# 1911 Encyclopædia Britannica/Stoichiometry

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​STOICHIOMETRY (Gr. στοιχεῖα , fundamental parts, or elements, μέτρον, measure), in chemistry, a term introduced by ​ Benjamin Richter to denote the determination of the relative amounts in which acids and bases neutralize one another; but this definition may be extended to include the determination of the masses participating in any chemical reaction. The work of Richter and others who explored this field is treated under Element; here we discuss a particular branch of the subject, viz. the determination of equivalent and atomic weights of elements, and the molecular weights of elements and compounds. Reference to Chemistry, Atom and Element will explain the principles involved. Every element has an " equivalent weight " which is usually defined as the amount of the element which combines with or replaces unit weight of hydrogen; the " atomic weight " may be regarded as the smallest weight of an element which can be present in a chemical compound, and the " molecular weight " is the weight of the least part of an element or compound which can exist alone. The atomic weight is therefore some multiple of the equivalent weight, and the determining factor is termed the valency (q.v.) of the element. We have mentioned hydrogen as our standard element, which was originally chosen as being the lightest known substance; but Berzelius, whose stoichiometric researches are classical, having pointed out that few elements formed stable compounds with hydrogen, and even these presented difficulties to exact analysis, proposed to take oxygen as the standard. This suggestion has been adopted by the International Committee of Atomic Weights, who take the atomic weight of oxygen as 16.000, hydrogen being 1.0087.
Deferring the discussion of gaseous elements and compounds we will consider the modus operandi of determining, first, the equivalent weight of an element which forms solid compounds, and, secondly, its atomic weight. Suppose we can cause our element in known quantity to combine with oxygen to form a definite compound, which can be accurately weighed, or, conversely, decompose a known weight of the oxide into its constituents, of which the element can be weighed, then the equivalent weight of the element may be exactly determined. For if x grams of the element yield y grains of the oxide, and if W be the equivalent of the element, we have x grams of the element equivalent to y–x grams of oxygen, and hence the equivalent weight W, which corresponds to 8 grams of oxygen, is given by the proportion y–x : x :: 8 : W, i.e. W = 8x/(y–x). For example, Lavoisier found that 45 parts of red oxide of mercury on heating yielded 411/2 parts of mercury; hence 411/2 parts of mercury is equivalent to 45–411/2 = 31/2 parts of oxygen, and the equivalent of mercury in this oxide is therefore 8 $\times$ 411/2 ÷ 31/2 = 95. The question now arises: is this value the true equivalent, i.e. half the amount of mercury which combines with one atom of oxygen (for one atom of oxygen is equivalent to two atoms of hydrogen)? Before considering this matter, however, we will show how it is possible to obtain the equivalent of elements whose oxides are not suitable for exact analysis. No better example can be found than Stas's classical determination of the atomic weight of silver and of other elements. It will be seen that the routine necessary to the chemical determination of equivalents consists in employing only such substances as can be obtained perfectly pure and stable (under the experimental conditions), and that the reactions chosen must be such as to yield a series of values by which any particular value can be checked or corrected.
Stas's experiments can be classified in five series. The object of the first series was to obtain the ratio Ag:O by means of the ratios KCl:O and Ag:KCl. The ratio KCl:O was determined by decomposing a known weight of potassium chlorate (a) by direct heating, (b) by heating with hydrochloric acid and weighing the residual chloride. The reaction may be written for our purpose in the form: KClO$_3$ = KCl+3O; in case a the oxygen is liberated as such; in case b it oxidizes the hydrochloric acid to water and chlorine oxides. The equation shows that one KCl is equivalent to 3O, and hence if x grams of chlorate yields y grams of chloride, then the ratio KCl:O = y/(x–y). Taking O as 16 and the experimental value of x and y, Stas obtained KCl:O = 74.9502. To find the ratio of Ag:KCl, a known weight of silver was dissolved in nitric acid and the amount of potassium chloride necessary for its exact precipitation, was determined. The reaction may be written as AgNO$_3$+KCl = AgCl+KNO$_3$, which shows that one Ag is equivalent to one KCl. The value found was Ag:KCl = 1.447110. The ratio Ag:O is found by combining these values, for Ag:O = KCl:O $\times$ Ag:KCl = 74.9502 $\times$ 1.44710 = 107.9401.
