Chemistry Everyday for Everyone
The Pathway to the Ostwald Dilution Law John T. Stock Department of Chemistry, University of Connecticut, Storrs, CT 06269-3060 The study of “affinity”, including the relative strengths of various acids and bases, began early in the 18th century. Later it was realized that a chemical reaction does not necessarily go to completion, and that an excess of one of the reactants can drive an “incomplete” reaction further. The “effect of mass” was noted in 1801 by Claude Louis Berthollet (1748–1822). In 1850, Alexander Williamson (1824–1904) pointed out that the equilibrium is dynamic, the result of opposing changes that are proceeding with equal velocities. An example is the slow and incomplete general reaction alcohol + acid
ester + water
studied in 1862 by Marcellin Pierre Berthelot (1827–1907) and Leon Péan St. Gilles (1832–1863). They showed that the same equilibrium is reached when either alcohol and acid or ester and water are the starting materials. They also derived an expression for the velocity of the forward reaction. Noting this work, Cato Maximilian Guldberg (1836– 1902) and Peter Waage (1833–1900) derived the law of mass action in 1864 (1). Their underlying assumption was that chemical action is proportional to the active mass—that is, to the number of molecules in unit volume. An example of the law of mass action, the “Ostwald Dilution Law” mentioned in various texts, is expressed by the relationship K ≈ c α2 / (1 – α)
(1)
Here K is the dissociation (or ionization) constant of a given monobasic weak acid or monoacidic weak base, c is the (analytical) concentration of its solution in gram-equivalents (here the same as moles) per liter, and α is the fraction of the acid or base that is dissociated at this concentration. The word “dilution” reflects the practice of the times: experiments were usually performed by the progressive dilution of the most concentrated solution. Equation 1 is an approximation, because activity, rather than concentration, governs the equilibrium. However, resulting K values are usually in reasonable agreement with modern values. The simplicity of the equation belies the long and sometimes tedious path that led to the formulation. A key step was the evaluation of α. If the dissociated and undissociated species differ in color, optical evaluation might be possible. More general are colorblind electrochemical methods, the oldest of which is electrolytic conductance (2). This technique enabled Wilhelm Ostwald (1853–1932) (Fig. 1) to develop the dilution law. Instruments for the direct measurement of conductance are presently available. Usually, however, the actual measurement is of a resistance, by comparison with a standard of resistance by Wheatstone bridge circuitry. The conductance is then obtained by taking the reciprocal of the resistance. Unless otherwise stated, this approach is assumed in the present account. Fixed lengths of specified copper wire were early, but unsatisfactory, standards. The electrical resistance of copper has a large temperature coefficient and is quite sensitive to the presence of impurities. In 1860, Werner Siemens (1816–1892) proposed the use of the easily purified mercury
for making a standard. The “Siemens unit” was defined as the electrical resistance of a mercury thread of 1 mm2 cross section and one meter in length, at 0 °C. Later, the mercury standard was modified to represent the international ohm, a standard that persisted until 1948. The primary standard was used to calibrate working standards, usually made of suitable metal alloy wire. For precise work, these were maintained at a specified temperature. Direct current (dc) from a battery was used in early experiments on electrolytic conductance. Electrolytic de- Figure 1. Wilhelm Ostwald composition at the platinum (1853–1932). electrodes of the conductance cell causes polarization, that is, the development of an emf opposing that applied by the battery. Results were therefore erroneous and often erratic. A procedure aiming to overcome this difficulty, outlined by Charles Wheatstone (1802–1875), was developed independently by Eban Horsford (1818–1893) in 1847 (3). It was assumed that if the current was kept constant, the polarization effect would also be constant. The apparent resistance R1 of a column of a chosen solution of concentration c, known cross section A, and length L1 was measured. Without changing the current, the column length was increased to L2 when the apparent resistance increased to R2. Then (R2 – R1) should represent the “true” resistance of a column of length (L 2 – L1 ) and cross section A. From this, the specific resistance (i.e., the resistance between a pair of opposite faces of a 1-cm cube of the solution) and hence the specific conductance, S c, can be found. Horsford’s “trough” cell, shown in Figure 2, was a 30cm long varnished wooden trough with two parallel wooden partitions. The inner faces of these were entirely covered by platinum sheet electrodes. One partition was movable, so that the interelectrode distance could be altered. The solution was added or withdrawn to keep the depth, and hence the cross section, constant.
Figure 2. Horsford “trough” cell.
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Chemistry Everyday for Everyone
Figure 3. Kohlrausch conductance apparatus.
