A Step-by-step Dilution-Extraction Method for Laboratory Experiments Alda Jover, Francisco Meijide, Victor Mosquera, and J& Vazquez Tatol Universidade de Santiago. Colexlo Universitario de Lugo. 27002 Lugo, Spain A large number of experiments in chemistry involve the measurement of some physical quantity at different concentrations of a solute in a solvent. Normally this kind of experiment is carried out by dilution of an initial solution that has a higher concentration than the maximum one necessary for the experiment. Rarely are all the solutions prepared directly. In the present paper we describe a sequential procedure for dilution and present its application to three typical educational experiments in physical chemistry: determinations of the absorptivity molar coefficient, the pK. of a weak acid, and the critical micelle concentration (CMC) of a surfactant. The method is very useful to illustrate the difference between random and systematic errors. Step by Step Dllutlon-Extraction Method The method consists of two steps. First, u mL of solvent are added to V mL of solution of Co molar concentration. Accepting that for diluted solutions volume effects are negligible, the concentration of the resulting solution will be Second, u mL of the new solution is extracted. The subsequent repetition of the two previous sequential steps will yield solutions with concentrations that follow a geometrical progression. Defining from eq 1 a nondimensional volume factor, VF, the concentration for the nth term of this geometrical progression is easily deduced, C" = C,V,"
(3)
A simple theory for error propagation2 can be used to estimate the error on concentration after n dilution-extractions, AC,, giving AC,, = C0nVFn-'AV,
From this equation, the closer VF is to 1, the greater will the n, value be, and therefore unacceptably large values for f(n < n,) will be obtained. Consequently, acompromise between the minimum number of experimental points necessary for calculations and plots (which will be equal to the number of dilution-extractions steps necessary to obtain a particular final dilution) and the maximum acceptable absolute error in concentration, is required. A convenient dilution at each step is 10-20% (VF N 0.8-0.9). By using an average value of 0.85 for VF, n(max) is --6 and f(n,.,) n 2.7. If we accept a typical value of 3.7 X for AVF, deduced from the absorbance experiment (vide infra), the maximum typical absolute error in concentration would be, Using those values in eq 6, the typical relative error would be which gives a relative error of0.9% after 20 steps, resulting a final concentration that is 1/25 times the initial one. The method is excellent for illustrating the influence of systematic errors in laboratory experiments. Since these errors are not random errors, previous equations are not valid for analyzing them. We can define the relative systematic error in concentration at the nth step according to eq 9,
where the subscripts W and G are used to distinguish between those C, values calculated with "wrong" volumes and those calculated with gauged volumes of pipets used. Defining Vw/Vc = p and uwluc = q and using eqs 2,3, and 9, the following expression is obtained:
(4)
where AVF is
This equation shows that for the case plq = 1(i.e., Vw/Vc = uwluc), the systematicerror is always zero. The greater the
The relative error can be obtained by dividing eq 4 by eq 3, giving, AC,,/C,, = nAVJV, (6) Hence the relative error increases as n increases. For a given AVFvalue the error in the concentration at the nth step depends on the factor, f(n), f(n) = nV,"-'
(7)
which shows a maximum at
' Author to whom correspondenceshould be addressed.
White, J. M. Physical Chemistry Laboratory Experiments; Prenlice-Hall: New Jersey, 1075.
530
Journal of Chemical Education
difference plq from 1, the greater will the error be, i.e., the most dramatic effect will appear when simultaneously p > 1 (Vw > Vc) and q < 1(uw < uc), or vice versa. For the pipet used in the experiments described below for measuring, u, the gauged volume is uc = 1.9368 f 0.0031 mL. This value is not statistically compatible with the nominal value of 2 mL (accuracy
