LOGARITHMIC DIAGRAMS IN TRACE ANALYSIS J. GILLIS University of Ghent, Ghent, Belgium NUMERICAL ASPECTS O F ANALYSIS
Numerical relations are fundamental in analytical chemistry as well as in science in general. Each analytical problem may theoreticallybe considered as solved if the mathematical function is known which relates the mass of the substance in question to the magnitude of an appropriate physical or chemical property. Useful properties are, for example: in gravimetry, the'weight; in titrimetry, the volume; in densimetry, the specific gravity; in spectroscopy, spectrophotometry, and radioactivity, the emission or theabsorption of radiations; and so on for the other methods of analysis. The mathematical function may be expressed graphically, as is done in Figure 1, as a system of
quently expressed in parts per million by weight. I n other cases, especially solutions, the amount of constituent is given in weight of constituent per unit volumewhat the chemist calls the concentration. Now it is pertinent that, for dilute aqueous solutions, the figures expressing concentration and parts by weight are the same, because 1 ml. of solution weighs practically 1 g. The numerical expression of the amount of constituent may be given by a ratio in the form of a/b, a being the weight of constituent to be estimated and b the weight of the corresponding volume of solvent. COLOGARITHMIC FUNCTIONS
A mixture to be analyzed includes the following substances: (1) constituent A, the amount of which is a; (2) solvent B, the amount of which is b; (3) extraneous suhstances, called altogether C, the amount of which is c. Corresponding to the definition of the hydrogen-ion exponent pH. = -log C* we may use the functions : PA
= -log a
p~ = -lag b
pC = -log c
in order to give a spacial representation of all possible mixtures of the three substances A, B, and C. If PA, pB, and pC are placed on the three axes of a system of rectangular coordinates, each point of this system (Figure 2) will correspond to a given mixture, the origin
pc=-hgc rectangular coordinates, plotting as ordinate an appropriate function of what may be called "magnitude of response" and as abscissa a function of the amount of constituent. In spectroscopy a curve of this shape -I is obtained, having for ordinate the "density" of a given spectral line and for abscissa a function, namely the logarithm, of the amount of constituent concerned. -3 I n spectrophotometry the transmittancy or the absorbancy of a solution is related directly to the amount of Fimn 2 constituent. In this case there is a straight line, when Beer's law holds, followed by a curve when the concentration of the solution increases. I n many of the system being a mixture containing A, B, and C other cases the relation is a linear one throughout. inunit amounts. This occurs in gravimetric and volumetric analysis, where the amount of constituent is directly propor- DIAGRAM pA:pC AT CONSTANT pB If all possible mixtures of A and C are considered in tional to the ~veightof precipitate or to the volume of the presence of a constant quantity b of solvent B, standard solution. I n trace analysis the m o u n t of constituent is fre- a section through the spacial figure, where pB = -log
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b = constant, is obtained. If b is equal to unity, pB will become zero and Figure 3 is obtained representing all possible mixtures of A and C dissolved in one unit of solvent B. Each point may be considered t o lie on a line parallel to the first bisectrix, whose distance to this line is a function of the ratio c/a. (Log c/a = pA - PC.) Each point may also be considered to lie on a curve which is the geometric locus of mixtures having a constant sum a c. Such curves are desigc = lo-', lo-=, lo-%,and so nated in Figure 3 by a
+
+
a line parallel mith the first bisectrix, mrhich is the geometrical locus of all points corresponding to the relation:
Figure 3
on. The diagram enables us to find the ratio c/a, as as the ratio a / ( a c). Furthermore, if the quantity b of solvent B is exactly known and constant, the ratio a/b is given immediately by the difference between the coordinates pA and pB, thus
+
Diagrams of this kind are of interest in physical and analytical chemistry if one wishes to correlate definite properties such as solubility, conductibility, potential, etc., with the exact composition of mixtures of two components A and C, dissolved in a given quantity of solvent B. They are of real significance where dilute solutions or trace analyses are concerned, including mixtures in which both components A and C are mixed in nearly equal amounts, as well as mixtures with vanishing amounts of one component. They give not only the relative amounts of each constituent but also the percentage and the concrete amounts in each mixture. With a supplementary axis used to take account of a given physical or chemical property of themixtures, they are far superior, for dilute solutions or trace analysis, to the classical diagrams relating percentage to a given physico-chemical property. DIAGRAM pA:pB AT CONSTANT pC
The situation is similar for mixtures of constituent A with variable amounts of solvent B in the presence of a constant amount of substance C (Figure 4). Each point of this diagram, pA:pB at constant PC, lies on
The distance of this line to the first bisectrix is a function of the ratio a/b. If pA - pB is positive this line is situated above the first bisectrix and the ratio a/b will be smaller than one. This means that the amount a of constituent to be detected or estimated in the mixture is lower than the amount b in which it is dissolved. This usill always be the case in analytical chemistry and a fortiori in trace analysis, where dilute solutions of constituent A will have positive values for the differencepA - pB for a given point situated above the first bisectrix. Suppose that a definite analytical test on an amount b of solution containing an amount a of constituent to be detected is being used, and that the limit between positive and negative reaction is just being reached, the test being barely positive. To the concrete amounts a and b effectively used correspond the cologarithms pA and pB, giving a definite point in the diagram. Suppose that pA = 6 and pB = 2, as is the case in Figure 4. We see that this point lies on a line given by pA - pB = 4, which is the locus of all points corresponding to the ratio a/b = lo-" This ratio forms the limit of sensitivity of the test since, with further dilution, the test should become negative. All the solutions situated above the line pA - pB = 4 should thus give a negative test and all the solutions situated under this line should give a positive test. This line may be called the sensitivity line of the test in question. Two points on this line are especially - significant. The first may be called the limit of concentration, corresponding to the coordinates pA = 4 and pB = 0; the second mav be called the limit o f dilution. corre---sponding to thk coordinates pA =
