−=∆ LLL ][][ −=∆ MMM ][][ k ML M L k

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Appendix VI

163

APPENDIX VI

Determination of the relaxation time equation for a reaction at the equilibrium

The reaction between M and L is represented by the first order reaction respect to each reagents M_f L_f ML_T k_d k_f + (VI.1)

whose kinetic law can be written as

] [ ] ][ [ ] [ T d f f f f ML k L M k dt L d − = − (VI.2)

If [Mf]eq, [Lf]eq and [MLT]eq are equilibrium concentrations we define the de of the

reactants concentrations from equilibrium as ] [ ] [M_f _eq M_f M = − ∆ (VI.2) ] [ ] [L_f _eq L_f L= − ∆ (VI.3) ] [ ] [ML_T _eq ML_T ML= − ∆ (VI.4)

and the (VI.2) becomes

[ ]

(

)

₍

_{[ ]}

₎

₍

_{[ ]}

₎

₍

_[

_]

₎

ML ML k L L M M k dt L L d eq T d eq f eq f f eq f ∆ − − ∆ − ∆ − = − ∆ (VI.5) or

[ ] [ ]

M L k

[

ML

]

k M L k

(

[ ]

M L

[ ]

L M

)

k ML k dt L d d eq f eq f f f eq T d eq f eq f f − + ∆ ×∆ − ∆ + ∆ + ∆ = ∆ (VI.7)

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Appendix VI

164

The mass conservations can be written in differential form as 0 = ∆ + ∆M ML (VI.8) 0 = ∆ + ∆L ML (VI.9)

Introduction of equations (VI.8) and (VI.9) in (VI.7) reasonably assuming that product ∆M×∆L is negligible and taking into account that kf[Mf]eq[Lf]eq = kd[MLT]eq

equation (VI.7) reduces to

[ ] [ ]

M L k dt k L L d d eq f eq f f _⎭⎬⎫ ⎩ ⎨ ⎧ _⎟₊ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ = ∆ ∆ − _(VI.10)

Integration of equation (VI.10) yields

τ t e L L=∆ °⋅ − ∆ (VI.11)

where 1/τ (s-1) is the time constant of reaction (VI.1) and is expressed by equation (VI. 12)

(

f eq f eq

)

d f M L k k + + = [ ] [ ] 1 τ (VI.12)

Now, for CM ≥ 10CL(pseudo first order conditions), equation (VI.12) becomes

M f d k C k + = τ 1 (VI.13)