In the second series the ratios AgCl:O and AgCl:Ag were obtained, the first by decomposing the chlorate by heating, and the second by synthesizing the chloride by burning a known weight of the metal in chlorine gas and weighing the resulting chloride, and also by dissolving the metal in nitric acid and precipitating it with hydrochloric acid and ammonium chloride. These two sets yield the ratio Ag:O, and also the ratio Cl:O, which, combined with the ratio KCl:O obtained in the first series, gave the atomic weight of potassium. The third and fourth series resembled the second, only the bromate and bromide, and iodate and iodide were worked with. The experiments gave additional values for Ag:O and also the atomic weights of bromine and iodine.
The fifth series was concerned with the ratios Ag$_2$SO$_4$:Ag:Ag$_2$S:Ag and Ag$_2$S:O. The first was obtained by reducing silver sulphate to the metal by hydrogen at high temperatures; the second by the direct combination of silver and sulphur, and also by the interaction of silver and sulphuretted hydrogen; these ratios on combination gave the third ratio Ag$_2$S:O. These experiments besides giving values for Ag:O, yielded also the atomic weight of sulphur. There is no need to proceed any further with Stas's work, but it is sufficient to say that the general routine which he employed has been adopted in all chemical determinations of equivalent weights.
The derivation of the atomic from the equivalent weight may be effected in several ways. The simplest are perhaps by means of Dulong and Petit's law of atomic heats (and by Neumann's extension of this law), and by Mitscherlich's doctrine of isomorphism. Dulong and Petit's law may be stated in the form that the product of the specific heat and atomic weight is approximately 6.4, or that an approximate value of the atomic weight is 6.4 divided by the specific heat. This application may be illustrated in the case of mercury. We have seen above that the red oxide yields a value of about 95 for the equivalent; but a green oxide is known which contains twice as much metal for each part of oxygen, and therefore in this compound the equivalent is about 190. The specific heat of mercury, however, is 0.033, an d this number divided into 6.4 gives an approximate atomic weight of 194. More accurate analyses show that mercury has an equivalent of 100 in the red oxide and 200 in the green; Dulong and Petit's law shows us that the atomic weight is 200, and that the element is divalent in the red oxide and monovalent in the green. For exceptions to this law see Chemistry: § Physical.
The application of isomorphism follows from the fact that chemically similar substances crystallize in practically identical forms, and, more important, form mixed crystals. If two salts yield mixed crystals it may be assumed that they are similarly constituted, and if the formula of one be known, that of the other may be written down. For example gallium sulphate forms a salt with potassium sulphate which yields mixed crystals with potash alum; we therefore infer that gallium is trivalent like aluminium, and therefore its atomic weight is deduced by multiplying the equivalent weight (determined by converting the sulphate into oxide) by three. General chemical resemblances yield valuable information in fixing the atomic weight after the equivalent weight has been exactly determined.
