Fit For When Two Reactants are Consumed

  The fit for problem worked in the blog Increasing the Rate of Dissociation by Adding Acid is similar to that in the last blog. It's what one would have to do for the case where the two reactants are consumed in the reaction. There was a minor complication with the complex κ.

The estimate value for κ turned out to be close to the complex conjugate of the original κ. For z = x + i y the complex conjugate is z* = x - i y. It turns out that tanh(κ*) = tanh(κ) when the result is real. There is another formula for the tanh of a complex number involving ordinary and hyperbolic sines and cosines that helps explain the situation. Either the cosine or the sine of the imaginary part in the result has to be equal to zero.

Some data sets are more difficult to fit than others. One has to play with the numbers of terms in the empirical fit for best results. The empirical fit for the initial rate of change might be good but that doesn't guarantee that the fit for the formula will be good. The rounded corner at the lower left of the curve below seems to be hardest to get right. The procedure is sensitive to random errors in the first few data points so some method of adjusting errors might prove useful.

Supplemental (Nov 8): A weakness of the fit procedure is that an error in k'₀ propagates through as one calculates all the other fit constants so if one searches the neighborhood of k'₀ for the value which gives a minimum for the variance of the difference between A' and A then one will get consistently good fits to the data points. The resulting fit is a least squares fit of the data.