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The isokinetic relationship. VI. Equilibrium systems
Chemical Physics 114 (1987) 457-462 North-Holland, Amsterdam
THE ISOKINETIC
RELATIONSHIP.
VI. EQUILIBRIUM
SYSTEMS
Wolfgang LINERT Institute of Inorganic Chemistry, Technical University of Vienna, Getreidemarkt 9, A-1060 Vienna, Austria
Received 19 January 1987
The isoequilibrium relationship (IER), i.e. the occurrence of a common point of intersection of the van ‘t Hoff plots of a homologous reaction series, is investigated on the basis of a master equation theory describing forward and reverse reaction rates. The relation between the isoequilibrium temperature and the isokinetic temperatures of forward and reverse reactions is deduced theoretically and compared with experimental results.
1. Introduction
As pointed out in earlier papers [l-3], both the isokinetic (IKR) and the isoequilibrium (IER) relationships (i.e. the appearance of a common point of intersection in Arrhenius or van ‘t Hoff plots of a reaction series) are mathematically [4] (but not statistically [1,5]) equivalent to the so-called “compensation effect” (i.e. a proportionality between enthalpy and entropy values of members of a reaction series where catalysts, substituents, solvents, etc. are varied). The fact that the compensation effect may occur as an artefact due to errors based on the mutual dependence of the named parameters has been overcome on the basis of proper statistical analyses ‘[1,5-81. We define the IKR for forward (f) and reverse (r) reaction series
by
and
where kt,, are the rate constants and < is a parameter (or set of parameters) defining the members of the series. At a certain temperature TiI,,, (or rather its reciprocal xisO= l/RTi:,,,) the selectivity in the rate constant has vanished. In refs. [2,3,9], a model has been introduced describing the variation of the Arrhenius parame-
ters when changing the reactants within a homologous series of reactions. The result of this investigation was to show that the isokinetic relationship (IKR) and its characteristic parameter (i.e. the isokinetic temperature) may be determined by reactant-heat-bath interactions. The model has been based on a master equation theory [lo-131 describing the reaction rate via a stationary state of the particle flow from an (arbitrary) particle source (i.e. a reactant bath) over an energy barrier (taken as point of no return). It has been shown that resonant vibration-vibration interactions between heat bath and reactants are of crucial importance for the description of the physical background for isokinetic temperatures near or within the experimental temperature region. The corresponding vibrations could be found in infrared spectra for unimolecular [2,3], bimolecular [9] and for heterogeneous (electrochemical) reactions [14]. Furthermore, it should be pointed out that the theory is consistent with a general explanation for the compensation effect given by Conner [15], and Conner and Schwarz [16] which has been based on the availability and accessibility of energy. It appeared logical to generalize the model by describing the isoequilibrium relationship (IER) on the basis of K(5) = k,@‘)/k,@).
(3)
When investigating the same reaction series for the forward and reverse reaction, the parameter <
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
W. Linert / The isokinetic relationship. VI
458
is valid for both and therefore also for the IER. Whatever this parameter is (it may be identified with Hammett’s substituent parameters, a solvent parameter, or, on a physical basis, with degrees of freedom, vibration frequencies, energy barrier heights, etc.), there should appear (supposing the IKR appears in both the forward and the reverse reaction) an IER defined by 3 ln K(5)/G
I xpp,= 0,
(4
where K is the equilibrium constant. 2. General consideration of the isoequilibrium relationship The Arrhenius law for the forward (f) and the reverse (r) reaction is given by expkE,,r(S)/~~l
(5)
k,(5) = A,(4) exp[ -4&YRTl.
(6)
k&1
=4(E)
and
For the equilibrium constant the analogous van ‘ t Hoff expression is K( 0 = exp[ A$‘( $)/II - A@‘(
[)/RT] .
(7)
Provided that both forward and reverse reaction show isokinetic behaviour these equations can be rewritten as (using x = l/RT and Xiso= l/RTi~,) k,(5)
=A: exp[-E&)(x
- &JI
(8)
= 4 exp[ -4&)(x
- &]I.
