The Kinetic Background

2 The Kinetic Background INTRODUCTION Kinetic studies are of fundamental importance in any investigation of reaction mechanism. Kinetics may be approached either as a discipline in its own right or as a tool for the elucidation of reaction mechanism. The aim of the present chapter is to discuss some basic practical kinetic techniques which are useful in the diagnosis of reaction mechanism. In the application of kinetic techniques to the investigation of reaction mechanism it is generally more useful to obtain five rate constants with an accuracy of ± 5% than one constant with an accuracy of ± 1%. Experimental data obtained over a wide range of conditions often permit a clear cut demonstration of the effect of a variable on the reaction rate and often reveal some unexpected result which leads to important mechanistic conclusions. Rates, Rate Constants and Reaction Order In the simplest case the reaction of two molecules A and Β to give a product Ρ in solution (Equation 2.1) is proportional A + B->P

(2.1)

to the number of collisions of the two molecules with each other, and is therefore proportional to the concentration of each of the reactants. This relationship is defined in Equation 2.2. The second order rate constant k defines the Rate = ν = -d[A]/dt = -d[B]/dt = d[P]/dt = k[A] [B]

(2.2)

proportionality of the rate to the concentrations of the two reacting molecules. If the concentrations of the reactants are expressed on the molar scale, the units of this rate constant are M" time" . The reaction is first order in [A] and is first order in [B], and is overall second order. The rate of a first order reaction (Equation 2.3) is proportional to the concentration of a single species and is described by a first order rate constant with the units 1

1

A -> Ρ

(2.3)

ν = k[A]

(2.4)

of reciprocal time ( s ) (Equation 2.4). Kinetics give information regarding the rate determining step of a reaction. For the reaction (Equations 2.5 and 2.6) l

36

The kinetic background

[Ch. 2

Β

(2.5)

B+ C

(2.6)

ν = k[A] and the reaction rate is independent of the concentration of C. The reaction is said to be zero order with respect to [C]. First O r d e r Reactions Equation (2.4) for a first order reaction can be integrated from t to the time t of an experimental measurement (Equations 2.7 to 2.10) where A is the concentration of A at zero time 0

0

I - d [ A ] / [ A ] = k Γ dt

-ln[A] + l n [ A ]

0

(2.7)

= kt

(2.8) (2.9)

l n [ A ] / [ A ] = kt 0

[A] = [ A ] e *

(2.10)

0

The variables in these equations are [A] and r, so that the concentration of A decreases exponentially with time. A plot of log[A] versus t is linear with a slope of -k/2.303, Fig. 2.1.

Aoo-A,

time/min Fig. 2.1 Semilogarithmic plot of the change in absorbance (absorbance increase) for the base hydrolysis of a cobalt(III) complex in a pH 10.9 buffer. The reaction follows first order kinetics with a half life which is independent of the extent of reaction. The half life is 0.575 min. givingk = 2.0χ 1 0 s ' . 2

Ch. 2]

First Order Reactions

37

The half life of the reaction is the time at which the concentration of A has decreased to half its initial value, 0.5A . Substitution in Equation (2.9) gives Equation (2.11) and establishes that the half life of a first order reaction o

l n [ A ] / 0 . 5 [ A ] = kt o

o

1/2

= ln2 = 0.693

(2.11)

is directly related to the first order rate constant as shown in equation (2.12). k = 0.693/t

(2.12)

1/2

Practical Examples Aquation of c « - [ R u ( [ 1 4 ] a n e S ) C l ( H 0 ) ] 4

+

2

The aquation of c/s-[Ru([14]aneS )Cl(H 0)] may be used to illustrate some of the points discussed in previous sections. The ligand [14]aneS has the structure (2.1). When cw-[Ru([14]aneS )Cl(H 0)] (2.2) is dissolved in CF COOH (1.0 M) at 25°C the interval scan spectra for the aquation reaction maintained a set of isosbestic points at 321 and 371 nm for a period of 2 hours, Fig. 2.2. That this spectral change corresponded to the release of chloride was supported by the observation that the addition of A g N 0 to the final reaction solution instantaneously gave a white precipitate of AgCl. Volhard's titration (addition of an excess of standard silver nitrate solution, followed by back titration with standard thiocyanate solution) confirmed that the release of CI" was quantitative. +

4

2

4

+

4

2

3

3

300

Fig. 2.2

350

400

wavelength/nm

Spectral changes for the aquation of c«-[Ru([14]aneS )CI(H 0)] in 1.0M CF COOH at 25°C +

