Enthalpy-Entropy Compensation: An Example of the Misuse of Least

10 Enthalpy-Entropy Compensation: An Example of the Misuse of Least Squares and Correlation Analysis R. R. KRUG

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Chevron Research Co., Richmond, CA W. G. HUNTER Statistics Department and Engineering Experiment Station, University of Wisconsin, Madison, WI 53706 R. A. GRIEGER-BLOCK Chemical Engineering Department, University of Wisconsin, Madison, WI 53706

Whether or not a linear functional relationship exists between reaction or equilibrium enthalpies and entropies has been the subject of chemical investiga tions for many years. Hinshelwood collected lots of data during the early years of modern kinetic theory to probe for possible functional dependencies between the Arrhenius parameters (1-3). Many of these and subsequent experimental investigations have led to findings that estimated enthalpies varied linearly with estimated entropies. Many chemical theories have been proposed to explain, in chemical terms, why such linear correlations should occur. Linear enthalpyentropy compensation is now widely accepted as occur ring because of chemical factors and is mentioned in many standard chemistry tests (4-8). In the past few decades, first chemists (9-16) and later statisticians (17-24) have begun to doubt that all enthalpy-entropy compensations arise as a result of chemical factors alone. In particular as the compensation temperature, the slope of a compen sation line inΔΗ-ΔScoordinates, approached the range of experimental temperatures, the chemical causality of such correlations was questioned. The debate over which observed c o r r e l a t i o n s were caused by chemical factors and which were caused by nonchemical factors ( i . e . data handling a r t i f a c t s that r e s u l t from the propagation of errors) apparently has not been adequately resolved to date because enthalpyentropy compensations are s t i l l reported and j u s t i f i e d merely by the s i g n i f i c a n c e of the estimated c o r r e l a t i o n c o e f f i c i e n t . In t h i s a r t i c l e we summarize and general i z e our e a r l i e r r e s u l t s ( 2 5 - 2 7 ) that indicate that the s i g n i f i c a n c e of an estimated c o r r e l a t i o n c o e f f i c i e n t i n the enthalpy-entropy plane i s not j u s t i f i c a t i o n f o r 192

In Chemometrics: Theory and Application; Kowalski, B.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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the detection of a chemically caused compensation, but the s i g n i f i c a n c e of an estimated c o r r e l a t i o n c o e f f i c i e n t i n the enthalpy-free energy plane with estimates evaluated a t the harmonic mean of the experi mental temperatures i s strong j u s t i f i c a t i o n f o r the detection of a chemical e f f e c t . This conclusion r e s u l t s from the f a c t that there i s a l i n e a r s t a t i s t i c a l compensation e f f e c t that i s confounded with what ever chemical compensation that might be detected i n the enthalpy-entropy plane. We a l s o present the regression algorithm f o r the estimation of the chemical compensation temperature from an observed c o r r e l a t i o n in AH-AG coordinates. hm Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0052.ch010

T

Chemical Theory The rigorous thermodynamic and s t a t i s t i c a l mechan i c a l arguments of L a i d l e r (23) , Hammett (5) , L e f f 1er (7,16), and R i t c h i e and Sager (29) a l l suggest a gen e r a l l y nonlinear f u n c t i o n a l r e l a t i o n s h i p between enthalpies and entropies. To i l l u s t r a t e t h i s r e s u l t , we c a l l upon the s t a t i s t i c a l mechanical d e f i n i t i o n s used by R i t c h i e and Sager. The entropy of a system and the enthalpy of a system can be written i n terms of the sums of energy states that the system occupies. , Zg (i: /kT)exp(-e /kT) S = 1ηΣ χρ(-ε Τ) + R \ [ ^ . ^ %