(9)
and k(E)
The isoequilibrium relationship is given by K(t)
= exp[ -AH’(O(
x - xZ)l.
00)
Using K= k,/k,, a comparison of temperaturedependent and temperature-independent terms leads to
A more detailed investigation of the relation between isoequilibrium and isokinetic temperatures needs a more developed theory of the IKR which is outlined in the next section. 3.Thetnodel 3.1. Master equation chemical reaction
theory for
steady
state
For each direction of the reaction (f and r are not given explicitly in this section) the reactant distribution to quantized eigenstates (denoted by index I) connected to a heat bath with the timedependent densities u,(t) can be described by a master equation theory [2,3,9-131. In the case that the energy level spacings are much smaller and the heigth of the energy barrier is larger than the corresponding thermal energy (i.e. kT) and furthermore using a continuous probability function for the transition probabilities between the energy levels, this theory yields k, = R(sN)( hw/kT)2
exp( -Aos,/kT),
(13)
for the first-order rate constant k, and k, = R(s,~)R(~~~)E~tzw,x~Aw,x~E, xexp(-2Ex)[g(+_,)
exp(E,)]-‘,
(14)
for the second-order rate constant k, [3,9]. Here E = s,ho is the energy barrier between collision complex and products, w the energy spacing of levels in the reactant potential wells, E, the barrier between reactants A and B and the collision complex and R(sN) the temperature-independent transition probability at the highest level in the potential well denoted by sN. g(l) represents a source term describing the energy distribution or reactants when forming the collision complex. The functional form of the transition probabilities R(l) is approximated [12,17-261 by P(l) = R,l
(15)
for transitions induced by gas phase collisions. In condensed phase reactions (or in gas phase reactions involving large molecules) vibrationally induced translations are dominant. For this case P( 1) = Rbl exp( lo/v)
(16)
W. Linert / The isokinetic relationship.
is used [9], where v is an active heat-bath frequency. With this the first-order rate constant takes the form k, = R,s,(
~ZWX)~exp[ E(l/ttv
- x)]
(17)
and the second-order rate constant
dlnK],Z=((alnK/aE,)dE, +(a In K/aE,)
X exp[ E(l/AV* + l/i?v,
- 2x) - E,,x]
X(g%J’.
d In K 1xpp= (3 In K/aE,)
(22)
dE,
- (a In K/aE,)
Applying the definition of the IKR (eq. (l), identified with the energy barrier E = s,hw) we get for first-order reactions
dE,
+(a ln JWE,,,,)
dE,,,,,
(23)
and for case (iii)
= Xiso= 1/E + ~/AU.
09)
Using the IKR definition in the form (for more than one parameter)
d In K ) xc = (a In K/&F,) d E,
dE
+(a In k/aE,)
dE,,
for case (ii) (18)
d ln k I x,= = (a In k/llE)
459
say the forward reaction) is monomolecular, the other bimolecular. To evaluate the xiso value for the equilibrium constants the relations should be rewritten for case (i)
k, = R;EE,,hq,x2Ao,x2
1/R~,
VI
dE,,=O,
ln K/aEr,,)
-(a
In K,GiE,) dE,
+ (a ln WE,,,)
(20)
dE,,,
dE,,,.
(24)
Neglecting the contribution of E,, in cases (ii) and (iii) for condensed phase reactions, we get for all three cases
one gets for the second-order reaction l/‘R7’i,, = Xiso dE,,l 1 dE E,,+j;;;;;+fiv,
-0
1
(xi’, - x;:)
dE, - (xi’s0- x2)
dE, = 0
(25)
or X(2-
S)-l.
(21)
Eqs. (19) and (21) have been derived and applied in part V [9] and part VII [3] of this series for the case of bimolecular and unimolecular experimental examples, respectively. It has been shown that for reaction series performed in condensed phase, E- and E,,-dependent terms may be neglected as the vibrational reactant-heat-bath interactions are dominant. According to this the isokinetic temperature is found to correspond to vibrational frequencies of the reactant surrounding. 3.2. Application of the theory to the IER Three cases may occur, namely (i) both forward and reverse reactions are monomolecular reactions (ii) both reactions are bimolecular and (iii) one of the reactions (without loss of generality we may
X9
1scl=
40 -
xif,, d EJd E, 1 - dE,/dE, ’
(26)
Finally, when reactants and products are similar (e.g. an isomerization reaction), dEJdE, may be taken as approximately equal to - 1, giving the isokinetic temperature Ti:,, as l/RTSO = x;z = (xif,, + x:&2.
