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The isokinetic relationship. IX. Connections to linear free energy relationships
265
Chemical Physics 119 (1988) 265-274 North-Holland. Amsterdam
THE ISOKINETIC IX. CONNECTIONS Wolfgang Institute
RELATIONSHIP. TO LINEAR FREE
ENERGY
RELATIONSHIPS
LINERT
of Inorganic
Chemistry,
Technical University of Vienna, Getreidemarkt
9, A-1060
Vienna, Austria
and Valentin
N. SAPUNOV
Chemical Technological Received
21 August
Mendeleev
Institute,
Moscow,
USSR
1987
The mutual interrelations between common points of intersection in various temperature-dependent relationships (LFERs) are analysed and compared with experimental examples. With this approach isokinetic relationships are brought under the umbrella of one treatment. Both Hammett and Broensted used for comparison with experiment.
1. Introduction In linear free-energy relationships [1,2] (LFERs) rate constants or equilibrium constants (or related parameters describing the kinetic or thermodynamic chemical behaviour) of a series of reactions are plotted versus a characteristic quantity measured by means of another reaction series. Typical examples are Broensted [3,4] and Hammett [5] relationships, or correlations with solvent parameters such as Gutmann’s donor number [6] and Reichhardt’s E, values [7], etc. Whenever a LFER is found, some quanitity must be present, which appears in the same functional form in the freeenergy terms of both, the “test reaction series” and the “reference reaction series” [8]. To handle the named relations in a generalized form, we substitute this quantity by a parameter 5 which obviously may be taken as linear for both reaction series. It has been shown that for reaction series, in which only one reaction mechanism occurs, isokinetic behaviour is found [9-141. The isokinetic relationship (IKR) produces, in its general form, a common point of intersection in the Arrhenius 0301-0104/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
linear free-energy isoparametric and reaction series are
(In k versus l/T) or the van ‘t Hoff (In K versus l/T) plane. It should be pointed out that a common point of intersection of straight lines is mathematically equivalent to a proportionality of their intercepts and slopes (i.e. the preexponential factor and activation energy in case of Arrhenius plots or the enthalpies and entropies in case of van ‘t Hoff plots). Therefore it has been referred to as the “compensation effect”. This is, however, not equivalent from the statistical point of view! (The proportionality may be an artifact due to measurement errors [14].) This has been discussed and resolved in detail in earlier papers [lo-171. Such common points of intersection are also found when the same data are depicted for different temperatures in the LFER plot. Whereas in the former case the characteristic intersection abscissa has the dimension of a reciprocal temperature (l/T& in the latter case a characteristic LFER parameter (EiJ is found. In other words, when an IKR is found an “isoparameter” appears and vice versa [2,10-141. A general description of such simultaneously arising common points of intersection is the purpose of the present paper.
B.V.
266
W. Linert, l! N. Sapunov / The isokinetic relationship. IX
2. Representation
of T-dependent LFER data
The common points of intersection are defined in differential form by means of the abscissa where the net differences in the reactivity (i.e. the selectivity) have vanished. For van ‘t Hoff or Arrhenius plots (i.e. In K or In k versus l/T) the IKR is given by a ln K/at I 1,T,, = 0.
(1)
For convenience, equilibrium constants and corresponding AG, AH and AS values are used throughout the paper. However, the same equations are valid for rate constants replacing AH by activation energies E, and AS by Arrhenius preexponentials In A. For a generalized LFER using the abovementioned parameter 5 we have: i3 In
fc/a(i/z-)
I(,, = 0.
(2)
It has been shown that the ordinates of this common point of intersection in different planes must be identical for a given reaction series, viz. ln K(5)
I XIW= In K(l/T)
1c,, = In Kiso.
(3)
The same relations must be true when In K is replaced by AG = -RT In K. In the following section we are concerned with some possible representations of such data measured at different temperatures for different reactions belonging to the same reaction series. The data may be repre-
a
sented in the form of In K (or In k) or as the corresponding free energies AG (or AG*). Assuming van ‘t Hoff or Arrhenius behavior such data yield either with l/T (In K represenation) or with T (AG representation) linear plots for each member of the reaction series. Putting it another way, the LFER may equally well be depicted in a In K versus 5 or in a AG versus 6 plot, both yielding linear plots for each temperature. 2.1. The In k versus .$ plane The left-hand side of fig. 1 shows an arbitrary LFER for different temperatures. The straight lines are represented by In K(t,
X)
=ln
K,(x)
for T,, T,, . . . , 7;,
+5 a In K(t, with x = 1/RT.
x)/at, (4
For example, if one identifies 5 with the Hammett parameter u, then the term a In K(a, x)/ax would represent the Hammett slope p. The term In K, refers to a reaction used as a reference (i.e. defining 4 = 0). For Hammett reaction series this would correspond to unsubstituted reactants. Using condition (2) for the common point of intersection we get Siso,H
a In K,(x)/ax aa In K(x)/axag
=
AH, = - am(~)
(9
b
x c
In Lo
~*~~~~~~~~_____
---
tnK
is.0
T T2
/
G
5
52
53
fiso,H f
Fig. 1. Schematic plot of equilibrium constants (or rate constants) of a series of three different reactions measured at three different temperatures in (a) the LFER-plane and (b) the van ‘t Hoff (Arrhenius) plane.
W. Linert, KN. Sapunov / The isokinetic relationship.
z
IX
261
cl
a
--.
A%0
-------
-- --
Tiso
A%o
TI -r2
-+
5
Fig. 2. Schematic plot of free energies (or free energies of activation) of a series of three different reactions measured at three different temperatures in (a) the LFER plane and (b) in the AG versus T plane.