Gases.—The generalization due to Avogadro—that equal volumes of gases under the same conditions of temperature and pressure contain equal numbers of molecules—may be stated in the form that the densities of gases are proportional to their molecular weights. It therefore follows that a comparison of the density of any gas with that of hydrogen gives the ratio of the molecular weights of the two gases, and if the molecular contents of the gases be known then the atomic weight is determinable. Gas reactions are available in many cases for solving the question whether a molecule is monatomic, diatomic, &c. Thus from the combination of equal volumes of hydrogen and chlorine to form twice the volume of hydrochloric acid, it may ​be deduced that the molecule of hydrogen and of chlorine con- tains two atoms (see Atom); and similar considerations show that oxygen, nitrogen, fluorine, &c, are also diatomic. Physical methods may also be employed. For instance, in monatomic gases the ratio of the specific heat at constant pressure to the specific heat at constant volume is 1.66; in diatomic gases 1.42; with other values for more complex molecules (see Molecule). This ratio may be determined directly by finding the velocity of sound in the gas (Kundt) or by other methods, or indirectly by finding the specific heats separately and then taking the ratio. It is found that the gases just mentioned are diatomic, whereas argon, helium, neon and the related gases, and also mercury and some other metals when in the gaseous condition, are monatomic. A knowledge of the atomicity of a gas combined with its density (compared with oxygen and hydrogen) would therefore give its atomic weight if Avogadro's law were rigorously true. But this is not so, except under extremely low pressures, and it is necessary to correct the observed densities. The correction involves a detailed study of the behaviour of the gas over a large range of pressure (presuming the densities are already corrected to 0°), and may be conveniently written in the form $a$ = ${\frac {1}{pv}}$${\frac {d(pv)}{dp}}$ Thus if D be the observed relative densities of a gas to hydrogen at 0° and under normal atmospheric pressure, a$_X$ and a$_H$ the coefficients of the gas and hydrogen, then the true density, or ratio of molecular weights, is D $\times$ (1 + a$_X$ )/(1 + a$_H$).
Lord Rayleigh and D. Berthelot have corrected several molecular weights in this fashion. The importance is well shown in the modification of Morley's observed density of oxygen, viz. 15.90, which, with Rayleigh's values of a$_O$ = –0.00094 and a$_H$ = +0.00053, gives the corrected density as 15.88. And this value is the atomic weight, for both hydrogen and oxygen molecules contain two atoms. Compound gases can also be experimented with. For example Gray (Journ. Chem. Soc., 1905, 87, p. 1601) found that it was easier to prepare perfectly pure nitric oxide than to obtain pure nitrogen, and he therefore determined the density of this gas from which the atomic weight of 14.012, or, corrected for deviations from Avogadro's law, 14.006, was deduced.
The principle indicated here is applicable to the determination of the molecular weight of any vaporizable substance by the so-called method of vapour-density (see Density).
Solutes.—The theory of solution permits the investigation of the molecular weights of substances which dissolve in water or some other solvent. It is shown in Solution that a solute lowers the freezing point and raises the boiling point of the solvent in a regular manner as long as dilute solutions are dealt with. It has been shown that if one gram molecule of a solute be dissolved in 100 grams of solvent then the boiling point is raised by 0.02 T$^2$/w. (say D) degrees, where T is the absolute boiling point and w the latent heat of vaporization of the solvent; this constant is known as the molecular rise of the boiling point, and varies from solvent to solvent. If we dissolve say m grams of a substance of molecular weight M in 100 grams of the solvent and observe the elevation in the boiling point, then M is given by M = mD/d. Similar considerations apply to the freezing points of solutions. In this case D = 0.02 T$^2$/w, where T is the absolute freezing point of the pure solvent and w the latent heat of solidification. To apply these principles it is only necessary therefore to determine the freezing (or boiling) point of the solvent (of which a known weight is taken), add a known weight of the solute, allow it to dissolve and then notice the fall (or rise) in the freezing (or boiling point), from which values, if the molecular depression (or elevation) be known, the molecular weight of the dissolved substance is readily calculated.
The following are the molecular depressions and elevations (with the freezing and boiling points in brackets) of the commoner solvents.
Molecular depressions: aniline (6°), 58.7; benzene (5.4°), 50.0; acetic acid (17.0°), 39.0; nitrobenzene (5.3°), 70.0; phenol (40 ), 72; water (0°), 18.5.
Molecular elevations: acetic acid (118.1°), 25.3; acetone (56°), 17.1; alcohol (78°), 11.7; ether (35°), 21.7; benzene (79°), 26.7; chloroform (61°), 35.9; pyridine (115°), 29.5; water (100°), 5.1.