Κ

9 ι β

i

ι Α

i

g

i

e

x

p

(

/

k

Eg j(εj/kT)exp(-ε j/kT) Eg^xpi-Cj/kT) If we take as the system a chemical plus i t s solvent undergoing reaction or equilibrium, two systematic v a r i a t i o n s that w i l l cause c o i n c i d e n t a l v a r i a t i o n s i n enthalpies and entropies are homologous v a r i a t i o n s of e i t h e r solvent composition (e.g., from polar to nonpolar) or substituents (e.g., from electron releasing to electron withdrawing). Passing through the homologous s e r i e s the energy states occupied by the system w i l l vary i n a systematic manner. Since the same energy states define a l l thermodynamic functions of the system, the thermodynamic param eters (including enthalpy and entropy) w i l l also vary i n a systematic manner such that a p l o t of enthalpy versus entropy, say, would reveal a system a t i c v a r i a t i o n . That a systematic v a r i a t i o n should be l i n e a r i s not obvious from the d e f i n i t i o n s , however. We may assume that i f a resultant v a r i a t i o n


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CHEMOMETRICS: THEORY AND APPLICATION

i s over a s p e c i a l region or i s s u f f i c i e n t l y short, the p l o t t e d v a r i a t i o n may appear to be l i n e a r . I t i s important to note that t h i s would be a l i n e a r segment of an otherwise nonlinear function. Such a l i n e a r v a r i a t i o n of enthalpy-enthropy p a i r s (AH,AS) i s generally summarized as


AH = BAS + AGg where the slope, β, has the dimension of temperature and i s a l t e r n a t e l y c a l l e d the compensation temperature, i s o k i n e t i c temperature or isoequilibrium temperature depending on whether the thermodynamic parameters were estimated from k i n e t i c or equilibrium data. The physical s i g n i f i c a n c e of the compensation temperature i s that at t h i s temperature a v a r i a t i o n i n enthalpy i s e n t i r e l y compensated f o r by a corresponding v a r i a t i o n i n entropy such that the free energy i s a con stant. To be consistent with the Gibbs equation, the intercept of such a l i n e a r r e l a t i o n s h i p i s the free energy at the compensation temperature, AGg (9.fJL3) . S t a t i s t i c a l Theory H i s t o r i c a l l y , compensation temperatures have been determined by least squares (or best graphical f i t , which i s e s s e n t i a l l y l e a s t squares without the computa t i o n a l rigor) and the goodness of f i t has been j u s t i f i e d by the high s i g n i f i c a n c e of the estimated c o r r e l a t i o n c o e f f i c i e n t s between the enthalpy and entropy estimates. Both of these procedures are i n c o r r e c t and, p a r t i c u l a r l y i n t h i s case, often lead to grossly i n c o r r e c t r e s u l t s . I t i s important to remember that thg enthalpy-entropy data pairs are a c t u a l l y estimates (AH,A§) not o r i g i n a l data that can be treated as either independent or as being r e l a t i v e l y free from error as might be r a t i o n a l i z e d f o r o r i g i n a l laboratory data, f o r example, k i n e t i c rate constants-temperature data (k>T) or chemical equilibrium constants-tempera ture data (Κ,Τ). The enthalpy and entropy estimates, AH and AS, both contain uncertainty, and hence l e a s t squares i s an improper technique f o r regression of a functional dependence of one on the other. What i s worse, a c t u a l l y , i s that these estimates are highly c o r r e l a t e d with one another due to t h e i r functional r e l a t i o n s h i p s with the k i n e t i c or equilibrium constants and the experimental ranges over which the data were sampled. Hence a c o r r e l a t i o n analysis might detect a s i g n i f i c a n t A

Λ



10.

KRUG ET AL.

