Temperature dependence of the rate and activation parameters for tert-butyl chloride solvolysis: Monte Carlo simulation of confidence intervals

Article abstract of DOI:10.1016/j.cplett.2004.05.093

The solvolysis rate constants (k_obs) of tert-butyl chloride are measured in 20%(v/v) 2-PrOH-H₂O mixture at 15 temperatures ranging from 0 to 39°C. Examination of the temperature dependence of the rate constants by the weighted least squares fitting to two to four terms equations has led to the three-term form, lnk_obs=a₁+a ₂T^-1+a₃lnT, as the best expression. The activation parameters, ΔH^? and ΔS ^?, calculated by using three constants a₁, a ₂ and a₃ revealed the steady decrease of ≈1 kJmol ^-1 per degree and 3.5 JK^-1mol^-1 per degree, respectively, as the temperature rises. The sign change of ΔS ^? at ≈20.0°C and the large negative heat capacity of activation, ΔC_p^?=-1020 JK ^-1mol^-1, derived are interpreted to indicate an S _N1 mechanism and a net change from water structure breaking to electrostrictive solvation due to the partially ionic transition state. Confidence intervals estimated by the Monte Carlo method are far more precise than those by the conventional method.

Full text of DOI:10.1016/j.cplett.2004.05.093

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Chemical Physics Letters 392 (2004) 378–382
www.elsevier.com/locate/cplett
Temperature dependence of the rate and activation parameters
for tert-butyl chloride solvolysis: Monte Carlo simulation
of confidence intervals
a,
b
c,
d
Dae Dong Sung *, Jong-Youl Kim , Ikchoon Lee *, Sung Sik Chung ,
e
Kwon Ha Park
a
Department of Chemistry, Dong-A University, Busan 604-714, Republic of Korea
b
Yangsan College, Yangsan, Kyungnam 626-740, Republic of Korea
Department of Chemistry, Inha University, Inchon 402-751, Republic of Korea
c
d
Department of Mechanical Engineering, Dong-A University, Busan 604-714, Republic of Korea
e
Division of Mechanical and Information Engineering, Korea Maritime University, Busan 606-791, Republic of Korea
Received 30 April 2004; in final form 21 May 2004
Available online 15 June 2004
Abstract
The solvolysis rate constants (kobs) of tert-butyl chloride are measured in 20% (v/v) 2-PrOH–H2O mixture at 15 temperatures
ranging from 0 to 39 °C. Examination of the temperature dependence of the rate constants by the weighted least squares fitting to
two to four terms equations has led to the three-term form, ln kobs ¼ a1 þ a2Tꢀ1 þ a3 ln T, as the best expression. The activation
parameters, DHz and DSz, calculated by using three constants a1, a2 and a3 revealed the steady decrease of ꢁ1 kJ molꢀ1 per degree
and 3.5 J Kꢀ1 molꢀ1 per degree, respectively, as the temperature rises. The sign change of DSz at ꢁ20.0 °C and the large negative heat
capacity of activation, DCpz ¼ ꢀ1020 J Kꢀ1 molꢀ1, derived are interpreted to indicate an SN1 mechanism and a net change from
water structure breaking to electrostrictive solvation due to the partially ionic transition state. Confidence intervals estimated by the
Monte Carlo method are far more precise than those by the conventional method.
Ó 2004 Elsevier B.V. All rights reserved.
DHz ¼ A þ DCpz:T;
DSz ¼ B þ DCpz: ln T;
where A and B are constants and DCpzð¼ CpTS ꢀ CpISÞ is
the heat capacity of activation [3]. Since for many liq-
uids, in particular for water, heat capacity (Cp) is nearly
independent of the temperature, the DCpz value for a
reaction in solution is practically constant.
Substitution of Eqs. (2) into (1) leads to a three-term
expression, (3), with correlations (4) between the
parameters.
ð2aÞ
ð2bÞ
1. Introduction
The influence of temperature on the rates (k) of
chemical reactions is complex. The absolute rate theory
expresses the temperature dependence of the rate in
terms of enthalpy and entropy of activation [1], (1),
where k is the rate constant and k is the Boltzmann
constant. However, the two activation parameters
themselves are also dependent on the temperature.
ln k ¼ lnðkT =hÞ ꢀ DHz=RT þ DSz=R:
For liquids, there are well-known empirical relations
[2,3], (2) which are applicable over quite a wide range of
temperature.