4

2

3

38

[Ch. 2

The kinetic background

(2.1)

(2.2)

The reaction stoichiometry was thus established to be the acid hydrolysis of cis[Ru([14]aneS )Cl(H 0)] , cK-[Ru([14]aneS )Cl(H 0)] + H 0 -> c/s-[Ru([14]aneS ( H 0 ) ] + CI". The stereoretentive nature of this reaction was supported by the observation that the addition of excess CI" to the final solution reversed the reaction retracing the same set of isosbestic points. The aquation reaction was monitored spectrophotometrically at 352 nm (absorption increase). Plots of 1η(Α -Α,) versus time were linear for at least three half lives ( A is the final absorbance, A, is the absorbance at time t), Fig. 2.3. The rate constant k was derived from the slope of such plots. The temperature dependence of k was determined at four temperatures, Table 2.1. +

4

2

+

4

2

2 +

2

4

2

2

Μ

K

aq

aq

for c w - [ R u ( [ 1 4 ] a n e S ) C I ( H 0 ) ]

Table 2.1 Temperature dependence of k 0.1 M C F j C O O H

4

Temp (°C)

10 k

25

7.8

(s"') 7.75

30

10.7

11.0

35

16.0

15.5

40

21.3

21.6

4

aq

10 k (calcy

(s"')

t Values calculated using AHt = 50.5 kJ mol" and ASt = -135 JK-1 mol'l 1

4

aq

2

2+

in

39

First Order Reactions

Ch. 2]

400

800

1200 time/s

Fig 2.3 Plot of l n ( A - A ) versus time for the aquation reaction at 25°C in 1.0M CF3COOH. The œ

t

reaction was monitored at 352 nm. A plot of In (k/T) versus 1/T is linear, Fig. 2.4. Least squares analysis of the data gives AH* = 50.5 kJ mol" and A S ^ g = - 1 3 5 JK" mol" with a correlation coefficient of 0.9980. Using these activation parameters gives the calculated values of k (calc) quoted in Table 2.1. There are frequently many errors in the activation parameters quoted in the literature and it is often advisable to check the values using the quoted rate constants. 1

1

1

aq

40

The kinetic background

[Ch. 2

Fig. 2.4 Eyring plot for the aquation of c;j-[Ru([14]aneS4)Cl(H 0)] in 0.1 M CF C0 H 2+

2

3

2

Ionic Strength Ionic strength is defined in terms of the equation, I (or μ) = 0.5 Ί,ζτ where I (or μ) is the ionic strength, c is the molar concentration of the ion and ζ its charge . The ionic strength must be controlled in studies of the reactions of metal complexes. For a 1:1 electrolyte such as 0.1 M NaC10 the ionic strength is equal to the molarity of the electrolyte. For a 2:1 electrolyte (e.g. 0.1M CaCl ), 4

2

I = 0.5 [0.1M(+2) + 0.2M(-1) ] = 0.3M 2

2

Normally the ionic strength is maintained using NaC10 , as C 1 0 is a weakly coordinating anion. If solubility problems arise (due to precipitation of the perchlorate salt of the complex) possible alternatives are K N 0 or KC1, but of course CI" may interfere depending upon the system under consideration. The primary salt effect is described by the Brônsted-Bjerrum Equation (2.13) _

4

4

3

log k = log k + z z 0

A

B

Λ/Ϊ

(2.13)

where k is the rate constant at infinite dilution and k is the experimental rate constant at 0

some specified ionic strength. On the basis of this equation a plot of log (k/k ) versus ·\/ϊ 0

Ch. 2]

41

Ionic Strength

should be linear of slope z z . Generally quite good agreement with the experimental data is observed, Fig. 2.5 and Table 2.2. A

B

Table 2.2 Effects of ionic strength Reaction

Slope = Z

(A) [ C o ( N H ) B r ] + Hg(II) - » [ C o ( N H ) O H ] + HgBr 2+

3

3+

5

3

5

A B Z

+4

+

2

+2

(B)

S O " + 2I~-> 2 S 0 ~ + I

(C)

C H 0 C N = N0 ~ + O H "

(D)

[Cr(urea) ] + H 0 -> [Cr(urea)OH ] + urea

(E)

H 0 + 2 H + 2Br" -> 2 H 0 + Br

(F)

[Co(NH ) Br] + O H " - > [ C o ( N H ) O H ] + Br"

-2

(G)

