The laws of gases, it is known, are in accord with the two simple assumptions of the kinetic theory. The first assumption is that [p027] gases consist of ultimate discrete particles (molecules), which move in all directions through the space filled by the gas and, at ordinary pressures, are so far apart, that the forces of molecular attraction between them are negligible; the pressure of the gas is simply the net result of the impacts of these flying particles upon the walls of the containing vessel. The second assumption of the kinetic theory is that temperature is a function of the mean kinetic energy of the moving molecules, and that the molecules of gases of the same temperature have the same mean kinetic energy. The kinetic energy of particles is a function of their mass m and their velocity u (K.E. = ½ m u²). When a gas is heated, the kinetic energy of its molecules is increased, and, since their masses remain unchanged, their velocity must increase. As a result, the number and the force of their impacts against the walls of a given space increase, and thus the pressure is increased.

We may ask, whether this theory cannot be used to explain the connection between osmotic and gaseous pressure. If temperature is a function of the kinetic energy of the molecules of which a substance consists,—and the whole behavior of gases confirms such a conception,—then one must conclude, that the mean kinetic energy of molecules, at a given temperature, must always be the same, irrespective of whether they are present in gaseous, or liquid, or solid form, or even in solution.39 The tendency of the molecules to move, resulting from the kinetic energy inherent at a given temperature, may be largely balanced (liquids), or overcome (solids), by molecular attractions of surrounding particles, but such conditions are altogether in harmony with the conception of a definite mean molecular kinetic energy, persisting at a given temperature, irrespective of the physical surroundings of the molecule. According to the kinetic theory, then, when we have a dilute solution, say of alcohol in water, the molecules of alcohol, at a given temperature, would have a given mean kinetic energy, [p028] and would be tending to move in all directions with a mass40 and velocity, the same as if the alcohol were present as a gas or vapor at the same temperature. If the solution is sufficiently dilute, the dissolved alcohol molecules are sufficiently far apart, for average time, to make the molecular attractions between them negligible, just as is assumed for gases. As far as the alcohol (solute) molecules alone are concerned, they may, evidently, be assumed to be present in the solution, in the same condition, as to number, mean kinetic energy and mean velocity, as they would be in alcohol vapor of the same concentration and temperature. We may ask, now, whether the osmotic pressure of the solution may not result from the pressure on the solvent, growing out of its bombardment by the solute molecules. And we may ask, further, what numerical relation would subsist between such a pressure and the pressure of the solute, if the latter were present as a gas, under the same conditions of temperature and concentration. In order to be prepared to answer these questions, we must consider, in what way the presence of the solvent must modify the motions and the forces of impact of solute molecules.

One great difference between the dissolved substance and the gas would be, that, in the solution, the solute is in intimate contact with the solvent. A decided attraction must exist between the solute molecules and the solvent molecules, since we could not otherwise understand how a solvent, like water, in dissolving a nonvolatile substance like sugar, could overcome those molecular attractions between the sugar molecules, which make sugar a solid. But we note, that all the solute molecules in a solution, except those at the surface, are surrounded on all sides equally by the solvent. The attractive forces, exerted upon the single molecules of the solute by the solvent molecules, thus sum up to zero, and need not be considered further. Only the small number of solute molecules, which are at the surface of the liquid, would involve a minor correction in the application of the kinetic theory, and this need not be considered here.

A second point of difference between a substance in solution, and the same substance as a gas or vapor at the same temperature [p029] in the same volume, lies in the fact that a gas molecule will go a much greater distance without colliding with some second molecule and changing its path, than would a solute molecule, the latter molecule being closely surrounded by the molecules of the solvent. The mean free path, as it is called, will be very much shorter for a solute molecule than for a gas molecule, and we note, as a matter of fact, how slow is the diffusion through a solvent (see exp. p. 8). But the shortness of the previous path does not affect the force of a blow resulting from the impact of a moving mass, the force of the impact being dependent only on the mass and the change in speed of the striking particle, at the moment of impact. Thus the short free mean path of a dissolved molecule does not affect the mean force of the blow, delivered when it strikes the resisting medium.