ð1Þ
ln k ¼ a1 þ a2Tꢀ1 þ a3 ln T;
A ¼ ꢀRa2;
ð3Þ
ð4aÞ
ð4bÞ
* Corresponding authors. Fax: +82-32-8654855 (Ikchoon Lee); +82-
51-2007259 (Dae Dong Sung).
E-mail address: ilee@inha.ac.kr (I. Lee).
B ¼ ½a1 þ a3 ꢀ 1 ꢀ lnðk=hÞꢂ;
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2004.05.093

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D.D. Sung et al. / Chemical Physics Letters 392 (2004) 378–382
379
DCpz ¼ ða3 ꢀ 1ÞR:
ð4cÞ
method [11]. The values listed in Table 1 are the aver-
ages of 5–6 runs. The standard deviations of the kobs
(secꢀ1) values are also shown in Table 1. Standard errors
of the kobs (secꢀ1) values are in most cases less than 1%
and rarely exceed 2%. Temperature was controlled to
better than ꢃ0.002 °C. A Beckmann precision ther-
mometer (with a resolution of 10ꢀ4 K) was used
throughout and the ice point of the thermometer was
checked at frequent intervals. As pointed out by Rob-
ertson [3] temperature stability of this order is essential
for rate determinations accurate to errors less than 1%
or better.
A simpler, but more widely used form of the tem-
perature dependence of the rate is the two-term form,
(5), from the Arrhenius equation [4]. On the other hand,
a more complex, four-term form (6), which is a type of
Clarke and Glew treatment [5], has been proposed.
ln k ¼ a01 þ a02T ꢀ1
;
ð5Þ
ð6Þ
ln k ¼ a100 þ a200Tꢀ1 þ a300 ln T þ a4T:
The solvolysis of t-butyl chloride in aqueous solution
is a model SN1 reaction and the rate is correlated with
the solvent ionizing power (Y ) by the well known
Grunwald–Winstein equation, (7), with m ¼ 1:0 [6].
2.3. Determination of parameters and error analysis
The parameters, ai, ai0 and a0i0 in Eqs. (3), (5) and (6),
were determined by least squares fitting of the respective
equation to the 15 (ki, Ti) rate data sets in Table 1 (un-
weighted parameter values). Since the experimental er-
rors (ri) of the measured rates (ki) are not uniform but
differ at different temperature (Ti), we applied a
weighting factor (Wi) of the form (8) [12,13], which si-
multaneously takes into account individual errors and
the effect of the mathematical transformation of the rate
ki to ln ki in the actual least squares fits.
logðk=k0Þ ¼ mY :
ð7Þ
The ionizing power of 2-PrOH (Y ¼ ꢀ2:73) [7] is
much lower than that of water (Y ¼ ꢀ1:09) so that
solvolysis of t-butyl chloride in 20%(v/v) 2-propanol–
H2O mixture should practically proceed by water whose
structure is affected by the presence of cosolvent,
20%(v/v)-2-PrOH.
In the present work, we examine temperature de-
pendence (i) of the rate using solvolysis data of t-butyl
chloride in 20%(v/v) 2-PrOH–water solution obtained at
15 temperatures ranging from 0 to 39 °C, and (ii) of the
activation parameters, DHz and DSz for this reaction.
These, together with the heat capacity of activation,
DCpz, determined by (2), form the basis of our discussion
on the mechanism of the reaction and structural changes
of water as the temperature is varied.
Wi ¼ ki2=ri2=ð1=15ÞR1=r2i :
ð8Þ
The ‘goodness of fit’ is considered in terms of the
overall standard deviation of the fit, rfit, the confidence
interval (CI) [14,15] which is defined for small sample
numbers n ( ¼ 15) as tri, where t is the Student’s t values
[14,15], and Fischer’s F -test [12,16]. Standard errors are
calculated by the diagonal elements of covariance ma-
trix, which is the inverse of the curvature matrix, in the
least squares method but their accuracies are limited in
the fitting of experimental data to nonlinear functions
[17,18]. The MC method is widely used to determine CIs
We have unequivocally shown that the temperature
dependence of the rate is represented best by the three-
term form, Eq. (3), through fitting to the rate data and
error analysis involving Monte Carlo (MC) simulation
[8] of confidence intervals.