The apparatus used in cryoscopic measurements is usually that devised by Beckmann (Zeit. phys. Chem. ii. 307). The working part consists of a tube 2–3 cms. in diameter, bearing a side tube near the top; the tube is fitted with a cork through which pass a differential thermometer of a range of about 6° and graduated in 50ths or 100ths, and also a stout platinum wire to serve as a stirrer. The lower part of the tube is enclosed in a wider tube to serve as an air-jacket, and the whole is immersed in a large beaker. The thermometer is adjusted so that the freezing point of the pure solvent comes near the top of the scale. A weighed quantity of the solvent is placed in the inner tube, and the beaker is filled with a freezing mixture at a temperature a few degrees below the freezing point of the solvent. The thermometer is inserted and both solvent and freezing mixture are stirred. When the temperature is about 0.3° below the correct freezing point the tube is removed from the beaker and the stirring continued. There ensues a further fall in the thermometer reading until ice separates, whereupon the temperature rises to the correct freezing point. The ice is then melted and the operation repeated so as to obtain a mean value. A known weight of the substance is introduced through the side tube, and the freezing point determined as with the pure solvent. The difference of the readings gives the depression; and from this value, knowing the weight of the solute and solvent, and also the molecular depression, the molecular weight can be calculated from the formula given above.
In the boiling point apparatus of Beckmann the solvent is contained in a tube fitted with side tubes to which spiral condensers can be attached; the neck of the tube carries a stopper through which passes a delicate differential thermometer, whilst the bottom is perforated by a platinum wire and contains glass beads, garnets or platinum foil to ensure regular boiling. The tube is surrounded by a jacket mounted on an asbestos box, so that the heating is regular. In conducting a determination the thermometer is adjusted so that the boiling point of the pure solvent is near the bottom of the scale. A known weight of the solvent is placed in the tube, the thermometer is inserted (so that the liquid completely covers the bulb), and the condensers put into position. The liquid is now cautiously heated, and when the thermometer becomes stationary the boiling point is reached. The temperature having been read, the apparatus is allowed to cool slightly, and the observation repeated. A known weight of the substance is now introduced, and the solution so obtained treated in the same fashion as the original solvent.
A different procedure wherein the boiling tube is heated, not directly, but by a stream of the vapour of the pure solvent, was proposed by Sakurai (Journ. Chem. Soc., 1892, 61, p. 994). Sakurai's apparatus has been considerably modified, and the form now principally used is essentially due to Landsberger (Ber., 1898, 31, p. 461). The boiling vessel is simply a flask fitted with a delivery tube, which is connected with the measuring tube. This consists of a graduated tube fitted with a stopper through which passes a thermometer and an inlet tube reaching nearly to the bottom. The measuring tube is surrounded by an outer tube which has an exit to a condenser at the side or bottom, communication being made between the measuring tube and jacket by a small hole near the top of the former. In outline the operation consists in placing some solvent in the measuring tube and passing in vapour until the condensed liquid falls at the rate of one drop per second or two seconds. The temperature is then read off. A known weight of the substance is introduced and the boiling point determined as before; but immediately the temperature is read the tube must be disconnected, so that no more vapour passes over and so alters the concentration of the solution. Two methods are in use for determining the quantity of the solvent. Landsberger weighed the tube; Walker and Lumsden (Journ. Chem. Soc., 1898, 73, p. 502) graduated the tube and thus measured the volume of the solvent; in W. E. S. Turner's apparatus (Journ. Chem. Soc., 1910. 97, p. 1184) both the weight and volume can be determined. Whilst the calculations in both the Beckmann and Sakurai-Landsberger methods are essentially the same the " molecular elevations " differ according as one deals with 100 grams or 100 ccs. of solvent. In all these methods it is necessary to carefully choose the solvent in order to avoid dissociation or association. For example, most salts are dissociated in aqueous solution ; and acids are bi-molecular in benzene but normal in acetic acid.
Other methods are available for dissolved substances such as measurements of the osmotic pressure, lowering of the vapour pressure and diminution of solubility, but these are little used. Mention may also be made of Ramsay and Shield's method of finding the molecular weights of liquids from surface tension measurements. (See Chemistry: §Physical.)
* ↑ We may here state that the equivalent weight of oxygen on this basis is 8.000, i.e. one half of its atomic weight. This matter is considered below.
* ↑ The formulae used in the following paragraph were established before Stas began his work; and as oxygen is taken as 16, the results are atomic and not equivalent weights.