Amplitude analysis of kinetic relaxation curves a) One single reaction

(1)

Appendix VII

133

APPENDIX VII

Amplitude analysis of kinetic relaxation curves a) One single reaction

Consider the reaction of a molecule D with a polymer site S:

D + S ' DS (VII.1)

which equilibrium constant is

(VII.2) Eq. (VII.2) can be expressed in logarithmic form

lnK = ln[DS] – ln[S] – ln[D] (VII.3)

Differentiation of eq. (VII.3) yields

dlnK = dln[DS] – dln[S] – dln[D] = d[DS]/[DS] – d[S]/[S] – d[D]/[D] (VII.4) Differentiating eqs. (II.3), (II.18) and (III.6) one obtains

d[S] = CP f ’(r) dr (VII.5)

d[DS] = - d[D] (VII.6)

dr = d[DS]/CP (VII.7)

Introduction of (VII.5), (VII.6) and (VII.7) in (VII.4) yields

[ ] [ ] [ ] [ ]

^DS¹ ^D¹ ^f^'^S^(r) ^d^DS

dlnK ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + −

= (VII.8)

that can be rewritten as [D][S]

K = [DS]

(2)

Amplitude analysis

134

[ ]

=

[ ] [ ]

+ −

[ ]

⁼^Γ

S (r) f' D 1 DS 1

1 dlnK

DS

d _(VII.9)

Γ is denoted as the amplitude factor. But the following equations also apply

[ ] [ ] [ ] [ ]

C T

T C C

T dT

dlnK dlnK

DS d dT

dK/K dK/K

DS d dT

dK dK

DS d dT

DS

d ⎟

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

=⎛

⎟⎠

⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

=⎛

⎟⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

=⎛ (VII.10)

2

C

RT

∆H dT

dlnK ⎟ =

⎠

⎜ ⎞

⎝

⎛

_(VII.11)

and therefore, from (VII.9), (VII.10) and (VII.11) one gets

[ ]

^dT

RT DS H

d =Γ ∆ ₂ (VII.12)

Now, for a relaxation curve recorded with absorbance detection, the change of DS concentration is related to the change of absorbance by eq. (VII.13)

dA = ∆ε d[DS] (VII.13)

Introduction of eq. (VII.12) in (VII.13) yields

RT dT Γ∆ε ∆H

dA= ₂ (VII.14)

For free site excess, CP>>CD, f’(r)/[S] →0 and the eq. (VII.9) becomes

[ ] [ ] [ ] [ ]

D DS C DS

D DS D DS

1 D

Γ ¹ 1 ^D

= ⋅

⋅

= + +

− = (VII.15)

The following equations also apply (from eq. (VII.2) and CD = [DS]+[D])

[ ] [ ]

[ ]

S K 1

S K C

DS

D = + (VII.16)

(3)

Appendix VII

135

[ ]

1 K

[ ]

S 1 C

D

D = + (VII.17)

then, from (VII.15), (VII.16) and (VII.17)

[ ]

( )

[ ] ( )

D P

2 P D

2

f(r)C KC

f(r) KC 1 C

S K

1 +

+ =

=

Γ

⁻¹ (VII.18)

By introducing eq. (VII.18) into the eq. (VII.14) one gets

( )

D P

2 2 P

f(r)C KC

f(r) KC 1

∆ε∆HdT RT dA

1 +

×

= ^(VII.19)

and, rearranging

(

1 KC f(r)

)

∆ε∆HdT RT dA

f(r) C C

P 2

D

P ⎟⎟⎠ × +

⎜⎜ ⎞

⎝

=⎛

⎟⎠

⎜ ⎞

⎝

⎛ ¹² ¹² (VII.20)

Setting B’ = (RT²/∆ε∆HdT)^½, eq. (VII.20) leads

f(r) KC B' dA B'

f(r) C C

P 12

D

P ⎟ = +

⎠

⎜ ⎞

⎝

⎛ (VII.21)

A plot of (CPCDf(r)/dA)^½ vs. CPf(r) is a straight line with intercept equal to B’ and slope B’K. Therefore, the ratio slope/intercept enables the equilibrium constant of the reaction to be obtained.

If the relaxation curve is produced by a known jump of temperature dT, being the R, T, ∆ε and dT values known, the enthalpy variation of the overall process is calculated from the B’ value.

(4)

Amplitude analysis

136