Table 1
Rate constants (kobs) for the solvolysis of t-butyl chloride in 20% (v/v)
2-PrOH–water with standard deviations (rk)
2. Experimental
2.1. Materials
Temperature (°C)
k (sꢀ1
)
rk (sꢀ1
)
0
5
1.511 ꢄ 10ꢀ5
3.043 ꢄ 10ꢀ5
6.773 ꢄ 10ꢀ5
1.364 ꢄ 10ꢀ4
2.747 ꢄ 10ꢀ4
4.528 ꢄ 10ꢀ4
5.287 ꢄ 10ꢀ4
6.113 ꢄ 10ꢀ4
8.251 ꢄ 10ꢀ4
9.331 ꢄ 10ꢀ4
1.008 ꢄ 10ꢀ3
1.360 ꢄ 10ꢀ3
1.700 ꢄ 10ꢀ3
2.243 ꢄ 10ꢀ3
3.028 ꢄ 10ꢀ3
3.72 ꢄ 10ꢀ7
1.94 ꢄ 10ꢀ7
1.15 ꢄ 10ꢀ7
7.42 ꢄ 10ꢀ6
5.19 ꢄ 10ꢀ6
2.34 ꢄ 10ꢀ6
2.41 ꢄ 10ꢀ6
1.17 ꢄ 10ꢀ6
1.09 ꢄ 10ꢀ6
1.03 ꢄ 10ꢀ6
8.43 ꢄ 10ꢀ5
7.29 ꢄ 10ꢀ5
7.31 ꢄ 10ꢀ5
7.54 ꢄ 10ꢀ5
7.43 ꢄ 10ꢀ5
2-Propanol (2-PrOH) was purified according to
Maryott’s method [9].
10
15
20
24
25
26
29
30
31
34
35
36
39
The HPLC grade 2-PrOH was dried over magnesium
and fractionally distilled in a 180 cm Dufton column.
The commercial t-butyl chloride was used after purifi-
cation as described previously [10]. Water was distilled
three times and then passed through Millipore ultra
water pure columns.
2.2. Kinetic procedure
The rates were measured conductometrically and the
kobs (secꢀ1) values were determined by the Guggenheim

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380
D.D. Sung et al. / Chemical Physics Letters 392 (2004) 378–382
of estimated coefficients obtained by linear and nonlin-
ear regression [17–21].
usually the regression improves with an increase in the
adjustable variables. The parameter a01 and a20 are similar
in the two methods of fits, but a1; a2 and a3 in the three-
term form differ considerably between the two methods
of fits.
This discrepancy is caused mainly by insufficient
number of experimental data sets and/or partially by
inaccuracy of the standard deviation of the experimental
value. We conclude here that (i) a better fit and hence
the more reliable parameters are obtained by weighting
procedure, and (ii) the three term form, (2), represents
the best temperature dependence of the rates.
In Table 2, we compared the CIs calculated based on
the covariance matrix (conventional method [16]) with
those on the MC simulation [17–21]. For unweighted
parameters, the CIs by the MC method are 2–3 times
smaller than those by the conventional method, which is
in good agreement with the results in the literature
[12,18]. For the weighted fit parameters, the CI values by
the MC method becomes with 7–10 times smaller than
those by the conventional method.
In this work, MC method was used to determine the
CIs of derived parameters. A new data set (ki0; Ti) is
generated from the original one (ki; Ti) by using Eq. (9),
where ei is a normally distributed assembly of random
number (mean ¼ 0, variance ¼ 1) which can be generated
using Box–Muller method [22] or rejection technique
[8,16].
ki0 ¼ ki þ riei:
ð9Þ
Regression analysis were then performed over each
synthetic data set to obtain optimized parameters, ai, ai0
and a0i0. The CIs were determined using t and ri values.
The MC simulation was repeated until essentially con-
stant estimates of the parameters (CIs ) are obtained. In
the present work the MC simulations were repeated 400
times. We found that the variation of parameters ai with
respect to the number of iteration oscillates widely until
ꢁ400 iterations are reached.
The temperature dependence of activation enthalpy,
DHz, and entropy, DSz, calculated by substitution of
Eqs. (4) into Eqs. (2) are given in Tables 3 and 4,
respectively.