195


c o r r e l a t i o n that r e s u l t s from these computations as data handling a r t i f a c t s , even i n the absence of any chemical e f f e c t . The s t a t i s t i c a l and chemical compen sations need not be hopelessly confounded, however, because the s c i e n t i s t has knowledge of both the chemi c a l i d e n t i t i e s and h i s choice of experimental sampling points p r i o r to analysis. Using the fundamental d e f i n i t i o n s of chemical k i n e t i c s and regression a n a l y s i s , we w i l l now show (1) that enthalpy-entropy estimates are highly corre lated, (2) that the s t a t i s t i c a l compensation equation i s f u n c t i o n a l l y i d e n t i c a l to the chemical compensation equation, (3) how to separate the chemical from the s t a t i s t i c a l e f f e c t , and (4) how to estimate the chemi c a l compensation temperature and i t s (1-α) confidence i n t e r v a l from k i n e t i c or equilibrium data. To avoid redundancy, we w i l l r e s t r i c t t h i s discussion to the case of k i n e t i c data, but f o r completeness we w i l l include the computational d e t a i l s f o r equilibrium data as well i n the Regression Algorithm. In t h i s discussion, we must make the usual assump tions that errors associated with the dependent v a r i able, the logarithm of the k i n e t i c observations, Zi lu ! i i ' normally and independently d i s t r i b u t e d =

a

r

e

2

with zero mean and constant variance, £ ^ Ν Ι Ο ( 0 , σ ) , and that the independent v a r i a b l e , the inverse experi mental temperatures, x^ = 1/T^, have no uncertainty. That i s , i n practice the experimental temperatures are determined with much greater p r e c i s i o n and accuracy than are the rate constants. To formalize t h i s analy s i s we consider data taken at 1 < i £ η temperatures for 1 £ j X and i s found to a good approximation to be 1

x

2


10.

KBUG ET AL.


30

201


lw{k/h)

20 >AS*

AG*

- f «0

EXPERIMENTAL TEMPERATURE RANGE -tOJ

ft-

Figure 2. Geometric interpretation of the parameter estimates. The indicated lengths are proportional to AG" and AS- and the indicated slope is proportional to —ΔΗ-. These data for the oximation of methyl thymyl ketone (30) indicate a strong dependence of the intercept estimate on the slope estimate because the data were taken over a very small temperature range far from the origin. The three data points are designated by dots. Reprinted with copyright permission by Nature (25) and the Journal of Physical Chemistry (26).



202 a/b = • λι/λ

2

« / η/Σ(1/Τ-)

z

For the usual experimental temperature ranges of organic chemistry t h i s r a t i o i s usually of the order of 10**. Hence, the j o i n t p r o b a b i l i t y regions appear as l i n e segments and are w e l l characterized by the l i n e that describes the major axis of the e l l i p s e . A canonical analysis of the dispersion matrix Z"" X XZ" reveals that t h i s l i n e i s 1


—

,

1

T

" hmAS + ΔΟ

and d i f f e r s from the extrathermodynamic equation, AH = $AS + AGg, only i n the value of the slope param eter. Hence, of the estimated compensation tempera ture β i s near the harmonic mean of the experimental temperatures T the compensation that i s detected h m

might only be the s t a t i s t i c a l compensation between parameter estimates that occurs because the range of the independent v a r i a b l e was too small to v a l i d a t e the extrathermodynamic model i n t h i s parameter space. Separation of the Chemical from the S t a t i s t i c a l Compensation. Because any extrathermodynamic e f f e c t i s strongly confounded with the s t a t i s t i c a l compensa t i o n e f f e c t i n the enthalpy-entropy parameter space f o r the usual ranges of experimental temperatures used i n organic chemistry, biochemistry, and even hetero geneous c a t a l y s i s , some s t a t i s t i c i a n s have attempted to solve the problem f o r the value of the compensation temperature i n the o r i g i n a l In k versus 1/T space (19, 20,22,24). The r e s u l t i n g normal equations y i e l d unwieldy nonlinear solutions that are better f o r the detection of the presence of an extrathermodynamic e f f e c t than f o r obtaining good numerical values of a compensation temperature. Others (15,31-34) have proposed c r i t e r i a to determine i f an observed com pensation i s of chemical o r i g i n or i s j u s t the s t a t i s t i c a l a r t i f a c t . We f i n d that the two compensations are separable through a t r a n s l a t i o n of the i n t e r c e p t and that the compensation temperature and i t s c o n f i dence i n t e r v a l can be solved f o r exactly using l i k e l i hood theory and the chemical Maxwell equations. The problem i s to choose an i n t e r c e p t f o r which the slope and i n t e r c e p t estimates are not c o r r e l a t e d . The i n t e r c e p t a t the arithmetic mean of the independent v a r i a b l e has t h i s property. Thus, we rewrite the l i n e a r i z e d Arrhenius equation i n the form


KRUG ET AL.

