distances

from these

and angles of such

tolerance t in this distance. Thus we

assume that the steric factor is given by

with d,,(H,i) being the distance of hydrogen atom H to lonepair position i, do being the optimal distance, for

720 which we fix the value of 2SA, and t denoting the tolerance for ketones indicate that 0.7A is a reasonable

in the distance. Quantitative

guess for the tolerance t.

The lonepair specific factor w,n’ varies by more than one order of magnitude

between the different

of oxygen atoms in organic compounds.

Initially we started with the expectation

expressed

properties

as a function

local frontier position,

of the electronic

orbital reactivity

descriptors

but none of these hypotheses

we had to introduce

of

(Klamt,

was able to explain satisfactorily

17 is needed for an optimal description

i = 1P

of the reactivity is observable

our previous finding (Klamt,

To avoid additional

parameters

considered

wlP

’ =

features, Instead,

data. Surprisingly,

and its best description of hydroxy

is given by a

groups,

while no

for ether oxygens. The latter fact is consistent

with

by the original MOOH method.

above. Thus, the sp3-oxygen

in a ester group may be

due to its two carbon neighbors. The few data available for carbonic acids can

by treating the carbonyl oxygen of the acid as a ketone, i.e. by applying

17, and by treating

the

tried to describe some further common situations of oxygen

by the factors introduced

as a ‘ether-oxygen

be described consistently

we successfully

at the lonepair

of a ketonic carbonyl

of the experimental

1993) that ethers are quite well described

atoms in organic compounds

potential

the experimental

factor of wlp i = 1.5. The same value appears to be valid for lonepairs enhancement

that this factor could be

For the lonepairs

carbonyl oxygen in a ether group appears to be by far less enhancing

significant

classes

the lonepair, e.g. by a partial charge, some of the

1993) or by the electrostatic

a constant factor for each class of oxygens.

oxygen a value of w

considerations

the hydroxy

special handling of the H-abstraction

Finally,

oxygen as before (wlPi=lS).

the factor

in combination

with the

described below, for aldehydic oxygens the same factor as for ketones

can be applied.

As will be shown below, the above assignment description

of the vast majority

Nevertheless, descriptors

of the available

experimental

due to the lack of a systematic expression it is impossible

to extrapolate

or even to other hydrogen-bond described

of the lonepair specific factors

intermediate

specific functional

hydrogen-bond

heteroatoms,

OH-adduct

data for oxygenated

scheme to other functional although

understanding

for such a progress. within

of oxygens

data for a

group are available, we can not assign a factor for the lonepairs of this group and thus

origin for the large difference

compounds

situations

As long as no experimental

we are not satisfied by this situation, but

have have to accept it as state of the art for the time being. Hopefully,

elucidated

compounds.

we may expect, that even for these the

is of importance.

no reliable estimate of the rate constant can be given. Obviously,

and by an improved

organic

of these factors as a function some molecular orbital

this estimation

accepting

wlp’ yields a quite good

of the reaction mechanisms

in the enhancement Nevertheless,

the MOOH approach

findings than the incremental

approach

in future by additional measurements

this lack can be overcome.

the

factors of ketonic and ester carbonyl oxygens has to be

even in its present state the proposed allows for a more systematic

(Atkinson,

Especially

1986 and 1988).

treatment

description

of oxgenated

of the experimental

721 3. CLASS

In this chapter a special discussion be given. For shortness reference.

Experimental

SPECIFIC

SECTION

for each of the classes of oxygenated

throughout

compounds

taken into account will

this chapter all rate constants are in units of IO-‘* cm3/s without special

data are from Atkinson (1989) if not indicated differently.

3.1 Ketones For 23 ketonic compounds extended magnitude,

MOOH

method

the experimental

and those calculated

by the above presented

are given in table 1. The data span a range of more than three orders of

starting with CH,COCF,

2,6-dimethyl-4-heptanone

rate constants

(k .,=27.5).