3. Results and discussion
The F -test at the 95% confidence level by comparing
the calculated values with the tabulated F -values indi-
cates that the number of significant terms in the best fit
equation is not the same in the unweighted (two-terms,
since Fcal ¼ 0:038 is smaller than Ftab ¼ 4:75) and
weighted (three-terms, since Fcal ¼ 18:16 is greater than
Ftab ¼ 4:75) regression. It is noteworthy that the fourth
term added in the four-term equation (a010 ¼
ꢀ2:960 ꢄ 103, a020 ¼ 5:115 ꢄ 104, a030 ¼ 5:479 ꢄ 102 and
a4 ¼ ꢀ1:144 in Eq. (6) by weighted method) aggravates
the fit since the calculated F -value ()8.37) is negative
(Ftab ¼ 4:84). The F -test results are supported by the rfit
values. The rfit values with unweighted method are
0.0655 and 0.0681 for the two- and three-term form,
respectively, whereas those with weighted method are
0.0297, 0.0192 and 0.0415, respectively, for the two-,
three-, and four-term forms. The best fit with the
smallest rfit value is predicted for the three-term equa-
tion. The fit is the worst with the four-term expression,
Eq. (6). This is rather an unexpected outcome, since
Both DHz and DSz values are seen to decrease with
the rise in temperature by ꢁ1 kJ molꢀ1 per degree and 3.
5 kJ Kꢀ1 molꢀ1 per degree, respectively. The confidence
intervals for DHz and DSz values by the MC method
become much smaller, by 5–10 times, than those by the
conventional method. Interestingly, the CIs by both
methods get wider, i,e., the errors in the estimated DHz
and DSz values increase, as the temperature rises or
drops from 20.0 °C, at which the CI values exhibit a
minimum. This could be related to the stability of the
reaction temperature since obviously the temperature
control will be the most precise around the room tem-
perature. In general the errors in the estimated values of
DHz and DSz are seen to be small around the room
temperature range of 15–26 °C. There is notable change
in the sign of DSz at 20 °C from positive to negative.
Relatively small positive or negative DSz values have
been considered as a characteristic of the SN1 solvolysis
[3], particularly in aqueous solution, associated with the
Table 2
Comparison of CIs estimated by the conventional (Conv.) and MC methods
Unweighted
Weighted
Parameter
CI
Parameter
CI
Conv.
MC
Conv.
MC
a10
a20
a1
a2
a3
30.87
)1.147 ꢄ 104
0.43
0.013 ꢄ 104
0.183
0.005 ꢄ 104
30.32
0.29
0.027
)1.130 ꢄ 104
8.441 ꢄ 102
)4.693 ꢄ 104
)1.218 ꢄ 102
0.009 ꢄ 104
1.861 ꢄ 102
0.815 ꢄ 104
0.279 ꢄ 102
0.0008 ꢄ 104
0.252 ꢄ 102
0.110 ꢄ 104
0.037 ꢄ 102
63.14
)1.288 ꢄ 104
)4.831
164.1
0.718 ꢄ 104
24.57
64.5
0.281 ꢄ 104
9.673

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D.D. Sung et al. / Chemical Physics Letters 392 (2004) 378–382
381
Table 3
The activation enthalpies (DHz) and their CIs
rich composition as the size of nonelectrolyte (or
cosolvent) increases [23,25], e.g. for benzyl chloride
solvolysis, the depth of the minimum in the activation
volume, DV z, increases and the position approaches
pure water as the cosolvent is changed from methyl
through ethyl and 2-propyl to t-butyl alcohol [23]. The
solvent mixture in the present work 20% (v/v) 2-PrOH–
H2O, (0.05 mole fraction), might just correspond to such
a composition at which an extremum in the water
structure has reached. In contrast to the water structure
making effect of nonelectrolytes, the presence of an ion,
or electrolyte, alters water structure by tending to de-
stroy, or break, the normal water structure in the im-
mediate neighborhood by electrostriction and modifying
water structure in the more remote area [3,26]. Tem-
perature rise also results in the destruction of water
structure with the increase in the concentration of
unbound water molecules, and as a result the electro-
striction of the unbound water by ion or electrolyte
increases with the temperature [3].
a
Temperature (°C)
DHz (kJ molꢀ1
)
CIbConv.