Horsford’s method was referred to by later workers, including Ostwald. However, the elimination of polarization, rather than attempted compensation, was an obvious goal. In 1869, Friedrich Kohlrausch (1840–1910) and Wilhelm Nippoldt (1843–1904) used alternating current (ac) to show that Ohm’s law is valid for solutions of electrolytes. Kohlrausch was professor of physics at Darmstadt. He moved to Würzburg in 1875. These workers then showed that precise measurements were possible by their approach (4). In a succession of papers, Kohlrausch showed that ac from a small induction coil was satisfactory, and that the Wheatstone bridge could be balanced audibly by use of a telephone earpiece. (A conventional galvanometer does not respond to ac.) Figure 3 shows the general arrangement. The electrodes of the cell were platinized to increase the effective surface areas, so that electrodes of quite small geometric area could be used. The observed conductance, S obs, of a given solution obviously depends upon parameters such as the shape of the cell and the size and separation of the electrodes. The “cell constant”, θ, must therefore be determined, so that the specific conductance, Sc, may be obtained as the product θ ? Sobs. In 1898, Kohlrausch and his coworkers used special cells of known length and cross section for the direct measurement of the specific conductances of several solutions of KCl of known concentrations (5). Such solutions can be used to evaluate θ for a given cell. Kohlrausch’s values have been refined by others, but the general principle remains. Dealing with solutions of mineral acids and metallic salts, Kohlrausch developed the concepts of equivalent conductance and ionic conductance. The equivalent conductance, Λv, of a solution of the chosen electrolyte at dilution v is given by Λv = (1000 Sv) ? v
(2)
where Sv refers to a solution of dilution v and v = 1/c (see eq 1). As v approaches infinity, Λv increases to a maximum value Λ∞, which is specific for the chosen electrolyte. Because Λv has a positive thermal coefficient of about 2%, the temperature must be specified. Ionic conductances λ ∞ can be handled arithmetically and, if tabulated, can be used to enumerate Λ∞ for a given binary electrolyte through the relationship Λ ∞ = λ∞+ + λ∞{
(3)
where the plus and minus signs are for the appropriate cation and anion. Because the ionic conductance of H+ (or H3 O+) is some 5 times larger than that of Na+, HCl is an even better conductor than an equivalent solution of NaCl. Kohlrausch was
866
puzzled because CH3 COONa in solution is a good conductor, whereas CH3 COOH, whose cation is H+, is not. Ostwald entered the University of Dorpat in 1872; at first he gave more time to the arts than to the sciences. This changed when, in order to graduate, he had to present a thesis on an original investigation. His choice of topic, the mass action of water, presaged his future path. He quantitatively studied the hydrolysis of BiCl3 and showed that the extent of this effect increased with dilution. While at Dorpat and later at Riga, Ostwald was studying the “affinities” of various organic monobasic acids by observations such as the change in volume on reaction with a base, or the kinetics of acid-catalyzed hydrolyses such as that of methyl acetate. He noted, but could not explain, the parallelism between the values of his “affinities” and the equivalent conductances of some acids that had been measured by Kohlrausch. In 1884, Ostwald became acquainted with the then unknown Svante Arrhenius (1850–1927), who had used conductance measurements to develop what is now known as the ionic theory. Ostwald immediately improvised a conductance apparatus and found that the equivalent conductances of his organic acids were nearly proportional to his previously determined “affinities”. In 1886, Ostwald summarized his conclusions (6). He had examined the effect on Λv of successive twofold dilutions of solutions of strong monobasic acids; the subscript v emphasizes his use of “dilutions”. He noted that Λv seemed to reach a maximum value close to 90 in his chosen units, at a dilution of 512 liters per gram equivalent. A slight decrease on further dilution was attributed to impurities in the distilled water then in use. Nowadays, a conductance monitor is used to indicate the quality of distilled or deionized water. The presence of a mere trace of an electrolyte greatly raises the conductance. Such traces can arise from the absorption of CO2, etc., from the air, or by dissolution of alkali from glass vessels. In 1891, Kohlrausch began his investigation of such sources of contamination and, with Adolf Heydweiller (1856–1926), carried out the heroic 42 vacuum distillations needed to produce “pure” water (7), which still had a slight conductance. This is true of the even purer water produced nowadays by ion-exchange methods. On the slight self-ionization of water rests the concept of its ionic product and, in fact, classical acid–base theory. Returning to Ostwald, he next examined solutions of various weak to moderately strong acids, noting the increase in Λv at each dilution step (6). The maximum increase δ was essentially the same in all cases. However, the dilution range for δ depended upon the solute: for example, for CH2 ClCOOH the step was 256 → 512 liters, whereas for the stronger acid CHCl2COOH it was 8 → 16 liters. Ostwald concluded that the dilutions at which the Λv values of the various acids are equal always bear a constant relation to one another. Ostwald assumed that Λv could range from zero to 90, the same upper limit that he accepted for strong acids. He then derived the expression tan Λ v = (v/ v 0)0.4124
(4)
where v0 , specific for the particular acid, is the dilution at which Λv reaches half-maximum value, that is, 45. The index depends only on the units chosen. Ostwald found some divergences greater than experimental error and was forced to admit that eq 4 had no rational foundation. At this stage, he began to examine some dibasic acids, which seemed to behave as mixtures of two monobasic acids of differing strengths. He found that the relationships for weak acids also applied to weak bases.