AH = - ElIn K([)/ax). ti, H is therefore the negative reciprocal of the rela’tive change of the reaction enthalpy with the LFER parameter [ and using AH, as a reference.
(with
2.3. The AG versus t plane The left-hand side of fig. 2 corresponds to AGfS, 7’) = AG,‘(T) + < aAG(& 2-)/X, for r,, jr,, . . . , Tl
2.2. The In k versus I / T plane
w
with the common point of intersection given by The right-hand side of fig. 1 represents Arrhenius or van ‘t Hoff plots and is described by:
aAG(&, T)/aT
In K(S,
This leads to the condition
x) =ln
&(c)
for &, 52, ---,
+x
3 In K($, x)/ikx,
tj*
(6)
~iso,S
In a more convenient form following the van ‘t Hoff law this can be rewritten as In K(5,
x) = AS(t)
- AH(<)x
(7)
which again rewrites including the IKR (IER) as In K([,
x) = In Kiso - AH(E)(x
- xi,,)
63)
where AS = In K, is the entropy, AH = 3 In K/ax the enthalpy of the reaction c and In K,, the ordinate of the common point of intersection. Condition (1) leads with this to 1 xiso
=
RT,so
aAS(l =
R aAH(~),‘at
aAS = R aAH
- (9)
refers therefore to the proportionality factor between AH and AS, Tiso, which is already known from the compensation effect. xiso
=
I &,, = 0.
~AG~~T)/aT T)/iilT
%AG(&
=-.-
(19
i3$
ASU aA.scwa4~
(12)
is therefore the reciprocal negative relative change of the reaction entropy with AS, taken as reference. Obviously this isop~~et~c value is not identical with that resulting from the In k versus < plots. (As one is enthalpy and the other entropy governed, subscripts H and S are used in eqs. (5) and (12).) tiso,S
2.4. The AG versus T plane The right-hand side of fig. 2 can be represented by AG(5, 7’) = AG,(5) + T aAG(E, T),‘W, for &, c2, .a.3
Ej-
(13)
268
x C
fist, H -5
Fig. 3. LFER plot using both the In
Again, this can be identified with AG(f, T) = A.W(E) + rA.s(O for an arbitrary parameter 5.
04)
Applying the condition for a common point of intersection which is now
K
and AG scale (see text)
mental range of 5 (e.g., when rate constants of such reactions are investigated). However, as mentioned earlier, all these plots should have the same interception ordinate either in the In K (In k) or the free-energy scale (AG;, = AHC - Z&S, = - RT,, In K). Thus enthalpic and entropic isoparameters are connected as follows
aAG,% I r,, = 0, this leads to
q,, = aAW$‘aA.s(O
06)
It can be seen that eqs. (9) and (16) are equivalent in contrast to eqs. (5) and (12). Therefore only one isokinetic temperature is found, independent of the ordinate used whereas two different isokinetic parameters (tiso,H and Eiso,s) appear. Fig. 3 shows this state of affairs by depicting both the In K versus E and the AG versus 5 plots on one graph using the same ordinate units. Assuming that no negative activation energies and no positive activation entropies should appear in an elementary reaction series, $iso,H and ,$i,O,smight be interpreted as limiting values for the experi-
From this AGi, can be identified easily knowing that it must be independent of the reactants, i.e. from .$’by using the (if necessary extrapolated) AG value of an arbitrary t for the isokinetic temperature. The latter is either known from temperaturedependent experiments by using a proper statistical isoparameter (isokinetic) analysis, or, as will be seen in the next section, by using some LFER information.
W. Linert, V.N.
Sapunov / The &kinetic relationship. IX
3. T-dependent slopes in LFERs
ing that for both the test and the reference reaction system an IKR (IER) exists, i.e.
As pointed out in section 1, 4 represents a temperature-independent parameter which defines some reactants, substituents, catalysts, solvents, etc. and which is linearly related to both the testand the reference-reaction series of a LFER investigation. In most cases we do not know the physical meaning of .$ so that AG or In K values of the reference series are used. As 4 should be independent of temperature, the corresponding AG or In K values are taken at a certain reference temperature (for example 298 K in case of the Hammett relation). The isoparameters given above (i.e. the parameters defining the reaction independent from temperature) as well as the LFER slopes appear in the appropriate LFER scale and obviously depend upon the chosen reference reaction series and its reference temperature). In this section relations are presented enabling us to transform between different LFER-scales and to interpret the physical meaning of the LFER slopes. Using the Leffler operator 6 111, a linear freeenergy relationship between the reaction series I and II (LFER(T,, T,,)) can be written as aACt*,, = ax,,x,, AGn,,,,,
(18)
with SAG, = AGSY, - AGX,,
(191
(22)
6AG1.X= 6 AHi(x - Xiso,i) and
(23)
SAG,,., = 6AHti (x - Xiso,ti)+ we get for the LFER slopes in such a plot
(24) With the temperature-independent
slope LYE
(25) this may be rewritten as ffx=ffox_
x-
xi*0
’
I
(26)
.
xiso.II
Eq. (25) shows that a LFER plot with both axes taken in their temperature-dependent form exhibit parallel lines. Physically this slope can be interpreted as the relative changes of the reaction enthalpies in test- and reference-reaction series. Eq. (26) gives the transformation to the common LFER plot using values for the reference reaction series at a constant temperature. 3.2. Test and reference reucticrns taken at different temperatures
and SAG,, = AG,,,, - AGU,,,
269
(20)
where s represents an arbitrary member of the series (i.e. a substituent, solvent, catalyst, etc.) and u is one member of the series taken as a reference.