10.

In t , = ij where


203

{ I n A - E / R T , }, - { E / R } . ( 1/T.-) + ε. . nm j 3 1 13

the independent

v a r i a b l e i s now

(1/T^-).

The s l o p e i s s t i l l a measure o f t h e e n t h a l p y , b u t t h e i n t e r c e p t i s now a m e a s u r e o f t h e f r e e e n e r g y a t t h e h a r m o n i c mean o f t h e e x p e r i m e n t a l temperatures. A

G

The

?

T

= h

-RT

m

model

t o

{lnA-E/RT

+

>

RT

h m

ln(kT

h m

e/h)

-

RT^

i s now η.


t o

= XQ. = Wr.,

"2

^2

y. = η 2

"2

w h e r e t h e p a r a m e t e r v e c t o r i s ζ' . = {-E/R}.) a n d t h e d e ssiiggnn m a t r i x 2 i -E/R}^ 1

+ ε .

"2

"2

( { l n A - E / R T ^ } ·, s

1

1/T -

1/T -

1/T -

X

n

2

The s l o p e a n d i n t e r c e p t p a r a m e t e r s thermodynamic parameters by

are related

to the

4\ = Αζ_ + Β where t h e thermodynamic p a r a m e t e r v e c t o r i s (AG^ ,ΔΗ^) the a d d i t i v e constant vector T h m

B

1

=

(RT

h m

ln(kT

h m

e/h)-RT

" A

R T

h m

hm

Proceeding correlation

consideration

ν(Ψ)

=

,-RT)

= ^

s

and

Ο

= -R

0

no

2

f

a s b e f o r e , we d e t e r m i n e t h a t t h e r e i s b e t w e e n AG^m a n d ΔΗ^ e s t i m a t e s a f t e r — i-hm — of the variance-covariance matrix

Γ Tm.2£ £ ( l / T - < 1 / T > r m

0

2

2

R a m|W W| e

0

η


a


204 Cov(AÔ£ Ρ

=

,ΔΗ^)

-j

. = . •Vuâjg ) ν ( Δ ά Π T nE(l/T-) —hm v

= 0 2

hm

The r a t i o of variances between these estimates i s found to be a constant that depends only on the choice of experimental temperatures σ Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0052.ch010

2

A

λ. = σ

H

2 Δ

J

=

£ J

(Σ1/Τ)

2

ηΣ(1/Τ-)

T

2

hm.

such that i f the same experimental temperatures are chosen f o r a l l experiments, a l l such estimate p a i r s w i l l have the same r a t i o of variances ( i . e . , Xj = λ for a l l j i f T ^

= T± f o r a l l j ) .

Since the Maxwell

equations are l i n e a r r e l a t i o n s h i p s between the thermo dynamic p o t e n t i a l s H, G, Ε , and A and the properties S, Τ, P, and V, an extrathermodynamic l i n e a r r e l a t i o n ship between any two must also be r e f l e c t e d by a l i n e a r extrathermodynamic r e l a t i o n s h i p between any other two. In p a r t i c u l a r , i f an extrathermodynamic relationship AH = 3AS + AGg e x i s t s , then by s u b s t i t u t i o n with the Gibbs equation, AH = yAG + (l-Y)AGg where the diagnostic parameter γ i s r e l a t e d to the compensation temperature β by γ = l/(l-T/$) and AGg = AH - $AS = AH + YT AS/(l-y) hm


10.

KRUG ET AL.