(27)
3.3. Test reaction series The isomerization of substituted 5-amino1,2,34etrazoles [20,21] (see scheme 1) gives an ideal example for the investigation of connections between IKR and IER because temperature-dependent data for both kinetic and equilibrium are available for a series of substituents. Forward and reverse reactions as well as the equilibrium data
W. Linert / The isokinetic relationship. VI
460
show both IKR and IER with well defined Xiso values. Figs. 1 and 2 show the Arrhenius plots of the forward and the reverse reactions and fig. 3 gives the van ‘t Hoff plots of the equilibrium constants. The results of the statistical analyses of the data according to the method described in refs. [1,5,7] are summarized in table 1. From the small F values it can be seen that all of them show highly significant [l] IKR or IER (at a significance level of (Y= 0.01 the F table values, which must be larger than the values from experiment to accept the IKR or IER, are about 15). The reaction with the phenyl substituent had, however, to be excluded because in all three diagrams this reaction is not consistent with the behaviour of the other members of the series. This would suggest that a different reaction mechanism is involved for this substituent. Using the approximation of eq. (27), xfz is calculated from & and xir, as 1.94 X lob4 mol J-i. To evaluate the value of d E f/d E’ the activation energies of the forward and the reverse reactions are ploted in fig. 4 yielding a value of - 1.28.
N’N\C-NHR II NScheme
II C-C6H5
1.
Inserting this into eq. (26) yields a value of 1.89 x 10e4 mol J-l for xl’,“,.Both values may be compared with the experimental value of 1.82 X lop4 mol JJ’. From eq. (19) the active heat-bath frequencies neglecting the term l/E are calculated. Corresponding to _-&, and xiSOheat-bath frequencies of 509.7 and 374.8 cm-’ are found, respectively. It is pleasing to note that the IR spectra of ethyleneglycol which has been used as the solvent for these reactions, show, within this region, only two bands [22] at 510 and 360 cm-‘. Summarizing the results of the previous parts of this series and of the present paper it is shown that both the IKR and the IER are real phenom-
2(
0.3
IO
0.2
P v;
E
0
01
-10
0.5
I
1.0
I
I
1.5
2.0
0 2.5
1000 KIT Fig. 1. Arrhenius plots of the forward rate constants of the isomerization of substituted 5-aminotetrazoles. Curved lines correspond to the sum of squares of deviations s, of the constrained Arrhenius lines from experimental points. (Substituents: (1) 3-Cl, (2) 4-CH,; (3) 4-CH,O; (4) C,H,-CH,.)
W. Linert / The isokinetic relationship.
VI
461 0.3
0.2
v;
0.1
-5
-15' 0.5
I
I
10
1.5
0
2 .o
2.5
1000 K/T Fig. 2. Arrhenius plots of the reverse reaction of the isomerization of substituted 5-aminotetrazoles. sum of squares of deviations S, of the constrained Arrhenius lines from experimental points.
Curved lines correspond (Substituents see fig. 1.)
to the
0.3
0.2
x
l/Y
5
0.1
-5
05
1.0
0
1.5
2.0
2.5
1000K/T Fig. 3. Van ‘t Hoff plots for the isomerization deviations s, of the constrained
of substituted 5-aminotetrazoles. Curved lines correspond to the sum of squares van ‘ t Hoff lines from experimental points (substituents; see fig. 1.)
