APPENDIX IX

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133

APPENDIX IX

Derivation of the fast relaxation time at [H+_{] > 0.04 M according to reactions} (4.1)-(4.2)-(4.5) of Chapter 4

Let´s consider the reaction scheme (charges omitted)

KH Fe ⇄ FeOH + H (IX.1) kD 2 FeOH ⇄ D (IX.2) k-D k’D FeOH + Fe ⇄ D + H (IX.2’) k’-D D + FeOH ⇄ T + H (IX.3)

under conditions of relatively high acidity, step (IX.3) can be neglected, so the reacting system is represented by steps (IX.1), (IX.2) and (IX.2’). If the displacement of the equilibria is small, as it occurs in the case of a dilution jump, then the chemical relaxation analysis can be applied to the reacting system as follows. The mass conservation equation in differential form is given by:

δFe + δFeOH + 2δD = 0 (IX.4)

The protonation transfer step (IX.1) is very fast with respect to the steps of dimer formation. Moreover, since the H+_{concentration is sufficiently high to buffering the}

system, the equilibrium expression for reaction (IX.1) can be written in differential form as eq (IX.5):

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Appendix IX

134

Introduction of eq (IX.5) into eq (IX.4) yields eq (IX.6):

€

δFeOH = −2

K

H

[H] + K

H

δD

(IX.6)

Therefore, the rate of the dimer formation in differential from is given by eq (IX.7):

(IX.7)

Introduction of eqs (IX.5) and (IX.6) into eq (IX.7) yields eq (IX.8):

(IX.8)

Integration of eq (IX.8) yields the expression for the fast relaxation time in the form of eq (IX.9) € 1 τf = 4 kD+ k'D [H] K_H ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ KH [H] + KH ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + k

(

−D+ k'−D[H]

)

(IX.9)