Pseudo First Order Reaction
A and B react to produce P:
If the initial concentration of the reactant A is much larger than the concentration of B, the concentration of A will not change appreciably during the course of the reaction The concentration of the reactant in excess will remain almost constant. Thus the rate's dependence on B can be isolated and the rate law can be written
Equation (1) represents the differential form of the rate law. Integration of this equation and evaluation of the integration constant C produces the corresponding integrated law.
Substituting [ B ] = c into equation (1) yields
Integrating equation (2) gives: The constant of integration C can be evaluated by using boundary conditions. At t = 0, the concentration of B is co.
Therefore Accordingly, equation (3) can be rewritten as follows:
If the decrease in concentration of B is followed by photometric measurement the Beer' Law must be taken into account.
Combining equation (5) and Beer' Law
A = Absorbance, e = Molar absorbtivity with units of L · mol -1 · cm -1
c = Concentration of the absorbing species in solution, expressed in mol · L -1, d = Path length through the sample
Io = Intensity of the initial light beam, I = Intensity of the transmitted light
gives the relationship between k' and lnA:
According to equation (7), a plot of lnA versus time should lead to a straight line whose slope is the pseudo-first order rate constant k'. The value of k' can then be divided by the known, constant concentration of the excess compound to obtain the true constant second order k:
The pseudo-first order rate constant k' can be also determined from the half-life t 1 / 2.
Measuring a second order reaction rate (see below) can be problematic: the concentrations of the two reactants must be followed simultaneously, which is more difficult; or measure one of them and calculate the other as a difference, which is less precise. A common solution for that problem is the pseudo first order approximation. As a general rule, a minimum of a 20-fold stoichiometric excess is necessary. A 50-fold or 100-fold stoichiometric excess is preferable.