a factor of 1.8. This is somewhat

(k,,-

-0

015, Wallington

The logarithmic

et. al., 1988) at the low end and ending with

standard error of the calculated

data corresponds

larger than the factor of 1.5 achieved for the non-oxygenated

within the original MOOH method. A great deal of this error arises from two compounds carbon in P-position

to the carbonyl as well as from 2,5-hexanedione,

to

compounds

with a tertiary

which is the strongest outher. In the

Table

1: Calculated and experimental rate constants. Experimental data are taken from Atkinson [3]. Mean values are taken if no recommendation is available. Those data marked by # are from [8]. All data are for room temperature. Only in the case of ally1 alcohole (**) the experimental value is at 440 K. The two compounds marked by * are left out in the evaluation of the overall error. aldebydes

formaldehyde acetaldehyde 1-propanal glycolaldehyde chloroformaldehyde dichloroformaldehyde trichloroformaldehyde fluorochloroformaldehyde fluorodichloroformaldehyde difluorochloroformaldehyde difluoroformaldehydehyde trifluorformaldehyde 1-butanal 2-methyl-1-propanal 1-pentanal 3-methyl- 1-butanal neopentanal benzaldehyd glyoxal methylglyoxal pentane- 1,5-dial acrolein crotonaldehyde methacrolein

alcoboles

9.77 15.50 19.60 11.35 10.00 6.95 3.09 2.39 3.75 1.46 1.51 2.08 2.63 1.09 1.20 0.89 0.81 1.54 2.53 0.60 0.47 17.51 23.50 22.87 26.30 18.19 28.50 19.24 27.40 28.30 26.50 11.40 12.90 25.17 12.79 17.17 21.18 24.37 23.82 17.70 19.90 29.92 35.97 27.79 33.49 17.50 10.64 16.55

methanol* ethanol 1-propanol I-butanol 2-propanol 2-methyl-2-propanol 3-pentanol cyclopentanol 2-chloroethanol 1,2-ethanediol 1,2_propanediol 1-pentanol 2-pentanol 3-methyl-2-butanol I-hexanol 2-hexanol 1-heptanol 2-methoxyethanol 2-hydroxymethanol 2-ethoxyethanol 3-ethoxy- 1-propanol 3-methoxy-1-butanol 2-butoxyethanol ally1 alcohol**

4.05 5.16 6.88 8.90 2.90 0.53 5.62 9.34 1.65 6.13 8.68 9.37 7.14 6.19 10.48 8.83 11.67 16.62 18.86 11.32 11.58 9.56 13.36 49.90

0.93 3.27 5.30 7.80 5.20 1.10 12.20 10.70 1.40 7.70 12.00 10.80 11.80 12.40 12.40 12.10 13.60 12.50 30.00 15.40 22.00 23.60 18.50 25.90

722 Table

1: (continued) esters

ketones

acetone 2-butanone 2-pentanone 3-pentanone 2-hexanone 3-hexanone 4-methyl-2-pentanone 2,6-dimethyl-4-heptanone 3,3_dimethylbutanone 2,4-dimethyl-3-pentanone cyclobutanone cyclopentanone cyclohexanone 1 , I,1 -trifluoropropanone# acetylchloride 2,3_butanedione 2,5hexanedione* hydroxyacetone methoxyacetone 2-heptanone 2-octanone 2-nonanone 2-decanone

two

former

cases

underestimation observed

the

0.18 1.65 5.79 3.23 10.57 10.09 3.92 14.9 I 1.66 3.30 0.80 3.51 5.09 0.03 0.04 0.30 1.89 4.51 9.91 13.10 15.30 18.10 18.00

0.23 1.15 4.90 2.00 9.10 6.90 14.10 27.50

1.20 5.40 0.87 2.94 6.40 0.01 0.06 0.24 7.10 3.00 6.80 8.70

carboxylic

1I .oo

0.69

0.34

1.94

1.60

2.54 0.87

3.40 3.40 5.50 0.05 0.23

1.39 0.04 0.43 1.17 1.87 2.62 1.51 3.36 0.91 I .95 2.76 2.37 3.26 4.07 8.36

1.02 2.38 3.12 3.04 4.20

I .oo 2.00 4.00 5.00 7.40 10.60 13.00

acids

formic acid acetic acid propionic acid butyric acid isobutyric acid

12.20 13.20

underestimation

methyl acetate ethyl acetate n-propyl acetate isopropyl acetate set-butyl acetate methyl trifluoroactate methyl formate ethyl formate n-propyl formate n-butyl formate methyl butyrate n-butyl acetate methyl propionate ethyl propionate n-propyl propionate ethyl butyrate n-propyl butyrate n-butyl butyrate 1-acetoxy-2-ethoxyethane

0.45 0.70 1.40 2.40 2.00

of the rate constants

most

0.70 0.73

1.35 3.11 1.36

probably

arises

from

a general

of the intrinsic reactivity k,” of the hydrogen atoms at tertiary carbons, which can also be

within the alkane data set of the original MOOH work (Klamt, 1993). Here this underestimation

is amplified H-bonded hexanedione

due to the dominating

influence of these hydrogens

adducts at the carbonyl we have no systematic

been measured

oxygen.