Fe

-6

2

2

2

g

2

5

4

2

2

C H OH + N 0 + C 0 ~ 2

2

2

5

2

3+

3

3+

6

2

2

2

2

2

2+

3

2+

2+

5

3

+ [Co(C 0 ) ] "-> F e 3

2

4

3

3+

0 -1

+

2

+1

5

+ 3 C 0 ' + Co 2

2

4

2 +

Fig. 2.5 Variation of the rate constant k with ionic strength I for ionic reactions illustrating the validity of the Brônsted-Bjerrum Equation (2.13)

42

The kinetic background

[Ch. 2

Buffers and pH In many reactions it is necessary to control the pH of the reaction medium. This is normally done by using buffer solutions. It is usually advisable to check that the buffer species themselves do not interfere with the reaction. Phosphate buffers can for example, interfere in some reactions and are best avoided. Buffer species should be poor ligands, and for this reason, sterically hindered bases such as 2,6-dimethylpyridine (2,6-lutidine) (2.3) and 2,4,6-trimethylpyridine (2,4,6-collidine) (2.4) are commonly used.

Me

The pK values relating to HA==>H + A" or B H ^ Β + H for a variety of buffers are listed in Table 2.3. For the ionisation of a weak acid (HA) we have +

+

+

a

HA

+ A-

the thermodynamic, ionisation constant ( K j ) is defined, K

= { H } {A-} / {HA} = { H } {A"} / [HA] +

T

+

where braces represent activities and brackets concentrations. Since the activity coefficient of an uncharged species is close to unity, {HA} ~ [HA]. Table 2.3 T h e r m o d y n a m i c ionisation constants for buffer acids at 25°C Acid

Acid

Oxalic (pK,)

1.27

PIPES

Glycine (pK,)

2.35

Ethylenediamine (pK )

6.85

Chloroacetic

2.88

BES

7.17 (20°C)

Citric (pK,)

3.13

MOPES

7.20 (20°C)

Formic

3.75

H P 0 (pK )

7.20

Succinic (pK,)

4.21

TES

7.50 (20°C)

Acetic

4.76

HEPES

7.55 (20°C)

5.96

Tris

8.06

Histidine (pK ) 2

6.80 (20°C) 2

3

4

«-BuNH

2

2

piperidine

10.64 11.12

Ch. 2]

Buffers and pH

For many purposes it is best to employ a practical ionisation constant K where p

K ={H }[A-]/[HA] +

p

i.e. - l o g { H } = - l o g K + log [A~] / [HA] +

p

and pH = p K + log [A~] / [HA] p

(pH is defined in terms of the hydrogen ion activity). The activity {H } = [H ] γ, whe γ, is the activity coefficient of a univalent electrolyte. Activity coefficients may be calculated if required, using the Davies equatii (Equation 2.14) +

-log γ = A z

+

(2.14)

17Γ -0.21

2

1+1

where A is the Debye-Huckel parameter, ζ is the charge on the ion and I is the ionic strength. At 25°C, A = 0.5115 giving γ, = 0.772. In many cases it is necessary tc calculate [OH"] from the pH. This is readily done, as K ={H }{OH-} +

w

[OH-] = K

{H } +

w /

Y l

and log [ O H ] = p H - p K - l o g -

w

Accepted values of p K and the Debye-Huckel parameter are given in Table 2.4. w

Table 2.4 Values of p K , the Debye-Huckel parameter and γ, w

Temp (°C)

γι

(α)

pK

w

15.0

0.5028

0.775

14.346

20.0

0.5070

0.774

14.166

25.0

0.5115

30.0

0.772

13.996

0.770

13.833

0.5262

0.766

13.534

0.5317

0.764

13.396

0.5161

40.0 45.0

3

6

5

0

8

0

( ) A t I = 0.1M. a

(b) Values of A and p K are taken from R.A. Robinson and R.H. Stokes "Electrolyte Solutions", 2nd Ed., Butterworths, London, 1959. w

[Ch. 2

The kinetic background

44

Elias [16] has described a series of 2,6-dimethylpyridines with substituents in the 3and/or 4-positions which can be used as essentially non-coordinating buffers in the pH range 3 to 8. The formation constants (K) for the 1:1 complexes with Mg(II), Ca(II), Ni(II), Cu(II) and Zn(II) are normally < 3. A further series of non-complexing tertiary amine buffers have been described by Rorabacher and coworkers [21]. Buffers can often be dispensed with if a pH-stat is employed. This technique, Fig. 2.6, allows the pH to be controlled within fine limits, and also allows the course of the reaction to be monitored by the consumption of acid or base.