MC
0
5
111.2
106.1
101.0
95.9
90.8
86.7
85.7
84.7
81.6
80.6
79.6
76.5
75.5
74.5
71.4
4.7
3.5
2.3
1.1
0.5
1.2
1.5
1.7
2.4
2.6
2.9
3.6
3.8
4.0
4.8
0.6
0.5
0.3
0.1
0.1
0.2
0.2
0.2
0.3
0.4
0.4
0.5
0.5
0.6
0.7
10
15
20
24
25
26
29
30
31
34
35
36
39
a Calculated using parameters, a1, a2 and a3 from the weighted least
squares fit.
b CIs calculated by conventional and Monte Carlo methods.
Table 4
The activation entropies (DSz) and their CIs
According to the Frank–Wen model [27–29] for ionic
hydration in aqueous solutions, there is a fault zone (B)
in which the water structure is broken, between an
electrostricted layer of water (A) and the unperturbed
bulk water (C). The zone (B) increases with the size of
the ions.
a
Temperature (°C)
DSz (J Kꢀ1 molꢀ1
)
CIb Conv.
MC
0
5
69.3
50.7
16.6
12.3
8.0
2.2
1.6
1.1
0.5
0.2
0.6
0.7
0.8
1.1
1.2
1.3
1.6
1.7
1.8
2.1
10
15
20
24
25
26
29
30
31
34
35
36
39
a; b
32.5
14.7
The reversal of the sign of DSz observed can be in-
terpreted as follows: the water structure, which is
probably at a maximum, breaks down at low tempera-
ture range (<20 °C) in the transition state due to the
partial charge development in the SNl solvolysis of
t-butyl chloride. This leads to liberation of water mole-
cules from the ice-like structure, which in turn tends to
increase the entropy. However, as the temperature rises
(>20 °C) the concentration of unbound water molecules
increases and hence the electrostrictive solvation of the
partially ionic transition state increases so that the en-
tropy turns to decrease. It seems therefore that at ꢁ20
°C the structure breaking (entropy gain) and electro-
striction (entropy loss) by the ionic transition state just
about balances to a maginal entropy change.
The strongest support for the increased structuredness
of water in the present solvent mixture relative to the
pure water comes from the large negative heat capacity
of activation, DCpz ¼ ꢀ1020 J Kꢀ1 molꢀ1, obtained in
this work. The reported value for t-butyl chloride sol-
volysis in water is DCpz ¼ ꢀ347 J Kꢀ1 molꢀ1 [3]. Since
DCpz is given as the difference, CpTS ꢀ CpIS, a large in-
crease in the initial state heat capacity (CpIS) due to a
more structured state of water in the 20%(v/v) 2-PrOH–
H2O is reflected in the larger negative DCpz than that in
pure water. The value obtained in the present work
(DCpz ¼ 1020 J Kꢀ1 molꢀ1) is the largest negative value
ever reported for the solvolysis reactions in aqueous so-
lution. A somewhat less negative value, DCpz ¼ ꢀ753
J Kꢀ1 molꢀ1, is reported by Moelwyn–Hughes [30]. In
4.0
1.7
)2.9
)16.8
)20.2
)23.6
)33.8
)37.2
)40.5
)50.6
)53.9
)57.2
)67.1
4.1
4.8
5.6
7.9
8.9
9.5
11.8
12.6
13.3
15.6
: See corresponding footnotes to Table 3.
solvent (water) structure breaking in the transition state.
Thus our results around 20 °C in Table 4 are consistent
with the fact that t-butyl chloride is generally accepted
to solvolyze by an SN1 mechanism [6,7].
Extremum behaviors are often observed in the de-
pendence of activation parameters on binary aqueous
solution composition [23]. For example, DHzð¼ HTS
ꢀ
HISÞ for solvolysis of t-butyl chloride in EtOH–H2O
mixtures is a minimum at ꢁ0.1 mole fraction EtOH,
which has been interpreted as a reflection of variation in
solvent structure with composition [24]. In the vicinity of
0.1 mole fraction ethanol, tetra-coordinated ice-like
water structure becomes a maximum due to the but-
tressing effect of the added cosolvent. As a result the
initial state enthalpy of formation (HIS) becomes a
maximum with little change in the transition state en-
thalpy of formation (HTS) at this composition. This type
of extremum has been shown to shift to a more water