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Chemistry Everyday for Everyone
Figure 4. Ostwald conductance cell.
Figure 5. Ostwald thermostatic bath.
Two major events occurred in 1887. Ostwald founded and became principal editor of the Zeitschrift für physicalische Chemie. He also became professor of physical chemistry at the University of Leipzig. While continuing his studies, he gave a detailed account of the conductance apparatus he was then using (8). The Ostwald conductance cell, shown in Figure 4, was used by many contemporary and later workers. The cell contains two platinized platinum discs that are mounted as shown. For constancy of temperature, Ostwald placed the cell in a thermostatic bath that he had devised. The bath, shown in Figure 5, is heated by a regulated gas flame. The bulb, the connecting tube, and the regulator above the U portion containing mercury are filled with 10% CaCl2 solution. This expands if the bath temperature rises, causing partial blockage of the gas supply. A small bypass hole prevents extinction of the flame. If warm air rising from the bath does not suffice to drive the stirrer, a small flame is mounted beneath the fan. Now adopting the Siemens resistance standard, Ostwald examined the sodium salts of numerous weak acids which, unlike the acids themselves, are strong electrolytes (9). An estimate of their likely infinite-dilution values, Λ∞, can thus be made. The same can be done for the strong electrolytes HCl and NaCl. Application of the concept of addition or subtraction of ionic conductances to these three estimates gives a value for Λ∞ of a particular weak acid, a value not obtainable by direct measurement. Ostwald reiterated the remarkable conclusion of the then new ionic theory of solutions (10). The laws governing the behavior of weak electrolytes are analogous to the gas laws. The reversible dissociation of a gas molecule into two portions is governed by the relationship p/ P 2 = constant
(5)
where p and P are the pressures of the undissociated and the dissociated portions, respectively. In a solution, the “pressures” are directly proportional to the amounts u (undissociated) and U (dissociated), and inversely proportional to v, the volume of the solution. Hence p :P = (u/ v):(U/ v) and eq 5 becomes u ? v/ U2 = constant
(6)
Arrhenius had shown that the amounts u and U can be calculated from conductance measurements. If Λv is the equivalent conductance at dilution, then v ? (Λ∞ – Λ v) / Λ v2 = constant
(7)
Except for change in symbols, this is Ostwald’s expression of the dilution law. Nowadays, “concentration” has replaced “dilution” and the conductance ratio, Λv/Λ∞ in the
above symbols, is now written as α = Λc/Λ0. Hence the “law” takes the form of eq 1. The dilution law breaks down when applied to solutions of strong electrolytes. A full understanding of the behavior of these was not developed until well into the 20th century. However, by 1900, Kohlrausch and Maltby had found experimentally that strong electrolytes tended to be governed by the “square root” rule Λc = Λ0 – Bc 1/2
(8)
where B is a constant for the given type of electrolyte (11). This relationship is valuable because Λ0 can be found by extrapolating a linear plot of conductance vs. c1/2. Equation 8 does not hold for weak electrolytes. During the next few years, Ostwald was greatly concerned with the Arrhenius ionic theory. That NaCl, formed from elements that combine so vigorously, should break up (i.e., ionize) when made into a solution, invited disbelief. Accordingly, Ostwald published a masterly account of the ionic theory (12). He showed that the theory was successful, nowhere contradicted the facts, and could explain some hitherto misunderstood or inexplicable relationships. Acknowledgment Part of this work was carried out under the Research Fellowship Program of the Science Museum, London. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Lund, E. W. J. Chem. Educ. 1965, 42, 548–550. Stock. J. T. Anal. Chem. 1984, 56, 561A–565A. Stock. J. T. J. Chem. Educ. 1988, 65, 700–701. Kohrausch, F.; Nippoldt, W. A. Ann. Phys. Chem. 1869, 138, 370–390. Kohlrausch, F.; Holborn, L.; Dieselhorst, H. Ann. Phys. Chem. 1898, 64, 417–455. Ostwald, W. Philosoph. Mag. 1886, 22, 104–118. Kohlrausch, F.; Heydweiller, A. Z. Phys. Chem. 1894, 14, 317–330. Ostwald, W. Z. Phys. Chem. 1888, 2, 561–567. Ostwald, W. Z. Phys. Chem. 1888, 2. 840–851. Ostwald, W. Z. Phys. Chem. 1888, 2, 36–37; 270–283. Kohlrausch. F.; Maltby, M. E. Wiss. Abhl. Phys. Techn. Reichanstalt 1990, 3, 156–227. Ostwald, W. Z. Phys. Chem. 1889, 3, 588–602.
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