Enthalpy-Entropy

205

Compensation

Thus, i f λ ? 1, the Gibbs equation i s i n s u f f i c i e n t to explain detected chemical behavior and an e x t r a thermodynamic e f f e c t i f detected. No s t a t i s t i c a l compensation e x i s t s between AG and AH where AH -—hm T

may be evaluated at any temperature, To t e s t the n u l l hypothesis, H :

including

Τ =

T . h m

γ = 1, AH must be

Q

regressed on AG to estimate the slope γ and i n t e r cept (l-y)AGg. Least squares i s an i n c o r r e c t pro cedure, because there i s uncertainty i n both variables. The errors are uncorrelated, however, and the r a t i o of variances i s known, see Figure 3a. The l i k e l i h o o d function i s maximized i n t h i s case by


T

min ™ a

'

Y

Σ

(AH.-a-yAcL ) hm. 3

2

T

τ— 2

-τ 2

j=l

(λ+γ ) T

hra.

This type of problem was f i r s t solved f o r the scope estimate by Lindley (35) and l a t e r commented on by others (36-38). To obtain a confidence i n t e r v a l for γ (and hence β) the d i s t r i b u t i o n of γ must be determined. Creasy (39.) solved t h i s type of problem i n transformed coordinates, which correspond i n our case to AH versus ^AG-, (see Figure 3b) , i n which hm the j o i n t p r o b a b i l i t y regions are c i r c u l a r , that i s , the errors propagate randomly with no p r e f e r e n t i a l direction. From the d i s t r i b u t i o n of the c o r r e l a t i o n c o e f f i c i e n t , the d i s t r i b u t i o n of the angle φ that a regression l i n e would make through such a plane i s determined. The d i s t r i b u t i o n of the slope γ i s deter mined from the r e l a t i o n s h i p between γ and φ. For our case, we extend t h i s l i n e of reasoning one more step and from the r e l a t i o n s h i p between γ and β, the d i s t r i b u t i o n and hence the maximum l i k e l i h o o d value and confidence i n t e r v a l of the compensation temperature β i s determined. A

The Regression Algorithm. Given k i n e t i c or equilibrium data, (k,T) or (Κ,Τ), the following algorithm may be used to obtain maximum l i k e l i h o o d estimates and t h e i r (1-a) confidence i n t e r v a l s f o r φ


#


9

Thm

Thm

Figure 3. (a) The linear regression problem in AH-AG coordinates is one for which joint confidence regions X*\y X) have a constant ratio λ of major to minor axes when the data are sampled at identical temperatures, (b) In AH- ^/\AG coordinates the maximum likelihood fit to a line is found by minimizing the sum of squares of residuals which are the perpendiculars to the regression line. Creasy (39) solved for the distribution of the slope estimate from the distribution of the correlation coefficient in similar coordinates.


Ρ g

Ε

10. KRUG ET AL.


207

γ, a, AGg and the compensation temperature β f o r a homologous s e r i e s of chemical data with 1 _< j _< η temperatures. 1. Regress = In k ^ or In onto (1/T -) i

to obtain parameter estimates squares s? η

and r e s i d u a l sum of ^

η Zy. .(1/T.-) i=l η Σ(1/Τ.-) i=l

Ά»

1 3

1


2

and .£(y j-Çij-C {l/T -}) i

s

i

T

-

2j

U^T)

— = η/Σ1/Τ

h m

i

±

= -

1

2. Then c a l c u l a t e enthalpy and free energy estimates from the slope and intercept estimates. For k i n e t i c data A a

?

R T

" - hm^j

h m

+

RT

ln

kT

i hm ( hme/h)-RT } hm

j AH^ = -RC2j - RT and f o r equilibrium

data

hm. 3 AHj

=

-R£

2

J

3. The data may be p l o t t e d with j o i n t confidence regions determined by the e l l i p t i c equation 1

1

1

2

(Ψ. - Ψ.)' A~ W W A " ( Ψ . - ψ.) = 2s F(2,n-2,1-α) ~1 ~1 ~1 ~3 3



208

or the data may be p l o t t e d along with standard devia tion increments from the maximum l i k e l i h o o d estimates from Step 2.