at the ideal positions for abstraction

For the strong underestimation

explanation.

of the rate constant

Instead we like to doubt the experimental

out of of

2,5-

value which has

in a series of related diones by Dagaut et. al. (1988). The reason for our doubt is the

apparent error in the value for 2,4_pentanedione, one of the most prominent

which is better known as acetylacetone.

examples of keto-enol tautomerism

takes the enol form to more than 90%. Estimating method for olephinic compounds

This compound

is

and it is well known that in the gas-phase

it

the reactivity

of this enol within the normal MOOH

we yield a value of 3 1. Therefore the reported value of 1.15 appears to be

very unlikely. Since Dagaut et al. did not realize the enolic nature of 2,4-pentanedione a series with the ketonic 2,3-butanedione entire series of data. For example,

and 2,5-hexanedione

an interchange

would bring both close to our expectations.

but discussed

it in

we can not be sure about the validity of the

of the values for 2,4-pentanedione

Thus we propose to reconsider

and 2,5-hexanedione

these experimental

data. Leaving

723 out this questionable

datum the standard error for the remaining

to what we had for the non-oxygenated

ketones corresponds

to a factor 1.6, close

compounds.

3.2. Esters Experimental

and calculated

again corresponds

rate constants

to a factor of 1.9. Again a considerable

intrinsic rate constants of hydrogens group. Leaving

for 19 carboxylic

esters are given in Table 1. The standard error part of this is due to underestimation

on tertiary carbons, which are in P-position

out the corresponding

compounds,

i.e. isopropylacetate

of the

with respect to the carbonyl

and set-butylacetat,

the standard

error reduces to a factor of 1.6.

3.3 Alkohols In addition to the direct and indirect H-abstraction the rate constants

mechanism

discussed so far, for an optimal estimation

of alcohols it is useful to make the plausible assumption

0

aldehydes

0

ketones

A

alkoholes

0

carboxylic

that the hydroxylic

H-atom does

ester

-2

acids

-1

0 log(K,,)

1 I talc.

Figure 2: Experimental vs. calculated rate constants: Dashed lines indicate a deviation factor of 2. The arrows left out in the calculation of standard errors, i.e. methanol

mark the two compounds and 2,5-pentanedione.

of

2

724 contribute

to the overall rate constants of alcohols. In quantitavite

[2,3] we assume a constant

contribution

lonepair factor of the hydroxylic constants

for 24 alcohols

overestimated

and diols are given in Table 1. Methanol,

compounds

this even for methanol.

reactivities,

is the strongest

corresponds

hydrogen.

The

and the corresponding

calculated

rate

outlier in this data set. Since we had to alkanes (Klamt, 1993), we may accept

approximately

within a factor of 2 and these give

to a factor of 1.6.

acids

Only for five carbonic

acids experimental

data are available. The data can be explained

the hydroxylic

oxygen and in addition a constant contribution

satisfactory

if we

oxygen, that of 1.5 from the alcohols for

adopt the lonepair factor of 17 from the ketones for the carbonylic

hydrogen.

of each hydroxylic

the rate constant of which is slightly

even in the case of the halogenated

All other alcohols are reproduced

a standard error, which again

3.4 Carbonic

from abstraction

oxygen is set to 1.5. Experimental

even by the intrinsic

exclude mono-carbon

of k,,,=0.2

agreement with the estimates of Atkinson

of kacid=0.7 for the abstraction

of the acidic

Then we achieve a standard error of 1.I.

3.5 Aldehydes For most aldehydes

the abstraction

of the aldehydic

reaction

is not covered

by the H-abstraction

mechanism

method, because it is an abstraction atoms. The recently

published

hydrogen

is the dominating

mechanism

descriptor

contributions

methods

used in (Klamt,

aldehydes

EGHx(0.18)

sufficient

by the molecular orbital

the mostly

relatively

small

where we use a lonepair factor of 17 in analogy to the

+ 14.96)

(3)

of the same local frontier orbital descriptor

abstraction

of hydrogens

reasonable

relationations

reactivity

descriptors.

Eq. 3 together

MOOH

rate constant kaldH comes out to be well descibed by

k a
Hence it is a function

of this mechanism

This

from sp3-carbon

(Rayez et al., 1994) now provide

1993). Taking into account

from direct and indirect H-abstraction,

ketones, the aldehydic

in the original

from an sp2-carbon in contrast to the usual abstraction

data for halogenated

diversity in the aldehyde data set to allow for the parametrisation reactivity

considered

reaction pathway.

from sp3-carbons.