of
W. Linert / The isokinetic relationship. VI
462 Table 1 Results of the statistical
IKR and IER analyses
of 5-aminotetrazoles 8)
10-4xiso (J mol-‘)
Yiso
;: 135.29 540.5 662.3
1.64 2.23 1.82
11.53 4.019 0.219
Reaction forward reverse equilibrium
of the isomerization
a) Natural logarithms of the isokinetic constant (dimensionless). b, For details of the statistical analyses
rate constants
(in min -‘)
for forward
[20,21]
F b,
fi b,
/2 b,
0.19 0.45 0.61
3 4 3
5 5 8
and reverse
reaction
and and of the isoequilibrium
see refs. [1,5].
under the project 5443 is gratefully acknowledged. Thanks are due to Professor V. Gutmann for many useful discussions. References
125
140
155
E,,/kJ.mol-’ Fig. 4. Relationship between the activation energies of forward reactions and reverse isomerization reactions of substituted 5-aminotetrazoles.
ena which physical background may be assigned to the interactions between reactants and their surroundings. For reaction series with reciprocal isokinetic temperatures significantly different from zero (i.e. series which are not isoentropic) vibrational interactions seem to dominate and the isokinetic temperature corresponds to vibrational frequencies which may be identified from the absorption bands of far infrared spectra of the reaction medium. Acknowledgement
Financial support by the Fonds zur FGrderung der Wissenschaftlichen Forschung in osterreich
[l] W. Liner& R.W. Soukup and R. Schmid, Computer Chem. 6 (1982) 47. [2] W. Linert and A.B. Kudjowtsev, Australian J. Chem. 37 (1984) 1134. [3] W. Linert, to be published. (41 J.E. Leffler and E. Grunwald, Rates and equilibria of organic reactions (Wiley, New York, 1963). [51 0. Exner, Collection &.ech. Chem. Commun. 40 (1975) 2762. [61 R.R. Krug, Indian Eng. Chem. Fund. 19 (1980) 50. [71 0. Exner, Progr. Phys. Org. Chem. 10 (1973) 411. PI W. Linert, Australian J. Chem. 39 (1986) 199. [91 W. Linert, Chem. Phys. 114 (1987) 449. 1101 B.J. McCoy and R.G. Carbonell, J. Chem. Phys. 66 (1977) 4564. WI R.J. Rubin and K.E. Shuler, Physik. Z. Sovietunion 10 (1936) 34. P21 K.N.C. Bray, J. Phys. B 1 (1968) 705; 3 (1970) 1515. P31 B.J. McCoy and R.G. Carbonell, Chem. Phys. 20 (1977) 1591. P41 W. Linert and J. Jaworski, to be published. 1151 W.C. Conner, J. CataI. 78 (1982) 238. WI W.C. Conner and J. Schwarz, J. Chem. Eng. Commun., to be published. P71 L. Landau and E. Teller, Physik. Z. Sovietunion 10 (1936) 34. WI R.N. Schwarz, Z.I. Slawsky and K.F. Herzfels, J. Chem. Phys. 20 (1952) 1591. J. Chem. Phys. 20 1191 D. Rapp and P. Englander-Golden, (1952) 1591. [201 I.L. Thomas, Chem. Phys. Letters 70 (1980) 413. and T.S. Cao, J. Am. 1211 E. Lieber, C.N. Romachandra Chem. Sot. 79 (1957) 5962. 1221 The Sadtler Standard Spectra, Sadtler Research Laboratories, New York (1976).
THE ISOKINETIC
RELATIONSHIP.
VI. EQUILIBRIUM
SYSTEMS
Wolfgang LINERT Institute of Inorganic Chemistry, Technical University of Vienna, Getreidemarkt 9, A-1060 Vienna, Austria
Received 19 January 1987
The isoequilibrium relationship (IER), i.e. the occurrence of a common point of intersection of the van ‘t Hoff plots of a homologous reaction series, is investigated on the basis of a master equation theory describing forward and reverse reaction rates. The relation between the isoequilibrium temperature and the isokinetic temperatures of forward and reverse reactions is deduced theoretically and compared with experimental results.