3

= ΛτίΔΗ.) = R s?/E(l/T-) /


•ΔΗ.

2

If a l i n e a r regression appears t h i s p l o t , then proceed with 4 f u n c t i o n a l i t y i s to be f i t t e d , weighted nonlinear technique. are s ^ =ν(ΔΗ^) from above.

to be j u s t i f i e d from and 5. I f a nonlinear use an appropriate The weighting factors

2

4. Calculate the following to f i n d maximum l i k e l i hood estimates using Lindley's s o l u t i o n (35). 2

2

λ = (ΣΙ/Τ) /(ηΣ(1/Τ-) ) S

S

GG

=

HH

=

E

E

= H

G

Θ

«

A

A

G

s

2

j/ j

H

s

2

2

( S

) / 2 S

X S

GG

2

(ΣΔΗ./3 ) /Σ1/5

ΣΔΗ .AG . / s ^ 3 3

/

2

δ

j/ j

HH-

2

- (ΣΔ0./ ) /Σ1/5

-

ΣΔΗ

2

2

/S ZAG

j

2

2

j

2

/s /ll/s

j

j

j

HG /

2

i

γ = Θ + Θ + 7 sgn( 0 +X ') = s g n ( s

) HG

u

71

Φ = tan-Μγ/^ϊ ) a =

Δ0

β

0

(EAH./S*-YZAG./S*)/E1/S*

3 D

=

â/(l-y)

-

w *

1

-

1

3

3

j

/ ^

5. F i n a l l y (l-a)100% confidence i n t e r v a l s may be calculated from the following upper and lower bound estimates using Creasy's s o l u t i o n (39).


10. KRUG ET AL.

1

t j

Enthalpy-Entropy

. ι sin"

Compensation λ(S

2t a/2 m-2

209 S

S

HH GG" HG

)

f

S

HG'

= v^Ttan^.

a„ = (ZAH /s?-Y ZAG./s?)/Zl/s L U j

2

L

3


AG

$U

= a„/(l-Y ) L U T

This regression algorithm gives maximum l i k e l i h o o d estimates and t h e i r confidence i n t e r v a l s even though there i s error i n both v a r i a b l e s , because an addi t i o n a l r e s t r a i n t i s placed on the system—the r a t i o of variances of dependent to independent variables i s a known constant, a function of the experimental tem peratures. This r e s t r a i n t holds so long as each system i s sampled a t i d e n t i c a l temperatures. I f this r a t i o λ becomes very large, the estimates w i l l con verge on the weighted l e a s t squares estimates. An i n t e r e s t i n g s i d e l i g h t i s the minimum l i k e l i hood estimate, the "worst" value of a parameter given the data. This estimate i s given by (35,36) 2

γ* = Θ + /Θ +λ

sgn (/ΡΤλ)

= -sgn(s ) HQ

$* = Τ /(1-1/γ*) ηπ

Because of the high c o r r e l a t i o n between enthalpyentropy estimates, the a p p l i c a t i o n of l e a s t squares to these enthalpy-entropy estimates w i l l y i e l d numerical values of the compensation temperature that are nearer the minimum l i k e l i h o o d value rather than the maximum l i k e l i h o o d value. Thus by misuse of l e a s t squares, a valuable s t a t i s t i c a l technique, the "worst" numerical value of a chemical parameter has usually been reported i n the l i t e r a t u r e rather than the "best" numerical value.


