Curve Fit for the Dissociation Rate Constants

  If one looks at the equation for the amount of the dissociated molecule as a function of time in the last blog one one can see that it can be rewritten as a linear function of A equal to the hyperbolic tangent of a linear function on the time t. Not all the constant coefficients in this equation are independent but we can find the first two as a function of the third by evaluating the equation at t = 0 and t = ∞ since the initial amount A₀ and equilibrium amount A_eq are known. Substituting the functions for μ and ν and separating out the coefficient of tanh(κ) we get two new functions of A, α and β, the sum of which is 1.

So we are left with two constants to be determined since α, β and t can be obtained from the experimental data. If κ is known we can compute λ so we can estimate the function λ(κ). One then has to search for the value of κ for which the square of the difference between the two sides of the equation is a minimum.

I did this for some of the data for A in the last blog and found a minimum for the variance V(κ) when κ = 0.458. With κ known we can go back and compute μ, ν and λ and consequently the equilibrium constant K and the dissociation rate constants k₀ and k₁.

The computed values for the fit are almost identical to the values chosen in the last blog since we used values for A that were nearly exact and so there was negligible error. The round-off errors and the estimate for λ(κ) contributed some error to the results. As we can see the variance changes very slowly near the optimum value for κ.

edit (Oct 26): Corrected the formula for λ(κ) in the derivation by removing the 2nd atanh that crept in with a paste. The correct formula was used for the fit so fortunately the error in the derivation didn't affect the results. The number of data points determines the accuracy of the fit but one gets a reasonably fit with just 20 data points. The calculation above used 100 data points and 200 came closest to the original values. Measurement errors also would affect the fit and consequently an error in κ will propagate through to the dissociation constants.