pH METER TITRATOR

and CONTROL

(RADIOMETER TTT60)

f

·-

COMBINATION ELECTRODE

NiTROGEN IN

REACTION SOLUTION

Fig. 2.6 Schematic diagram of a pH-stat. Using a peristaltic pump and a flow-through cell it is possible to monitor the reaction both by pH-stat and spectrophotometrically The pH stat technique is particularly useful for studying base hydrolysis reactions [17]. Thus for the reaction c«[Co(en) (NH Me)Cl] + OH" -> [Co(en) (NH Me)OH] + Cl" plots of In (V^-V,) versus time are linear of slope = - k ( V is the the final volume of base consumed and V, is the volume consumed at time t), Fig. 2.7. 2+

2

2+

2

2

o b 5

x

2

Ch. 2]

45

Buffers and pH

time/min Fig. 2.7 Kinetic plots for the base hydrolysis of m-[Co(en)2(NH Me)Cl] at 25°C, I=0.1M and 2+

2

pH A, 9.20; B, 9.40; and C, 9.60 In this case rate = k - [Complex] [OH~] = k [Complex] a n d k = koH-[OH-] i.e. k - = k / [ O H - ] 0 H

obs

o b s

O H

obs

Typical kinetic results for this reaction are summarised in Table 2.5, giving k - = 12.8 ± 0.2 M" s" at 25°C and I = 0.1 M. Values of the hydroxide ion concentration were calculated from the pH using the Equation (2.14). Hydroxide ion concentrations derived from the pH at 25°C and I = 0.1 mol dm" in the pH range 9-10 are shown in Table 2.6. This Table can be used for other pH ranges, thus at pH 7.51, the [OH"] is clearly 4.22 χ 10" mol dm" . 0 H

1

1

3

7

3

46

[Ch. 2

The kinetic background

Table 2.5 Kinetic data for the base hydrolysis of ci$-[Co(en)j(NH Me)Cl] at 25°C a n d I = 0.1M 2+

2

PH

10 [OH-] (M)

10 k (s-')

9.20 9.40 9.50 9.60 9.80 10.00

2.07 3.28 4.12 5.19 8.23 13.04

2.68 4.23 5.23 6.55 10.48 16.66

5

4

obs

W = W[OH-] (M"' s"') 12.9 12.9 12.7 12.6 12.7 12.8

Table 2.6 Values of [ O H ] at various pH values at 25°C and I = 0.1 mol dm" -

pH

10 [OH-]

pH

10 [OH-]

9.00 9.01 9.02 9.03 9.04 9.05 9.06 9.07 9.08 9.09 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25

1.30 1.33 1.37 1.40 1.43 1.46 1.50 1.53 1.57 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.89 1.93 1.97 2.02 2.07 2.12 2.16 2.22 2.27 2.32

2.66 2.73 2.79 2.85 2.92 2.99 3.06 3.13 3.20 3.28 3.35 3.43 3.51 3.59 3.68 3.76 3.85 3.94 4.03 4.12 4.22 4.32 4.42 4.52 4.63 4.74

9.26 9.27 9.28 9.29 9.30

2.37 2.43 2.47 2.54 2.60

9.31 9.32 9.33 9.34 9.35 9.36 9.37 9.38 9.39 9.40 9.41 9.42 9.43 9.44 9.45 9.46 9.47 9.48 9.49 9.50 9.51 9.52 9.53 9.54 9.55 9.56 9.57 9.58 9.59 9.60 9.61

5

5

4.85 4.96 5.07 5.19 5.20

3

47

Base Hydrolysis of (Co(pydpt)Brj

Ch. 2]

10 [OH]

PH

5

9.62 9.63 9.64 9.65 9.66 9.67 9.68 9.69 9.70 9.71 9.72 9.73 9.74 9.75 9.76 9.77 9.78 9.79 9.80 9.81 9.82

5.44 5.56 5.69 5.83 5.96 6.10 6.24 6.39 6.54 6.69 6.85 7.00 7.17 7.33 7.51 7.68 7.86 8.04 8.23 8.42 8.62

PH

10 [OH]

9.83 9.84 9.85 9.86 9.87 9.88 9.89 9.90 9.91 9.92 9.93 9.94 9.95 9.96 9.97 .9.98 9.99 10.00

