If the variance is constant at each point C, a fit of In C uersus time would require weighting each point proportional to the reciprocal of the variance, approximately C2 in the case of constant CV. With constant variance, a zero-order plot would require no weighting.
From similar considerations, an estimate of u2 (equal to C V in the case of constant CV and a first-order reaction) can be obtained from replicate determinations of the rate constant using the formula:
Where Ew is the weighted mean, n is the nu_mber of replications and w is Z(t -5)2. (Term w is eqgal to k2Z(tt)2in the case of In k). A shortcut formula for Zw(kk ) 2 is Zwk' -Zw(kw)*.
An example of some calculations for the combination ab follows. The duplic@e determinations of the rate constants were 5.86 and 8.22, and Z(tt ) l was 0.0125 for each d_etermination:
(1) The weighted average, k,, = [(0.0125)5.86 + (0.0125)8.22]/(0.025 + 0.025) = 7.04 (since the weights are equal, the weighted average equals the ordinary arithmetic average). u2 (One degree of freedom) from the duplicates = [Zwk2 -(2) Bw(Ew)2]/(n -1) = [(0.0125)(5.86)2 + (0.0125)(8.22)2 -0.025 (7.04)2]/1 = 0.0348 (3) The weighted average of In k = In k , = [(5.86)2(0.0125) In 5.86 + (8.22)2(0.0125)1n 8.22]/[(5.86)2(0.0125) + (6.22)2(0.0125)] = 1.99
The approximations inherent in the method should be kept in mind, namely the transformation to logs and the fact that observed values are used rather than true values in the weighting. However, if the CV is not large, the reliability of any conclusions from this analysis should not be in doubt. 1403 (1980).