For the case in which the temperature of the test-reaction series and the reference-reaction series do not coincide, we get in the in K scale
and for CX~,,~,, 3.1. Test and reference series used at equal
XI
-xiso,II
=
x -
xiso,II
x -
xiso,ll
temperatures
f-G, ,x,, = a0
For the case in which equal temperatures are used for both the test and the reference series (LFER(T,, ‘I’,,) with TX = T,, = l/Rx) and assum-
In common LFER plots the reference-reaction series is taken at a constant xtI (or Tt,) value yielding the slopes &,,+,,,. For example in the
(28)
ax XII -
xiso,II
W. Linert, V. N. Sapunov / The isokinetic relationship. IX
270
common Hammett plot 7’,, is 298 K with a temperature-dependent Hammett slope or. The temperature-dependent slopes CY~,,~,,can be calculated from CY, (resulting from a plot with both series taken at the same temperature) and using the isokinetic temperature of the reference reaction series. It is therefore possible to predict the temperature dependence of the test reaction series. The isokinetic reactant of the test reaction series according to .$‘t,,, is defined by a ln(K,,,/K,,,)/~x From
this definition
I S=E,M.H = 0.
(29)
- xiso,,,)
AH, = AHu(l
- I/‘Siso,H).
Tii,, may then be calculated
AGi,, = T,,,AS,
(30)
(33) from
+ AH,)
which can be used to calculate ing to AS, = 6A HJT,,,
we get the condition
ln( k .T,,,..,,/ku.,,),,, = ‘AH,,(x,,
knowing AH, and AS, for the unsubstituted reactant in a Hammett series). In this way the enthalpies of the test-reaction series may be calculated using eq. (17)
3.3. Experimental
(34) AS, values, accord-
+ AS,, .
(35)
examples
which equals, with eq. (25) ln(k
.L.II&I),,,
= (‘AffI/a,)(xII
-xiso,~I).
(31) The enthalpy A Hiso,, must equal zero for the test reaction with the reactant according to ,$‘t,,, with leads to ln(k r,,,,,,,/k”.,,),,,
= (AHu.,/a~)(x,,
- xiso,,,). (32)
In this equation the ratio ln(ks!,,,,/ku,,,)x,, gives the abscissa of the common point of intersection in a T-dependent LFER with a reference reaction series taken at a constant temperature T,, = l/Rx,,. It is seen that an isoparametric relationship (for example the iso-Hammett relation) only exists when the reference-reaction series is itself temperature dependent and when an isokinetic temperature Tiso,,, = l/RXiso,,, exists. In this case eq. (32) gives, if this condition is fulfilled, a simple prediction of the isokinetic parameters ,$,,O,H and Inserting this using eq. (17) 5is0,s is obtained. result (in form of ln( k,/k,) in eq. (27) gives the ordinate of the common point of intersection, i.e. of the In Kiso,l. In this way, the T dependence whole test-reaction series is described by knowing the temperature dependence of the reference-reaction series, the LFER slope at one temperature and the temperature dependence of one member of the test-reaction series (for example when
For the Hammett relations it has been in parts III and IV of this publication [13,14] that
shown series
A H,,/263Oa,
(36)
p. = A Hi/( AH,” - A Hs’lis,,)
(37)
=
and
is valid, when Tiso ,, = - 255 K and T,, = 298 K is used for the ionization of the benzoic acids and assuming T, = T,,. AH,’ is the enthalpy value of the test reaction with unsubstituted reactants; AH,” and AH,” are the ionization enthalpies of the parent benzoic acid and that bearing the isokinetic substituent, respectively. This is consistent with eq. (25). In this paper 16 different. Hammett reaction series were examined for their IKR behavior. Those reaction series showing statistical significant common points of intersection (i.e. excluding isoenthalpic series without a defined point of intersection) are used to compare experimental and calculated u,,, H values. The ordinate values in fig. 4 are calculated from eq. (36) with p. values taken from eq. (28) to allow for different mean experimental temperatures for test and reference reaction series. From the point of view that no information about the T dependence of the test-reaction series is used to calculate the common points of intersection, the agreement which is
H
I N
Nl-+-C~ \N II
I
&--C----N Scheme 1.
-10
Fig, 4, ExperimentaX versus c&Aated
biw,H vafues of Hammett reaction series (for details see refs. [1J,14]).
fsund seems to be rather impressive
Further the IKR analyses have shuwn the equality of the intersection ordinates in the Hammett plots and the correspcCGng van ‘t Hoff or Arrhenius plots in agreement with eq. (3).
As another exam@ fop a LFER the generalized 33roensted relation using the isomerization of Saminotetrazoles according to the reaction shown in scheme f [l&19]. This reaction has been chosen because fur both LFER axes t~~~peraturedependent equilibrium constants and rate constants are available from the @xperimentaf data. The Arrhenius and the van ‘t Hoff plots are depicted in ref. [Xl] for the rate constants of the forward reactions and the e~~~~~b~~rnconstants. Both show highly significant PKR (IER) with cumm3on points of intersection at TiSo:lso.f = 765.3 K for the forward reactions and T,So,ec, = 662.3 K fur
0.6
04
v;”
0.2
Fig. 5, Broensted plot for the isomerkation of substituted S-aminotetrmoles [27,28j using temperature-dependent abscissa vah~s~ The curv4 line referes to the residual sum of square deviations of the measured points from constrained (i.e. assuming a common point of intersection) straighr lines; see r&s. f16,17].