It should be mentioned

for the rate constants of the halogenated

with the direct and indirect H-abstraction

description

of the 24 experimental

corresponds

to a factor of 1.4.

EGHH(0.18) which determines

that Rayez et. al. also found some

aldehydes

from sp3-carbon

data of aldehydes and dialdehydes

the H-

with semi-empircal

MO-based

atoms yields a rather accurate

given in Table 1. The

standard error

725 4. SUMMARY

Starting from the assumption

of H-bonded

have extended

estimation

compounds,

the MOOH

i.e. ketones,

for the abstraction

alcoholes,

method.

method

radical

to the most important

esters, carbonic acids, and aldehydes.

of the aldehydic hydrogen has been developed.

of the original MOOH. Altogether compounds

adduct states of the hydroxyl

classes

at oxygen lonepairs of oxygenated

Ethers have been in the range of validity

the standard error of this extended

MOOH method on

All severe outliers beyond a factor of two are underestimated.

as they sometimes

estimation

reported so far. Therefore, environmental

93 oxygenated

to that of the original

Thus, the MOOH method

method, as it has been before. No overestimations

occur within the incremental

organic

For the latter a special equation

given in Table 1 and plotted in Figure 2 is 1.6 and thus comparable

being a conservative

we

method, e.g. for halogenated

keeps

by orders of magnitude,

propanes and esters,

have been

the MOOH method is very well suited for a worst case estimate, as it is usual in

fate studies, since one can be quite sure to be on the save side if one applies a factor half to

the MOOH result.

Due to the lack of a systematic the classes of oxygenated experimental

compounds

of the lonepair factors the presented

which have been considered

explicitly.

method is restricted

deficiency

will be overcome

in some future by additional experimental

insight. The author would be greatful for corresponding

Like the original

MOOH the extended

request in form of a subroutine

In addition MOOH2 will be an integrated

semi-empirical

will be available MO-packages

from the author on

MOPAC and AMPAC.

part of the next MOPAC release.

REFERENCES

Atkinson,

R.(1986), Chem. Rev. 86, 69; (1988), Envir. Toxicol. Chem. 7, 435

Atkinson,

R., (1989), J. Phvs. Chem. Reference

Dagaut, P., T.J. Wallington,

Data. Monoeraoh

1

R. Liu, M.J. Kurylo (1988), J. Phvs. Chem. 92, 4375,

Dewar, M.J.S., E.G. Zoebisch,

Klamt, A. (1993), Chemosphere

E.F. Healy, J.J.P. Stewart (1985) , J. Am. Chem. Sot. 107, 3092

26, 1273

this

data or an improved

advice.

MOOH method (MOOH2)

to the well established

to

For other classes additional

data are needed to fix this factor before they can be included into the method. Hopefully

methodological mechanistic

understanding

726 Rayez, M.T., D.J. Scollard, J.J. Treaty, H.W. Sidebottom, Cm,

C. Balestra-Garcia,

S. Teton, G. I.e Bras (1994).

452

Wallington,

T.J., P. Dagout, M.J. Kurylo (1988), J. Phvs. Chem. 92, 5024

ADDENDUM

Although

being out of the context

present a small further extension data reported

for this class of compounds

(27.4/20.1),

standard error corresponds

in the review of Atkinson

and calculated butadiyne

data are: acetylene

(18.9/22.5),

I-pentyne

The author likes to use the opportunity

linear exponential

(0.9/1.0),

propyne

(11.2/l 1.3), and I-hexyne

(5.9/7.8),

I-butyne

(12.6/12.6).

The

2

of this article to correct two severe typing errors in the equations of

the original paper on the MOOH method [ 11. Eq. 2 should read:

7.23

kz = e

1 +exp( -5.35(EEH

-O.l65A&-2.72 ‘(2.22) +10.53))

Eq. 3 should read:

I apologize

[3] the following

to a factor 1.05.

ADDENDUM

kH ohs = ex

by this article the author likes to

+ 24.03}

experimental

(8.0/7.3), 2-butyne

covered

description:

= exp{2.15ECHc(0.58)

The corresponding

compounds

of the method, i.e. the extension to carbon- carbon triple bonds. On the 6

equation gives a very satisfactory

$L

of oxygenated

1

9.93 (1+exp{-2.18(ECHa(0.18)+11.0)})o”3

for eventual confusion.

-8.03

I