8.82 9.02 9.23 9.45 9.67 9.89 10.13 10.36 10.60 10.85 11.10 11.36 11.63 11.90 12.17 12.46 12.75 13.04

5

Base Hydrolysis of [ C o ( p y d p t ) B r ] The complex [Co(pydpt)Br] where pydpt is the pentadentate ligand (2.5) 2+

2+

N

C= N(CH ) N(CH ) Nz: CT 2

3

H

2

3

N

H (2.5)

can have a variety of structures which are shown diagramatically in (I-IV). Ή NMR data indicates that the isomer produced in the preparation is (IV) with the central tridentate amine component in the wjer-configuration. The complex undergoes very rapid base hydrolysis, and it was necessary to study the hydrolysis reaction using succinate buffers with pH <6. [18] The interval scan spectrum for the hydrolysis of the

48

The kinetic background

H

H

III

IV

[Ch. 2

complex is shown in Fig. 2.8, and the kinetic results are summarised in Table 2.7 A plot of k versus the hydroxide ion concentration is linear and passes through the origin indicating a first order dependence on [OH"] and k - = k /[OH"], Fig. 2.9. obs

0 H

obs

Table 2.7 Base hydrolysis of [Co(pydpt)Cl](C10 ) in aqueous succinate buffers at I = 0.1 mol dm" ( K N 0 ) 4

2

3

3

Temp (°C)

pH

10 [OH] (mol d m ) 9

10 k 3

(s" ) 1

obs

3

25

27

3.91 4.10 4.29 4.49 4.68 5.32 5.75 3.91 4.10 4.29 4.48 4.68 5.32 5.75

0.10 0.16 0.25 0.42 0.62 2.69 7.34 0.12 0.19 0.29 0.45 0.72 3.17 8.53

ΙΟ- ^(dm mol" s" ) 6

3

0.19 0.30 0.41 0.78 1.16 5.34 12.84 0.25 0.38 0.56 0.98 1.51 6.78 17.31

1

1.89 1.82 1.66 1.85 1.87 1.98 1.75 2.07 2.02 1.92 2.15 2.07 2.14 2.03

1

Ch. 2]

Base Hydrolysis of [Co(pydpt)Br]

Temp (°C)

pH

10 [ O H ] (mol dm" ) 9

10 k 3

49

J+

(s" )

10" k (dm mol" s )

1

obs

6

OH

3

3

1

30

3.86 4.09 4.26 4.49 4.69 5.43 5.89

0.13 0.23 0.35 0.59 0.94 5.13 15.04

0.37 0.54 0.95 1.58 2.54 13.82 36.75

2.66 2.34 2.74 2.68 2.69 2.68 2.34

40

3.81 3.42 4.73 4.51 4.70

0.24 0.47 0.76 1.21 1.91

1.35 2.25 4.02 6.32 8.24

5.61 4.70 5.28 5.22 4.30

1

ΔΗ+ = 50.1 kJ mol" ; ASt 9g = 43.0 JK" 1 mol" Data from R.W. Hay and N. Govan, J. Indian Chem. Soc, 69, 495, (1992). 1

1

2

wavelength/nm Fig. 2.8 Interval scan spectrum of [Co(pydpt)Br](C104)2 in succinate buffer pH 4.61 at 25°C (I = 0.1 KNO3). The time interval between scans is 30s.

[Ch. 2

The kinetic background

50

10 k 3

o b s

I

2

3

4

5

6

7

8

10 [OH"] 9

versus the hydroxide ion concentration for the base hydrolysis of Fig. 2.9 Plot o f k [Co(pydpt)Cl]+ at 27°C and I = 0.1 mol dm-3. 0 D S

Conductivity Measurements Conductivity changes can frequently be useful for studying the reactions of metal complexes in solution. Molar conductivities (Λ ) are normally determined using 1 χ 10" M solutions of the complexes. Typical ranges for different electrolyte types are listed in Table 2.8. 3

Μ

Table 2.8

Molar conductivity ranges for different electrolytes (ohm" c m mol" ) at 25°C 1

, Solvent Water Nitromethane Nitrobenzene Acetone Acetonitrile DMSO Methanol

Dielectric Constant

1

Electrolyte Type

/

78.4 35.9 34.8 20.7 36.2 36.7 32.6

2

v.