W. Linert, V.N. Sapunov / The isokinetic relationship.
272
IX
0
-10 -5
0
15
5
20
In K(T=423Kl Fig. 6. Broensted plot for the isomerization of substituted 5-aminotetrazoles [26,27] using abscissa values taken at a constant temperature. The curved line refers to the residual sum of square deviations of the measured points from constrained (i.e. assuming a common point of intersection) straight lines; see refs. [16,17].
the equilibria. F values obtained from the statistical analysis [17] are 0.19 (fi = 3, f2 = 5) and 0.61 ( fi = 3, f2 = 8). Plotting In kf versus In K straight lines are found in agreement with a generalized Broensted relation. Fig. 4 shows the plot of this relation with both temperature-dependent values on coordinates (curved lines rather to residual sum of squares associated with a common point of intersection). In agreement with the above theory (see eqs. (25) and (26)) these plots show merely parallel lines. In fig. 5 the same data are plotted using the equilibrium constants (used as a reference series) at a constant temperature. A common point of intersection is found at an ordinate value of In kiso,f = 8.5 f 1 and abscissa value of In Kiso,r = 16.0 f 1. The corresponding F value of 0.063 ( fi = 3, f2 = 8) shows that the isoparameter relation has to be accepted at a high significance level. Using eq. (39) (again without using any information about the T dependence of the rate constants treated as the test-reaction series) an isoparametric value of 17.3 is calulated for In Kiso,r in agreement with the above experimental value. The experimental examples given above show
that it is possible to predict the common point of intersection exhibited by reaction series plotted for different temperatures in the LFER plane. To calculate this point (and with this the temperature dependence of the whole reaction series) we need the LFER at one temperature, the temperature dependence (IKR or IER) of the reference-reaction series and the temperature dependence of one member of the considered test-reaction series.
4. Summary Fig. 7 shows the various possibilities to represent temperature-dependent LFER data (in the In K scale) as described in this paper. As the treatment of common points of intersection is used (it originally arises from the statistical analysis of the problems) one might speak about “invariant points” in chemical reaction series or about “isoparametric” relationships. Obviously the same is valid for AG plots. In the previous papers [2,9,17,20-221 a theoretical explanation of the isokinetic relationship has been given. As a main
W’. Liner?, V.N. fapunov
273
/ Tke isokinetic relationship. IX
Ln K, (T)
LFER IT,, Tn)
tn K,
in K, iT)
(T= constl
LFER (T;=const, Tn)
LFER (T,, T,=constl
1
lKRI
T
IKR,
I
Fig. 7. Various graphical representations of the temperature dependence of LFER plots.
conclusion, the isokinetic temperature can be interpreted as an active vibrational frequency of the surroundings of the reaction site. The frequencies fv) co~espon~ng to ~iso (Xiso - l/‘tiv) are found in the vibrational spectra of the solvent in case of liquid phase reactions [20-221 and of the catalyst in the case of heterogeneous reactions [23]. From this point of view the crux of the present paper is that different isoparametric relations can be deduced from the isokinetic relationship. The present paper also enables the prediction of the temperature dependence of chemical reaction series and the characterization of a manifold of experimental data using q:,,, and %iso. Acknowledgement to
Thanks are due to Professor V. Gutmarm and the “ Fonds zur Forde~ng der wissen-
schaftlichen Forschung project 5443.
in fisterreich”
under the
References [l] J.E. Leffler and E. Grunwald, Rates and equilibria of organic reactions (Wiley, New York, 1963). [2] W.C. Conner, J. Catal. ‘78 (1982) 238. [3] J.N. Broensted, Chem. Rev. 5 (1929) 231. ]4] R.D. Levine, J. Phys. Chem. 83 (1979) 159. [5] L.P. Hammett, Physikahsche organ&he Chemie (Vet-lag Chemie, Weinheim, 1973). [6] V. Gutmann, The donor acceptor approach to molecdar interactions (Plenum Press, New York, 1978). (73 C. Reichardt, Solvent effects in organic chemistry (Verlag Chemie, Weinheim, 1979). [8] R.P. Wells, Linear free energy relationships (Academic Press, New York, 1968). [9] 0. Exner, Collection Czech. Chem. Commun. 40 (1975) 2742.
274
W. Linert, KN. Sapunoo / The isokinetic relationship.
[lo] 0. Exner and V. Beranek, Collection Czech. Chem. Commun. 38 (1973) 781; 0. Exner, Progr. Phys. Org. Chem. 10 (1973) 411. [ll] W. Liner& A.B. Kudjawtsev and R. S&mid, Australian J. Chem. 36 (1983) 1903. [12] W. Linert and A.B. Kudjawtsev, Australian J. Chem. 37 (1984) 1139. [13] W. Liner& R. S&mid and A.B. Kudjawtsev, Australian J. Chem. 38 (1985) 677. [14] W. Linert, Australian J. Chem. 39 (1986) 199. [15] C.G. Swain, MS. Swain, A.L. Powell and S. Altmni, J. Am. Chem. Sot. 105 (1983) 502.
IX
[16] W. Linert, R.W. Soukup and R. S&mid, Computer Chem. 6 (1982) 47. [17] W. Linert, Inorg. Chim. Acta 141 (1988), to be published. [18] R.A. Henry, W.G. Finnegan and E. Lieher, J. Am. Chem. Sot. 77 (1955) 2264. 1191 E. Lieber, C.N. Romachandra and T.S. Cao, J. Am. Chem. Sot. 79 (1957) 5962. [20] W. Liner& Chem. Phys. 114 (1987) 457. [21) W. Linert, Chem. Phys. 114 (1987) 449. [22] W. Linert, Chem. Phys. 116 (1987) 381. [23] W.C. Conner and W. Linert, to be published.