210

CHEMOMETRICS: THEORY AND

APPLICATION

Our analysis of 37 reported enthalpy-entropy com pensations revealed that only three had compensation temperatures s i g n i f i c a n t l y d i f f e r e n t than the harmonic mean of the experimental temperatures by an analysis i n the A H - A S plane (26) and only 7 had detectable chemical compensations by an analysis i n the A H - A G plane (27). hm T


A p p l i c a t i o n to Chemical Examples To i l l u s t r a t e the necessity of the proper regres sion procedure and proper c o r r e l a t i o n analysis, we compare a data set that c l e a r l y has a l i n e a r chemical compensation with one that c l e a r l y does not show such an extrathermodynamic e f f e c t . The v a l i d i t y of such an e f f e c t f o r t h i s second example has been debated many times i n the l i t e r a t u r e (]_r2_*i§.*i§.) · We f i n d that data f o r the hydrolysis of e t h y l benzoate (1) display a l i n e a r extrathermodynamic e f f e c t but data f o r the hydrolysis of a l k y l thymyl ketones (30) do not. The r e s u l t s of a comparative c o r r e l a t i o n analysis are l i s t e d i n Table I . As expected from our previous arguments on the c o r r e l a tion c o e f f i c i e n t , both data sets display s i g n i f i c a n t c o r r e l a t i o n s r i n A H - A S coordinates that approximate the expected c o r r e l a t i o n c o e f f i c i e n t ρ due to the propagation of e r r o r s . Only the hydrolysis data has a s i g n i f i c a n t (AO) estimated c o r r e l a t i o n c o e f f i c i e n t in AH-AG coordinates, however. This finding hm indicates that the observed enthalpy-enthropy c o r r e l a t i o n f o r the oximation data i s a r e s u l t of only the propagation of measurement e r r o r s . T

Table I . C o r r e l a t i o n C o e f f i c i e n t s * AH^-AS^

AH^-AG^n

hm Reaction 1. 2. 3.

Oximation of a l k y l thymyl ketones (30) Same as (1) but deleting the methy lated compound Hydrolysis of ethyl benzoate (1)

m 0.9999 0.9724 0 -0.2273

7

0.9999 0.9988 0

0.0770

6

0.9988 0.9987 0

0.9929

12

•Reprinted with copyright permission by the Journal of P h y s i c a l Chemistry (2_7) .



10.

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211

The j o i n t confidence regions, Μψ|γ_,Χ) and £(ψ|γ_,χ), f o r the oximation and hydrolysis data are p l o t t e d i n Figures 4 and 5 f o r comparison. The oximation data have much greater uncertainty than do the hydrolysis data. I t i s t h i s greater uncertainty that i s largely responsible f o r the apparent compensa tion i n AH-AS coordinates. In f a c t the r a t i o of major to minor axes f o r the oximation data i n AH-AS coordinates i s a/b = 23252 causing the j o i n t con fidence regions to be w e l l represented by the l i n e s of t h e i r major axes. The compensation temperature and other parameter estimates are compared i n Table II f o r estimation by (a) the regression algorithm presented here, (b) weighted l e a s t squares of AH on A Ô using s j ^ as the T

weighting factors and (c) l e a s t squares of AH on As. Because λ >> 1 f o r both examples, the a p p l i c a t i o n of weighted l e a s t squares i n the AH-AG plane hm (b) gave estimates close to the maximum l i k e l i h o o d values (a). Also for both examples the minimum l i k e l i h o o d value of the compensation temperature β* i s near the harmonic mean of the experimental tempera tures T as expected. For both examples the value T