1:1

2:1

3:1

118-131 75-95 20-30 100-140 120-160 65-90 80-115

235-273 150-180 50-60 160-200 220-300 130-170 160-220

408-435 220-260 70-82

4:1

\

-560 290-330 90-100

340-420 200-240

An example of the use of this method is the hydrolysis of c/i-[Cr(Me cyclam)Cl ]Cl in water at 25°C 2

c/s-[Cr(Me cyclam)Cl ] -> c«-[Cr(Me cyclam)(OH ) ] + 2C1" +

2

2

3+

2

2

2

2

51

Activation Parameters AHt and ASt

Ch. 2]

The zero time value of A is ca. 125 ohm" c m mol" , Fig. 2.10, consistent with a 1:1 electrolyte and this value rises to 450 ohm" c m mol' over a period of time indicating loss of the two coordinated chlorides and the formation of a 3:1 electrolyte. 1

2

1

2

1

M

100

1

0

' 60

' 120

'— 180

1

1

240

300

360

420

480

time/min Fig. 2.10. Conductivity changes observed with c/s-[Cr(Me2cycIam)Cl2]Cl in water at 25°C Activation Parameters A H t and ASt The dependence of observed reaction rates on temperature follows the exponential Equations (2.15) and (2.16), where Ε is the Arrhenius activation energy and PZ k = PZe-Ea

7 1 1 7

=Ae-

E

(2.15)

/ R T a

Ink = - E / R T + In PZ

(2.16)

a

or A represents a "collision factor". The activation energy is given by Equation (2.17) dT

RT

and is usually evaluated from the integrated Equation (2.18), or from the slope of a plot of log k log 6

*1 k,

=__Ea_ (_L 4.575 \Ί2

0 TiJ

( v

2.18) '

versus 1/T (in degrees Kelvin), which gives -E /2.303 R. If this plot is non-linear or if equation (2.18) does not give constant values of the activation energy over different a

[Ch. 2

The kinetic background

52

temperature ranges, the reaction has a significant heat capacity of activation or there is a change in the nature of the rate determining step with changing temperature. The "collision factor" PZ (or A) is the so called temperature independent factor (although it is not completely independent of temperature) and has a normal value of about 3 χ 10" M" s" for a second order reaction. 1

1

The free energy of activation in transition state theory (AG ) has the usual relationship to the enthalpy of activation AH and the entropy of activation AS (Equation 2.19). AG " = A H t - T A S t

(2.19)

1

The relationship of AH to E is slightly different for different types of reaction, but for reactions in solution it differs only by the factor RT (Equation 2.20) a

AH = E -RT

(2.20)

+

a

The entropy of activation may be calculated from Equation (2.21) A S = 4.576 χ log (PZ/T)-49.203

(2.21)

f

t

At 25°C a convenient expression for the calculation of AS (in cal K mol ) is shown in Equation (2.22) ASt/4.576 = log k - 1 3 . 2 2 7 + E /1.364

(2.22)

a

where k is the rate constant at 25°C (s" units) and E is in kcal mol" (for conversion to SI units 1 cal = 4.184 J). The physical meaning of the entropy of activation provides an intriguing subject for interpretation (and overinterpretation). The entropy of activation may be regarded as a measure of the width of the saddle point of energy over which reacting molecules must pass as activated complexes. The enthalpy of activation is a measure of the energy barrier which must be overcome by reacting molecules. The enthalpy of activation is a measure of the proportion of the molecules having the energy requirement which can actually react. The entropy of activation includes steric and orientation requirements, the energy of dilution, concentration effects (which result from the choice of some standard state in which to express equilibrium and rate constants), and solvent effects. Other 1

1

a

things being equal, unimolecular reactions will have values of AS close to zero as no concentration or orientation requirements exist for such reactions. Bimolecular reactions +

will have negative values of AS as a result of the entropy requirement of bringing together two molecules to form a single activated complex. More negative entropies result from steric and orientation requirements, including losses of translational and rotational degrees of freedom in the transition state. The entropy of activation (like the volume of activation) is extemely sensitive to solvent effects. The orientation of solvent molecules around charges or developing charges results in a negative entropy of activation, and the accompanying électrostriction will give a negative volume change. These effects may be as large or larger than those resulting from the molecularity of the reaction.