1. Introduction
As pointed out in earlier papers [l-3], both the isokinetic (IKR) and the isoequilibrium (IER) relationships (i.e. the appearance of a common point of intersection in Arrhenius or van ‘t Hoff plots of a reaction series) are mathematically [4] (but not statistically [1,5]) equivalent to the so-called “compensation effect” (i.e. a proportionality between enthalpy and entropy values of members of a reaction series where catalysts, substituents, solvents, etc. are varied). The fact that the compensation effect may occur as an artefact due to errors based on the mutual dependence of the named parameters has been overcome on the basis of proper statistical analyses ‘[1,5-81. We define the IKR for forward (f) and reverse (r) reaction series
by
and
where kt,, are the rate constants and < is a parameter (or set of parameters) defining the members of the series. At a certain temperature TiI,,, (or rather its reciprocal xisO= l/RTi:,,,) the selectivity in the rate constant has vanished. In refs. [2,3,9], a model has been introduced describing the variation of the Arrhenius parame-
ters when changing the reactants within a homologous series of reactions. The result of this investigation was to show that the isokinetic relationship (IKR) and its characteristic parameter (i.e. the isokinetic temperature) may be determined by reactant-heat-bath interactions. The model has been based on a master equation theory [lo-131 describing the reaction rate via a stationary state of the particle flow from an (arbitrary) particle source (i.e. a reactant bath) over an energy barrier (taken as point of no return). It has been shown that resonant vibration-vibration interactions between heat bath and reactants are of crucial importance for the description of the physical background for isokinetic temperatures near or within the experimental temperature region. The corresponding vibrations could be found in infrared spectra for unimolecular [2,3], bimolecular [9] and for heterogeneous (electrochemical) reactions [14]. Furthermore, it should be pointed out that the theory is consistent with a general explanation for the compensation effect given by Conner [15], and Conner and Schwarz [16] which has been based on the availability and accessibility of energy. It appeared logical to generalize the model by describing the isoequilibrium relationship (IER) on the basis of K(5) = k,@‘)/k,@).
(3)
When investigating the same reaction series for the forward and reverse reaction, the parameter <
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
W. Linert / The isokinetic relationship. VI
458
is valid for both and therefore also for the IER. Whatever this parameter is (it may be identified with Hammett’s substituent parameters, a solvent parameter, or, on a physical basis, with degrees of freedom, vibration frequencies, energy barrier heights, etc.), there should appear (supposing the IKR appears in both the forward and the reverse reaction) an IER defined by 3 ln K(5)/G
I xpp,= 0,
(4
where K is the equilibrium constant. 2. General consideration of the isoequilibrium relationship The Arrhenius law for the forward (f) and the reverse (r) reaction is given by expkE,,r(S)/~~l
(5)
k,(5) = A,(4) exp[ -4&YRTl.
(6)
k&1
=4(E)
and
For the equilibrium constant the analogous van ‘ t Hoff expression is K( 0 = exp[ A$‘( $)/II - A@‘(
[)/RT] .
(7)
Provided that both forward and reverse reaction show isokinetic behaviour these equations can be rewritten as (using x = l/RT and Xiso= l/RTi~,) k,(5)
=A: exp[-E&)(x
- &JI
(8)
= 4 exp[ -4&)(x
- &]I.
(9)
and k(E)
The isoequilibrium relationship is given by K(t)
= exp[ -AH’(O(
x - xZ)l.
00)
Using K= k,/k,, a comparison of temperaturedependent and temperature-independent terms leads to
A more detailed investigation of the relation between isoequilibrium and isokinetic temperatures needs a more developed theory of the IKR which is outlined in the next section. 3.Thetnodel 3.1. Master equation chemical reaction
theory for
steady
state
For each direction of the reaction (f and r are not given explicitly in this section) the reactant distribution to quantized eigenstates (denoted by index I) connected to a heat bath with the timedependent densities u,(t) can be described by a master equation theory [2,3,9-131. In the case that the energy level spacings are much smaller and the heigth of the energy barrier is larger than the corresponding thermal energy (i.e. kT) and furthermore using a continuous probability function for the transition probabilities between the energy levels, this theory yields k, = R(sN)( hw/kT)2
exp( -Aos,/kT),
(13)
for the first-order rate constant k, and k, = R(s,~)R(~~~)E~tzw,x~Aw,x~E, xexp(-2Ex)[g(+_,)
exp(E,)]-‘,
(14)
for the second-order rate constant k, [3,9]. Here E = s,ho is the energy barrier between collision complex and products, w the energy spacing of levels in the reactant potential wells, E, the barrier between reactants A and B and the collision complex and R(sN) the temperature-independent transition probability at the highest level in the potential well denoted by sN. g(l) represents a source term describing the energy distribution or reactants when forming the collision complex. The functional form of the transition probabilities R(l) is approximated [12,17-261 by P(l) = R,l
(15)
for transitions induced by gas phase collisions. In condensed phase reactions (or in gas phase reactions involving large molecules) vibrationally induced translations are dominant. For this case P( 1) = Rbl exp( lo/v)
(16)