Chemical Physics 119 (1988) 265-274 North-Holland. Amsterdam
THE ISOKINETIC IX. CONNECTIONS Wolfgang Institute
RELATIONSHIP. TO LINEAR FREE
ENERGY
RELATIONSHIPS
LINERT
of Inorganic
Chemistry,
Technical University of Vienna, Getreidemarkt
9, A-1060
Vienna, Austria
and Valentin
N. SAPUNOV
Chemical Technological Received
21 August
Mendeleev
Institute,
Moscow,
USSR
1987
The mutual interrelations between common points of intersection in various temperature-dependent relationships (LFERs) are analysed and compared with experimental examples. With this approach isokinetic relationships are brought under the umbrella of one treatment. Both Hammett and Broensted used for comparison with experiment.
1. Introduction In linear free-energy relationships [1,2] (LFERs) rate constants or equilibrium constants (or related parameters describing the kinetic or thermodynamic chemical behaviour) of a series of reactions are plotted versus a characteristic quantity measured by means of another reaction series. Typical examples are Broensted [3,4] and Hammett [5] relationships, or correlations with solvent parameters such as Gutmann’s donor number [6] and Reichhardt’s E, values [7], etc. Whenever a LFER is found, some quanitity must be present, which appears in the same functional form in the freeenergy terms of both, the “test reaction series” and the “reference reaction series” [8]. To handle the named relations in a generalized form, we substitute this quantity by a parameter 5 which obviously may be taken as linear for both reaction series. It has been shown that for reaction series, in which only one reaction mechanism occurs, isokinetic behaviour is found [9-141. The isokinetic relationship (IKR) produces, in its general form, a common point of intersection in the Arrhenius 0301-0104/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
linear free-energy isoparametric and reaction series are
(In k versus l/T) or the van ‘t Hoff (In K versus l/T) plane. It should be pointed out that a common point of intersection of straight lines is mathematically equivalent to a proportionality of their intercepts and slopes (i.e. the preexponential factor and activation energy in case of Arrhenius plots or the enthalpies and entropies in case of van ‘t Hoff plots). Therefore it has been referred to as the “compensation effect”. This is, however, not equivalent from the statistical point of view! (The proportionality may be an artifact due to measurement errors [14].) This has been discussed and resolved in detail in earlier papers [lo-171. Such common points of intersection are also found when the same data are depicted for different temperatures in the LFER plot. Whereas in the former case the characteristic intersection abscissa has the dimension of a reciprocal temperature (l/T& in the latter case a characteristic LFER parameter (EiJ is found. In other words, when an IKR is found an “isoparameter” appears and vice versa [2,10-141. A general description of such simultaneously arising common points of intersection is the purpose of the present paper.
B.V.
266
W. Linert, l! N. Sapunov / The isokinetic relationship. IX
2. Representation
of T-dependent LFER data
The common points of intersection are defined in differential form by means of the abscissa where the net differences in the reactivity (i.e. the selectivity) have vanished. For van ‘t Hoff or Arrhenius plots (i.e. In K or In k versus l/T) the IKR is given by a ln K/at I 1,T,, = 0.
(1)
For convenience, equilibrium constants and corresponding AG, AH and AS values are used throughout the paper. However, the same equations are valid for rate constants replacing AH by activation energies E, and AS by Arrhenius preexponentials In A. For a generalized LFER using the abovementioned parameter 5 we have: i3 In
fc/a(i/z-)
I(,, = 0.
(2)
It has been shown that the ordinates of this common point of intersection in different planes must be identical for a given reaction series, viz. ln K(5)
I XIW= In K(l/T)
1c,, = In Kiso.
(3)
The same relations must be true when In K is replaced by AG = -RT In K. In the following section we are concerned with some possible representations of such data measured at different temperatures for different reactions belonging to the same reaction series. The data may be repre-
a
sented in the form of In K (or In k) or as the corresponding free energies AG (or AG*). Assuming van ‘t Hoff or Arrhenius behavior such data yield either with l/T (In K represenation) or with T (AG representation) linear plots for each member of the reaction series. Putting it another way, the LFER may equally well be depicted in a In K versus 5 or in a AG versus 6 plot, both yielding linear plots for each temperature. 2.1. The In k versus .$ plane The left-hand side of fig. 1 shows an arbitrary LFER for different temperatures. The straight lines are represented by In K(t,
X)
=ln
K,(x)
for T,, T,, . . . , 7;,
+5 a In K(t, with x = 1/RT.
x)/at, (4
For example, if one identifies 5 with the Hammett parameter u, then the term a In K(a, x)/ax would represent the Hammett slope p. The term In K, refers to a reaction used as a reference (i.e. defining 4 = 0). For Hammett reaction series this would correspond to unsubstituted reactants. Using condition (2) for the common point of intersection we get Siso,H
a In K,(x)/ax aa In K(x)/axag
=
AH, = - am(~)
(9
b
x c
In Lo
~*~~~~~~~~_____
---
tnK
is.0
T T2
/
G
5
52
53
fiso,H f
Fig. 1. Schematic plot of equilibrium constants (or rate constants) of a series of three different reactions measured at three different temperatures in (a) the LFER-plane and (b) the van ‘t Hoff (Arrhenius) plane.
W. Linert, KN. Sapunov / The isokinetic relationship.
z
IX
261
cl
a
--.
A%0
-------
-- --
Tiso
A%o
TI -r2
-+
5
Fig. 2. Schematic plot of free energies (or free energies of activation) of a series of three different reactions measured at three different temperatures in (a) the LFER plane and (b) in the AG versus T plane.