Ch. 2]

53

Volumes of Activation

Volumes of Activation A V t The effect of pressure P, on the rate of an elementary reaction at constant temperature is given by the Equation (2.23). fdlnkï I dp J

=

T

-AV + RT

The volume of activation AV* is the difference in partial molar volume between the transition state and the reactants. For

AV > 0, k decreases with pressure

For

AV < 0, k increases with pressure

Experimentally pressures up to 2 or 3 k bar are employed. Plots of In k versus Ρ may be non-linear which implies that AV is pressure dependent. Linear plots of In k versus Ρ indicate that AV is independent of pressure, and the volume of activation can be obtained directly from the slope. Some typical kinetic results for the pressure dependence of a second order rate constant are shown in Table 2.9. 1

f

Table 2.9 Pressure dependence of a second order rate constant at 25°C P/atm lO'k/M-'s" Ink

TO 9.58 -6.95

1

27Ô 11.1 -6.80

54Ô 13.1 -6.64

82Ô 15.3 -6.48

ÎÔ9Ô 17.9 -6.33

A plot of In k versus Ρ is shown in Fig. 2.11. The slope of the line is 5.78 χ 10' atm"' so that -AV"i"/RT = 5.78 χ 10'" atm" 4

1

and AV " = -5.78 χ 10' χ 0.082 χ 298 dm mol" = -1.41 χ ΙΟ" dm mol" = -14.1 cm mol' 1

4

3

2

3

3

1

1

1

Typical results for the pressure dependence of the reaction, [Fe(CN) (NH Me)] " + py -> [Fe(CN) py] " + MeNH 3

5

(2.24)

3

2

5

2

are shown in Table 2.10. Table 2.10 Pressure dependence of the reaction (2.1) at 40°C Pressure/MPa 5 25 50 75 100

k/s0.026 0.022 0.017 0.013 0.011 1

AHt = 103 ± 3 kJ mol" ; AS " = +54 JK" mol" Data from K.B. Reddy and R. van Eldik, Inorg. Chem., 30, 596, (1991). 1

1

1

1

3 + log k 1.415 1.342 1.231 1.114 1.041

54

[Ch. 2

The kinetic background

I

I

ι

ι

200

400

600

800

ι—

1000

pressure/atm Fig. 2.11 Pressure dependence of a reaction with Avt = -14.1 cnr* mol

1

A plot of log k versus the pressure is linear with a negative slope, Figure 2.12, indicating that AV is positive. Reaction (2.24) is known to follow a limiting D mechanism with iron-amine bond cleavage as the rate determining step [19] and a positive value of AV is therefore expected.

I

I

I

20

1

I

40

I

I

60

I

1

I

I

80

100

pressure/MPa Fig. 2.12 Pressure dependence of reaction (2.24) at 40°C, giving Avt = +24.4 cm^ m o l ' -

Ch. 2]

Volume Profiles

55

The volume of activation can be calculated from the slope of the plot as the slope = -AV / 2.303RT. For pressure in MPa and k in s

_ 1

then AV = -slope χ 19.138 χ T. The

slope of Fig. 2.12 is -4.067 χ 10" (correlation coefficient 0.9975) giving AV = +4.067 χ 10" x 1 9 . 1 3 8 x 3 1 3 = 24.4 cm mol' . 3

3

3

1

Interpretation of AVt The experimentally determined volume of activation can be considered to be made up of two parts, the intrinsic part and the solvation part, W

= AV\

f obs

+AV

na

f s o l v

The parameter A V ^ ^ reflects changes in volume when moving from the reactants to the transition state, and in principle provides information regarding the molecular (i.e. intrinsic) mechanism. The A V ^ term reflects reorganisation of the solvent. The contribution of these two terms to AV* is not always easy to determine. However, if the reaction involves no change of formal charge, ΔΝ ι is expected to be small and AV* = AV^nfr. When there are changes in charge, AV . becomes important and cannot be neglected. obs

+

50

ν

+

obs

S0

V

The reaction volume AV° is given by Equation (2.25) AV° = EV (products) - I V (reactants)

(2.25)

The reaction volume can be determined by dilatometry, by measuring the effect of pressure on the equilibrium constant K, or by a combination of separately determined partial molar volumes of all reactants and products using Equation (2.25). It is convenient for a simplified analysis of activation volumes to consider octahedral complexes as essentially incompressible spherical species with a characteristic average radius. It has been assumed that a five coordinate intermediate (ML ) arising from the dissociative release of a neutral ligand (L) will occupy the same volume as its sixcoordinate precursor M L . The volume of [ N i ( N H ) ] (138 cm mol" ) is identical to that calculated for the hypothetical [ N i ( N H ) ] cation. For the analogous cobalt(III) system, the intermediate [ C o ( N H ) ] has been estimated to have the same volume as [ C o ( N H ) ] (55 c m mol' ). In the case of an associative mechanism the seven coordinate intermediate (ML ) is believed to occupy the same intrinsic volume as its precursor M L . For neutral ligands L, the dissociative mechanism is expected to lead to a transition 5