W. Linert / The isokinetic relationship.
is used [9], where v is an active heat-bath frequency. With this the first-order rate constant takes the form k, = R,s,(
~ZWX)~exp[ E(l/ttv
- x)]
(17)
and the second-order rate constant
dlnK],Z=((alnK/aE,)dE, +(a In K/aE,)
X exp[ E(l/AV* + l/i?v,
- 2x) - E,,x]
X(g%J’.
d In K 1xpp= (3 In K/aE,)
(22)
dE,
- (a In K/aE,)
Applying the definition of the IKR (eq. (l), identified with the energy barrier E = s,hw) we get for first-order reactions
dE,
+(a ln JWE,,,,)
dE,,,,,
(23)
and for case (iii)
= Xiso= 1/E + ~/AU.
09)
Using the IKR definition in the form (for more than one parameter)
d In K ) xc = (a In K/&F,) d E,
dE
+(a In k/aE,)
dE,,
for case (ii) (18)
d ln k I x,= = (a In k/llE)
459
say the forward reaction) is monomolecular, the other bimolecular. To evaluate the xiso value for the equilibrium constants the relations should be rewritten for case (i)
k, = R;EE,,hq,x2Ao,x2
1/R~,
VI
dE,,=O,
ln K/aEr,,)
-(a
In K,GiE,) dE,
+ (a ln WE,,,)
(20)
dE,,,
dE,,,.
(24)
Neglecting the contribution of E,, in cases (ii) and (iii) for condensed phase reactions, we get for all three cases
one gets for the second-order reaction l/‘R7’i,, = Xiso dE,,l 1 dE E,,+j;;;;;+fiv,
-0
1
(xi’, - x;:)
dE, - (xi’s0- x2)
dE, = 0
(25)
or X(2-
S)-l.
(21)
Eqs. (19) and (21) have been derived and applied in part V [9] and part VII [3] of this series for the case of bimolecular and unimolecular experimental examples, respectively. It has been shown that for reaction series performed in condensed phase, E- and E,,-dependent terms may be neglected as the vibrational reactant-heat-bath interactions are dominant. According to this the isokinetic temperature is found to correspond to vibrational frequencies of the reactant surrounding. 3.2. Application of the theory to the IER Three cases may occur, namely (i) both forward and reverse reactions are monomolecular reactions (ii) both reactions are bimolecular and (iii) one of the reactions (without loss of generality we may
X9
1scl=
40 -
xif,, d EJd E, 1 - dE,/dE, ’
(26)
Finally, when reactants and products are similar (e.g. an isomerization reaction), dEJdE, may be taken as approximately equal to - 1, giving the isokinetic temperature Ti:,, as l/RTSO = x;z = (xif,, + x:&2.
(27)
3.3. Test reaction series The isomerization of substituted 5-amino1,2,34etrazoles [20,21] (see scheme 1) gives an ideal example for the investigation of connections between IKR and IER because temperature-dependent data for both kinetic and equilibrium are available for a series of substituents. Forward and reverse reactions as well as the equilibrium data
W. Linert / The isokinetic relationship. VI
460
show both IKR and IER with well defined Xiso values. Figs. 1 and 2 show the Arrhenius plots of the forward and the reverse reactions and fig. 3 gives the van ‘t Hoff plots of the equilibrium constants. The results of the statistical analyses of the data according to the method described in refs. [1,5,7] are summarized in table 1. From the small F values it can be seen that all of them show highly significant [l] IKR or IER (at a significance level of (Y= 0.01 the F table values, which must be larger than the values from experiment to accept the IKR or IER, are about 15). The reaction with the phenyl substituent had, however, to be excluded because in all three diagrams this reaction is not consistent with the behaviour of the other members of the series. This would suggest that a different reaction mechanism is involved for this substituent. Using the approximation of eq. (27), xfz is calculated from & and xir, as 1.94 X lob4 mol J-i. To evaluate the value of d E f/d E’ the activation energies of the forward and the reverse reactions are ploted in fig. 4 yielding a value of - 1.28.