AH = - ElIn K([)/ax). ti, H is therefore the negative reciprocal of the rela’tive change of the reaction enthalpy with the LFER parameter [ and using AH, as a reference.
(with
2.3. The AG versus t plane The left-hand side of fig. 2 corresponds to AGfS, 7’) = AG,‘(T) + < aAG(& 2-)/X, for r,, jr,, . . . , Tl
2.2. The In k versus I / T plane
w
with the common point of intersection given by The right-hand side of fig. 1 represents Arrhenius or van ‘t Hoff plots and is described by:
aAG(&, T)/aT
In K(S,
This leads to the condition
x) =ln
&(c)
for &, 52, ---,
+x
3 In K($, x)/ikx,
tj*
(6)
~iso,S
In a more convenient form following the van ‘t Hoff law this can be rewritten as In K(5,
x) = AS(t)
- AH(<)x
(7)
which again rewrites including the IKR (IER) as In K([,
x) = In Kiso - AH(E)(x
- xi,,)
63)
where AS = In K, is the entropy, AH = 3 In K/ax the enthalpy of the reaction c and In K,, the ordinate of the common point of intersection. Condition (1) leads with this to 1 xiso
=
RT,so
aAS(l =
R aAH(~),‘at
aAS = R aAH
- (9)
refers therefore to the proportionality factor between AH and AS, Tiso, which is already known from the compensation effect. xiso
=
I &,, = 0.
~AG~~T)/aT T)/iilT
%AG(&
=-.-
(19
i3$
ASU aA.scwa4~
(12)
is therefore the reciprocal negative relative change of the reaction entropy with AS, taken as reference. Obviously this isop~~et~c value is not identical with that resulting from the In k versus < plots. (As one is enthalpy and the other entropy governed, subscripts H and S are used in eqs. (5) and (12).) tiso,S
2.4. The AG versus T plane The right-hand side of fig. 2 can be represented by AG(5, 7’) = AG,(5) + T aAG(E, T),‘W, for &, c2, .a.3
Ej-
(13)
268
x C
fist, H -5
Fig. 3. LFER plot using both the In
Again, this can be identified with AG(f, T) = A.W(E) + rA.s(O for an arbitrary parameter 5.
04)
Applying the condition for a common point of intersection which is now
K
and AG scale (see text)
mental range of 5 (e.g., when rate constants of such reactions are investigated). However, as mentioned earlier, all these plots should have the same interception ordinate either in the In K (In k) or the free-energy scale (AG;, = AHC - Z&S, = - RT,, In K). Thus enthalpic and entropic isoparameters are connected as follows
aAG,% I r,, = 0, this leads to
q,, = aAW$‘aA.s(O
06)
It can be seen that eqs. (9) and (16) are equivalent in contrast to eqs. (5) and (12). Therefore only one isokinetic temperature is found, independent of the ordinate used whereas two different isokinetic parameters (tiso,H and Eiso,s) appear. Fig. 3 shows this state of affairs by depicting both the In K versus E and the AG versus 5 plots on one graph using the same ordinate units. Assuming that no negative activation energies and no positive activation entropies should appear in an elementary reaction series, $iso,H and ,$i,O,smight be interpreted as limiting values for the experi-
From this AGi, can be identified easily knowing that it must be independent of the reactants, i.e. from .$’by using the (if necessary extrapolated) AG value of an arbitrary t for the isokinetic temperature. The latter is either known from temperaturedependent experiments by using a proper statistical isoparameter (isokinetic) analysis, or, as will be seen in the next section, by using some LFER information.
W. Linert, V.N.
Sapunov / The &kinetic relationship. IX
3. T-dependent slopes in LFERs
ing that for both the test and the reference reaction system an IKR (IER) exists, i.e.
As pointed out in section 1, 4 represents a temperature-independent parameter which defines some reactants, substituents, catalysts, solvents, etc. and which is linearly related to both the testand the reference-reaction series of a LFER investigation. In most cases we do not know the physical meaning of .$ so that AG or In K values of the reference series are used. As 4 should be independent of temperature, the corresponding AG or In K values are taken at a certain reference temperature (for example 298 K in case of the Hammett relation). The isoparameters given above (i.e. the parameters defining the reaction independent from temperature) as well as the LFER slopes appear in the appropriate LFER scale and obviously depend upon the chosen reference reaction series and its reference temperature). In this section relations are presented enabling us to transform between different LFER-scales and to interpret the physical meaning of the LFER slopes. Using the Leffler operator 6 111, a linear freeenergy relationship between the reaction series I and II (LFER(T,, T,,)) can be written as aACt*,, = ax,,x,, AGn,,,,,
(18)
with SAG, = AGSY, - AGX,,
(191
(22)
6AG1.X= 6 AHi(x - Xiso,i) and
(23)
SAG,,., = 6AHti (x - Xiso,ti)+ we get for the LFER slopes in such a plot
(24) With the temperature-independent
slope LYE
(25) this may be rewritten as ffx=ffox_
x-
xi*0
’
I
(26)
.
xiso.II
Eq. (25) shows that a LFER plot with both axes taken in their temperature-dependent form exhibit parallel lines. Physically this slope can be interpreted as the relative changes of the reaction enthalpies in test- and reference-reaction series. Eq. (26) gives the transformation to the common LFER plot using values for the reference reaction series at a constant temperature. 3.2. Test and reference reucticrns taken at different temperatures
and SAG,, = AG,,,, - AGU,,,
269
(20)
where s represents an arbitrary member of the series (i.e. a substituent, solvent, catalyst, etc.) and u is one member of the series taken as a reference.