Λ

h m

of the compensation temperature as determined b y l e a s t squares of AH on AS (c) was biased toward β* as expected, but f o r the oximation example the value of the compensation temperature as estimated by (c) was numerically much closer to the minimum l i k e l i h o o d estimate β* than to the maximum l i k e l i h o o d estimate P . That the^confidence i n t e r v a l f o r β should appear to exclude β when no chemical compensation i s detected i s i l l u s t r a t e d i n Figure 6. I f the p r o b a b i l i t y d i s t r i b u t i o n f o r the diagnostic parameter γ overlaps unity ( r e c a l l Η : γ = 1, where Ύ = 1 f o r no l i n e a r extrathermodynamic effect) the p r o b a b i l i t y density for β i s t h i n l y d i s t r i b u t e d over a l l possible numbers such that the confidence i n t e r v a l f o r γ traces to a confidence i n t e r v a l f o r β that s t a r t s a t a f i n i t e value, extends to i n f i n i t y , returns from minus i n f i n i t y , and f i n a l l y returns to a f i n i t e value. The MLE of β i s then somewhere i n that i n t e r v a l . When t h i s happens H cannot be rejected and the p r o b a b i l i t y density of β i s d i s t r i b u t e d so t h i n l y that the p r o b a b i l i t y of detecting a compensation temperature i n a reasonably f i n i t e i n t e r v a l i s i n f i n i t e s i m a l , and hence the p r o b a b i l i t y that a l i n e a r extrathermodynamic e f f e c t i s detected i s e s s e n t i a l l y zero. A

Q



212


AS* (eu)

AG* ( M Nature Journal of Physical Chemistry

Figure 4. The 50% joint confidence regions for the oximation of alkyl thymyl ketones (30). The A//=-AS= ellipses are so narrow that they appear as lines. Departure of the methylated compound from a common AG= value causes it to fall off the statistical com pensation "line" between ΔΗ= and AS= estimates. Notice that AG= is estimated more precisely than ΔΗ=. All values were calculated for Τ — T = 308.1 Κ (25, 27). hm



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213


AS*(eu)

A6*(kcal) Journal of Physical Chemistry

Figure 5. Plots of thermodynamic parameter estimates and their respective 50% confi dence regions for the hydrolysis of ethyl benzoate (1). The linear structure in the ΔΗ=AG= plot indicates that a linear chemical compensation is detected. The data are evalu ated atT = T — 292.6 Κ (27). hm




Figure 6. Probability density functions for y and β. The bold line is the function f(y) through which the well-behaved probability density functions p(y\V T) are mapped into either well-behaved or skewed density functions pjf/?|k,T). The footnote "1" represents parameters obtained from the hydroly sis of ethyl benzoate (1), ana "T represents parameters obtained from the oximation of thymyl ketones (SO). Reprinted with copyright permission by the Journal of Physical Chemistry (27). t


10. KRUG ET AL.

Enthalpy-Entropy

215

Compensation


22

linear compensation line σ

ο

18

-ΗΧ

> 1, which i s the usual case. Such structured nonlinear functions are v i r t u a l l y impossible to d i s t i n g u i s h from random (unstructured) s c a t t e r i n AH-AS coordinates because of the dominant s t a t i s t i c a l compensation between parameter estimates i n those coordinates. T


216

CHEMOMETRICS:

THEORY AND APPLICATION


Conclusions The h i s t o r y of enthalpy-entropy compensation i s one characterized by a strong misuse of fundamental s t a t i s t i c a l tools (the method of l e a s t squares and c o r r e l a t i o n analysis) to reach i n c o r r e c t conclusions about the detection and v a l i d i t y of observed compensa tions between enthalpy and entropy estimates. Any chemical e f f e c t i n the enthalpy-entropy plane i s strongly confounded with the l i n e a r s t a t i s t i c a l com pensation pattern due to the l i m i t e d experimental temperature ranges suitable for most chemical i n v e s t i gations. Enthalpy and entropy estimates are s t i l l frequently p l o t t e d versus one another to display f a l s e c o r r e l a t i o n s (compensations) f o r organic, biochemical, and heterogeneous c a t a l y t i c reactions, however. Because both the relevant chemical i d e n t i t i e s and the choice of experimental temperatures are a v a i l a b l e to a s c i e n t i s t p r i o r to data analysis, he can and should choose to perform h i s c o r r e l a t i o n analysis and regression i n the AH-AGplane to detect the prehm sence of both l i n e a r and nonlinear r e l a t i o n s h i p s between any thermodynamic variables f o r both k i n e t i c and equilibrium data. T

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41.



Enthalpy-Entropy Compensation: An Example of the Misuse of Least

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