N’N\C-NHR II NScheme
II C-C6H5
1.
Inserting this into eq. (26) yields a value of 1.89 x 10e4 mol J-l for xl’,“,.Both values may be compared with the experimental value of 1.82 X lop4 mol JJ’. From eq. (19) the active heat-bath frequencies neglecting the term l/E are calculated. Corresponding to _-&, and xiSOheat-bath frequencies of 509.7 and 374.8 cm-’ are found, respectively. It is pleasing to note that the IR spectra of ethyleneglycol which has been used as the solvent for these reactions, show, within this region, only two bands [22] at 510 and 360 cm-‘. Summarizing the results of the previous parts of this series and of the present paper it is shown that both the IKR and the IER are real phenom-
2(
0.3
IO
0.2
P v;
E
0
01
-10
0.5
I
1.0
I
I
1.5
2.0
0 2.5
1000 KIT Fig. 1. Arrhenius plots of the forward rate constants of the isomerization of substituted 5-aminotetrazoles. Curved lines correspond to the sum of squares of deviations s, of the constrained Arrhenius lines from experimental points. (Substituents: (1) 3-Cl, (2) 4-CH,; (3) 4-CH,O; (4) C,H,-CH,.)
W. Linert / The isokinetic relationship.
VI
461 0.3
0.2
v;
0.1
-5
-15' 0.5
I
I
10
1.5
0
2 .o
2.5
1000 K/T Fig. 2. Arrhenius plots of the reverse reaction of the isomerization of substituted 5-aminotetrazoles. sum of squares of deviations S, of the constrained Arrhenius lines from experimental points.
Curved lines correspond (Substituents see fig. 1.)
to the
0.3
0.2
x
l/Y
5
0.1
-5
05
1.0
0
1.5
2.0
2.5
1000K/T Fig. 3. Van ‘t Hoff plots for the isomerization deviations s, of the constrained
of substituted 5-aminotetrazoles. Curved lines correspond to the sum of squares van ‘ t Hoff lines from experimental points (substituents; see fig. 1.)
of
W. Linert / The isokinetic relationship. VI
462 Table 1 Results of the statistical
IKR and IER analyses
of 5-aminotetrazoles 8)
10-4xiso (J mol-‘)
Yiso
;: 135.29 540.5 662.3
1.64 2.23 1.82
11.53 4.019 0.219
Reaction forward reverse equilibrium
of the isomerization
a) Natural logarithms of the isokinetic constant (dimensionless). b, For details of the statistical analyses
rate constants
(in min -‘)
for forward
[20,21]
F b,
fi b,
/2 b,
0.19 0.45 0.61
3 4 3
5 5 8
and reverse
reaction
and and of the isoequilibrium
see refs. [1,5].
under the project 5443 is gratefully acknowledged. Thanks are due to Professor V. Gutmann for many useful discussions. References
125
140
155
E,,/kJ.mol-’ Fig. 4. Relationship between the activation energies of forward reactions and reverse isomerization reactions of substituted 5-aminotetrazoles.
ena which physical background may be assigned to the interactions between reactants and their surroundings. For reaction series with reciprocal isokinetic temperatures significantly different from zero (i.e. series which are not isoentropic) vibrational interactions seem to dominate and the isokinetic temperature corresponds to vibrational frequencies which may be identified from the absorption bands of far infrared spectra of the reaction medium. Acknowledgement
Financial support by the Fonds zur FGrderung der Wissenschaftlichen Forschung in osterreich
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