For the case in which the temperature of the test-reaction series and the reference-reaction series do not coincide, we get in the in K scale
and for CX~,,~,, 3.1. Test and reference series used at equal
XI
-xiso,II
=
x -
xiso,II
x -
xiso,ll
temperatures
f-G, ,x,, = a0
For the case in which equal temperatures are used for both the test and the reference series (LFER(T,, ‘I’,,) with TX = T,, = l/Rx) and assum-
In common LFER plots the reference-reaction series is taken at a constant xtI (or Tt,) value yielding the slopes &,,+,,,. For example in the
(28)
ax XII -
xiso,II
W. Linert, V. N. Sapunov / The isokinetic relationship. IX
270
common Hammett plot 7’,, is 298 K with a temperature-dependent Hammett slope or. The temperature-dependent slopes CY~,,~,,can be calculated from CY, (resulting from a plot with both series taken at the same temperature) and using the isokinetic temperature of the reference reaction series. It is therefore possible to predict the temperature dependence of the test reaction series. The isokinetic reactant of the test reaction series according to .$‘t,,, is defined by a ln(K,,,/K,,,)/~x From
this definition
I S=E,M.H = 0.
(29)
- xiso,,,)
AH, = AHu(l
- I/‘Siso,H).
Tii,, may then be calculated
AGi,, = T,,,AS,
(30)
(33) from
+ AH,)
which can be used to calculate ing to AS, = 6A HJT,,,
we get the condition
ln( k .T,,,..,,/ku.,,),,, = ‘AH,,(x,,
knowing AH, and AS, for the unsubstituted reactant in a Hammett series). In this way the enthalpies of the test-reaction series may be calculated using eq. (17)
3.3. Experimental
(34) AS, values, accord-
+ AS,, .
(35)
examples
which equals, with eq. (25) ln(k
.L.II&I),,,
= (‘AffI/a,)(xII
-xiso,~I).
(31) The enthalpy A Hiso,, must equal zero for the test reaction with the reactant according to ,$‘t,,, with leads to ln(k r,,,,,,,/k”.,,),,,
= (AHu.,/a~)(x,,
- xiso,,,). (32)
In this equation the ratio ln(ks!,,,,/ku,,,)x,, gives the abscissa of the common point of intersection in a T-dependent LFER with a reference reaction series taken at a constant temperature T,, = l/Rx,,. It is seen that an isoparametric relationship (for example the iso-Hammett relation) only exists when the reference-reaction series is itself temperature dependent and when an isokinetic temperature Tiso,,, = l/RXiso,,, exists. In this case eq. (32) gives, if this condition is fulfilled, a simple prediction of the isokinetic parameters ,$,,O,H and Inserting this using eq. (17) 5is0,s is obtained. result (in form of ln( k,/k,) in eq. (27) gives the ordinate of the common point of intersection, i.e. of the In Kiso,l. In this way, the T dependence whole test-reaction series is described by knowing the temperature dependence of the reference-reaction series, the LFER slope at one temperature and the temperature dependence of one member of the test-reaction series (for example when
For the Hammett relations it has been in parts III and IV of this publication [13,14] that
shown series
A H,,/263Oa,
(36)
p. = A Hi/( AH,” - A Hs’lis,,)
(37)
=
and
is valid, when Tiso ,, = - 255 K and T,, = 298 K is used for the ionization of the benzoic acids and assuming T, = T,,. AH,’ is the enthalpy value of the test reaction with unsubstituted reactants; AH,” and AH,” are the ionization enthalpies of the parent benzoic acid and that bearing the isokinetic substituent, respectively. This is consistent with eq. (25). In this paper 16 different. Hammett reaction series were examined for their IKR behavior. Those reaction series showing statistical significant common points of intersection (i.e. excluding isoenthalpic series without a defined point of intersection) are used to compare experimental and calculated u,,, H values. The ordinate values in fig. 4 are calculated from eq. (36) with p. values taken from eq. (28) to allow for different mean experimental temperatures for test and reference reaction series. From the point of view that no information about the T dependence of the test-reaction series is used to calculate the common points of intersection, the agreement which is
H
I N
Nl-+-C~ \N II
I
&--C----N Scheme 1.
-10
Fig, 4, ExperimentaX versus c&Aated
biw,H vafues of Hammett reaction series (for details see refs. [1J,14]).
fsund seems to be rather impressive
Further the IKR analyses have shuwn the equality of the intersection ordinates in the Hammett plots and the correspcCGng van ‘t Hoff or Arrhenius plots in agreement with eq. (3).
As another exam@ fop a LFER the generalized 33roensted relation using the isomerization of Saminotetrazoles according to the reaction shown in scheme f [l&19]. This reaction has been chosen because fur both LFER axes t~~~peraturedependent equilibrium constants and rate constants are available from the @xperimentaf data. The Arrhenius and the van ‘t Hoff plots are depicted in ref. [Xl] for the rate constants of the forward reactions and the e~~~~~b~~rnconstants. Both show highly significant PKR (IER) with cumm3on points of intersection at TiSo:lso.f = 765.3 K for the forward reactions and T,So,ec, = 662.3 K fur
0.6
04
v;”
0.2
Fig. 5, Broensted plot for the isomerkation of substituted S-aminotetrmoles [27,28j using temperature-dependent abscissa vah~s~ The curv4 line referes to the residual sum of square deviations of the measured points from constrained (i.e. assuming a common point of intersection) straighr lines; see r&s. f16,17].