2+

6

3

3

1

6

2+

3

5

3+

3

3+

3

3

5

1

6

7

6

state (ML + L) of greater volume than the M L precursor (positive AV ). An associative mechanism should produce a transition state of less volume than the combined volumes t 5

6

of M L and L (negative AV ). 6

Volume Profiles The reaction of dimethylsulphoxide (Me SO) with [Pd(OH ) ] to give [Pd(OH ) (Me SO)] has been studied in detail and values of AV measured for both the forward and reverse reactions, [20]. Solvent effects are minimised by the use of uncharged ligands 2+

2

2+

2

3

2

2

+

4

[Ch. 2

The kinetic background

56

[Pd(OH ) l 2

4

2+

+ MezSO

AW° = -7.5 cm

3

[Pd(OH ) (Me2SO)] + H ( 2+

2

ΔΥ =-1.7 cm [Pd(OH) ) (Me2SO)] 2

3

3

2 +

4

Fig. 2.13 Volume profile for the reaction of [Pd(OH )4] + and Me SO in water 2

2

2

AV° for the process is not negligible but the negative AV* values support associative activation in both directions. The AS* values are also negative.

2

Bibliography and References

57

Bibliography and References Ion Solvation 1. S.F. Lincoln, Coord. Chem. Rev., 6, 309, (1971). 2. A. Fratiello, Progr. Inorg. Chem., 17, 57 (1972). Reviews of the determination of solvation numbers using NMR techniques. 3. A.I. Popov, PureAppl. Chem., 41, 275 (1975). A short conference lecture showing how N M R and IR-Raman spectroscopies give complimentary information on ion solvation. Solvation Numbers 4. J.F. Hinton and E.S. Amis, Chem. Rev., 71, 627 (1971). An exhaustive compilation of hydration and solvation numbers, but uncritical. Activation Volumes 5. S. Suvachittanont, J. Chem. Ed., 60, 150 (1983). 6. R. Van Eldik (ed), Inorganic High Pressure Chemistry, Elsevier, New York, (1986). The first article gives a good brief introduction to the use of activation volumes for the elucidation of reaction mechanisms. The latter text provides a definitative account of high pressure inorganic kinetics. 7. D.R. Stranks, Pure. Appl. Chem., 38, 303, (1974). 8. G.A. Lawrance and D.R. Stranks, Acc. Chem. Res., 12, 403, (1979). 9. D.A. Palmer and H. Kelm, Coord. Chem. Rev., 36, 89 (1981). 10. R. van Eldik, T. Asano and W.J. le Noble, Chem. Rev., 89, 549, (1989). Solvent Exchange 11. A.E. Merbach/uA-e. Appl. Chem., 21, 1479 (1982). Complex Formation 12 J. Burgess , Metal Ions in Solution, Ellis Horwood Ltd., (1978), Chapter 12. 13.R.G. Wilkins, Acc. Chem. Res., 3, 408, (1970); PureAppl. Chem., 33, 583, (1973); Comments Inorg. Chem., 2, 187, (1983). Key accounts by a pioneer in the field. 14.R.G. Wilkins and M. Eigen, Adv. Chem. Ser., 49, 55, (1965). The key original reference to the Eigen-Wilkins mechanism. Aqua Ions 15.D.T. Richens, Perspectives on Bioinorganic Chemistry, ed. R.W. Hay, J.R. Dilworth and K.B. Nolan, JAI Press Connecticut, (1993). D.T.Richens, The Chemistry of Aqua Ions, Wiley, New York, (1997). A full discussion of the chemistry of aqua ions and their reactivity References 16.U. Bips, H. Elias, M. Hauroder, G. Kleinhans, S. Pfeiferand K.J. Wannowius, Inorg. Chem., 22, 3862(1983). 17. R.W. Hay and P.R. Cropp,./. Chem. Soc. (A), 42, (1969). 18. R.W. Hay and N. Govan, J. Indian Chem. Soc, 19.H.E. Toma and J.M. Malin, Inorg. Chem., 12, 1030, (1973). 20. Y. Ducommun, A.E. Merbach, B. Hellquist and L.I. Elding, Inorg. Chem., 26, 1759, (1987). 2 1 . O.Y.Yu, A.Kandegedara, Y.P.Xu and D.B.Rorabacher, Anal.Biochem., 253, 50, (1997)