W. Linert, V.N. Sapunov / The isokinetic relationship.
272
IX
0
-10 -5
0
15
5
20
In K(T=423Kl Fig. 6. Broensted plot for the isomerization of substituted 5-aminotetrazoles [26,27] using abscissa values taken at a constant temperature. The curved line refers to the residual sum of square deviations of the measured points from constrained (i.e. assuming a common point of intersection) straight lines; see refs. [16,17].
the equilibria. F values obtained from the statistical analysis [17] are 0.19 (fi = 3, f2 = 5) and 0.61 ( fi = 3, f2 = 8). Plotting In kf versus In K straight lines are found in agreement with a generalized Broensted relation. Fig. 4 shows the plot of this relation with both temperature-dependent values on coordinates (curved lines rather to residual sum of squares associated with a common point of intersection). In agreement with the above theory (see eqs. (25) and (26)) these plots show merely parallel lines. In fig. 5 the same data are plotted using the equilibrium constants (used as a reference series) at a constant temperature. A common point of intersection is found at an ordinate value of In kiso,f = 8.5 f 1 and abscissa value of In Kiso,r = 16.0 f 1. The corresponding F value of 0.063 ( fi = 3, f2 = 8) shows that the isoparameter relation has to be accepted at a high significance level. Using eq. (39) (again without using any information about the T dependence of the rate constants treated as the test-reaction series) an isoparametric value of 17.3 is calulated for In Kiso,r in agreement with the above experimental value. The experimental examples given above show
that it is possible to predict the common point of intersection exhibited by reaction series plotted for different temperatures in the LFER plane. To calculate this point (and with this the temperature dependence of the whole reaction series) we need the LFER at one temperature, the temperature dependence (IKR or IER) of the reference-reaction series and the temperature dependence of one member of the considered test-reaction series.
4. Summary Fig. 7 shows the various possibilities to represent temperature-dependent LFER data (in the In K scale) as described in this paper. As the treatment of common points of intersection is used (it originally arises from the statistical analysis of the problems) one might speak about “invariant points” in chemical reaction series or about “isoparametric” relationships. Obviously the same is valid for AG plots. In the previous papers [2,9,17,20-221 a theoretical explanation of the isokinetic relationship has been given. As a main
W’. Liner?, V.N. fapunov
273
/ Tke isokinetic relationship. IX
Ln K, (T)
LFER IT,, Tn)
tn K,
in K, iT)
(T= constl
LFER (T;=const, Tn)
LFER (T,, T,=constl
1
lKRI
T
IKR,
I
Fig. 7. Various graphical representations of the temperature dependence of LFER plots.
conclusion, the isokinetic temperature can be interpreted as an active vibrational frequency of the surroundings of the reaction site. The frequencies fv) co~espon~ng to ~iso (Xiso - l/‘tiv) are found in the vibrational spectra of the solvent in case of liquid phase reactions [20-221 and of the catalyst in the case of heterogeneous reactions [23]. From this point of view the crux of the present paper is that different isoparametric relations can be deduced from the isokinetic relationship. The present paper also enables the prediction of the temperature dependence of chemical reaction series and the characterization of a manifold of experimental data using q:,,, and %iso. Acknowledgement to
Thanks are due to Professor V. Gutmarm and the “ Fonds zur Forde~ng der wissen-
schaftlichen Forschung project 5443.
in fisterreich”
under the
References [l] J.E. Leffler and E. Grunwald, Rates and equilibria of organic reactions (Wiley, New York, 1963). [2] W.C. Conner, J. Catal. ‘78 (1982) 238. [3] J.N. Broensted, Chem. Rev. 5 (1929) 231. ]4] R.D. Levine, J. Phys. Chem. 83 (1979) 159. [5] L.P. Hammett, Physikahsche organ&he Chemie (Vet-lag Chemie, Weinheim, 1973). [6] V. Gutmann, The donor acceptor approach to molecdar interactions (Plenum Press, New York, 1978). (73 C. Reichardt, Solvent effects in organic chemistry (Verlag Chemie, Weinheim, 1979). [8] R.P. Wells, Linear free energy relationships (Academic Press, New York, 1968). [9] 0. Exner, Collection Czech. Chem. Commun. 40 (1975) 2742.
274
W. Linert, KN. Sapunoo / The isokinetic relationship.
[lo] 0. Exner and V. Beranek, Collection Czech. Chem. Commun. 38 (1973) 781; 0. Exner, Progr. Phys. Org. Chem. 10 (1973) 411. [ll] W. Liner& A.B. Kudjawtsev and R. S&mid, Australian J. Chem. 36 (1983) 1903. [12] W. Linert and A.B. Kudjawtsev, Australian J. Chem. 37 (1984) 1139. [13] W. Liner& R. S&mid and A.B. Kudjawtsev, Australian J. Chem. 38 (1985) 677. [14] W. Linert, Australian J. Chem. 39 (1986) 199. [15] C.G. Swain, MS. Swain, A.L. Powell and S. Altmni, J. Am. Chem. Sot. 105 (1983) 502.
IX
[16] W. Linert, R.W. Soukup and R. S&mid, Computer Chem. 6 (1982) 47. [17] W. Linert, Inorg. Chim. Acta 141 (1988), to be published. [18] R.A. Henry, W.G. Finnegan and E. Lieher, J. Am. Chem. Sot. 77 (1955) 2264. 1191 E. Lieber, C.N. Romachandra and T.S. Cao, J. Am. Chem. Sot. 79 (1957) 5962. [20] W. Liner& Chem. Phys. 114 (1987) 457. [21) W. Linert, Chem. Phys. 114 (1987) 449. [22] W. Linert, Chem. Phys. 116 (1987) 381. [23] W.C. Conner and W. Linert, to be published.