But a third question, of fundamental importance in the comparison of the condition of a substance existing as a gas and its condition in a solution of the same concentration and temperature, results from a consideration of the frequency of the impacts of the solute molecules against the solvent, growing out of the reduction of the lengths of the mean free paths of the solute molecules.41 In order to be able to take this fact properly into account, it will be necessary to consider somewhat more precisely the manner in which, according to the kinetic theory, gas pressure is produced.

We may consider that we have in a cube of unit volume (1 c.c.) n molecules of a gas, each of mass m and average velocity u cm. per second. We may assume that one-third of the total number [p030] of molecules moves in each of the three dimensional directions.42 A single molecule of mass m, striking the surface with a velocity u and rebounding with the same velocity in the opposite direction, will exert on the surface a force of 2 m u units. But, with a velocity of u cm. per second, it will reach the opposite wall and return to the surface we are considering, u / 2 times in one second. A single molecule will consequently exert a force 2 m u × u / 2 or m u² on the surface, and the n / 3 molecules moving in the same direction will exert a force n / 3 × m u² on the unit surface. This represents, therefore, the pressure of such a gas, as calculated on the basis of the assumptions of the kinetic theory. Now, when a gas is so strongly compressed, that the bulk of the molecules is not negligible in comparison with the total volume of the gas, the number of impacts on unit surface in unit time becomes sensibly greater than n / 3 × u / 2, since the distance to be covered between successive blows on the surface will be sensibly less than 2 cm., in a cube of unit volume. If we imagine, for the sake of a rough illustration, that one-third of the molecules in 1 c.c. are united into one spherical mass (indicated by A in Fig. 7), moving upwards and downwards, it is obvious that the distance covered between two successive blows on a surface is not 2 cm., but that distance diminished by twice the diameter of the sphere. For strongly compressed gases, the total number of impacts on unit surface is therefore sensibly greater than n / 3 × u / 2, and the pressure is proportionately greater. According to van der Waal's correction for this effect, P = n / 3 × m u² / (1 − b), where b represents the volume actually occupied by the molecules in 1 c.c. of the gas.43

Now, for solute molecules, the "free space" of movement, as we may call it, is, similarly, very considerably reduced by the presence [p031] of the solvent, and the reduction of this free space, as Nernst has shown, will have the same effect on the pressure produced against unit surface of the solvent by the bombardment of the solvent by the solute, as the reduction of the free space has on the gas pressure when a gas is strongly compressed. The resulting pressure on unit surface of the solution must thus be increased, from the pressure P_gas, which would be exerted by the solute against the walls of a vessel, if it were present as a gas of the same concentration, at the same temperature, to P_gas / (1 − v), where v represents the real volume occupied by the solvent and (1 − v) the free space for the solute molecules in unit volume of solution.44 If osmotic pressure is the result of such a bombardment of the solvent by the molecules of the solute, one might, therefore, expect to find the osmotic pressure very much greater than the gas pressure of the same substance in the same volume at the same temperature. However, in all the experimental determinations (by means of semipermeable membrane, vapor pressure, boiling-point and freezing-point measurements) of the osmotic pressure as defined on p. 10, this corrective factor cancels out again.45 According to the kinetic theory, the osmotic pressure of a substance in dilute solution should, consequently, be found by experiment to be equal to the gas pressure which a gas, of the same molecular concentration, would exert at the same temperature.46

We find thus that the significant coincidence between the osmotic pressure of a substance in dilute solution, as defined and measured according to van 't Hoff, and the gas pressure which the substance would exert, if it were present as a gas in the same volume and at the same temperature, is in agreement with the fundamental assumptions of the kinetic theory. This theory, consequently, gives us an adequate theoretical explanation of [p032] osmotic pressure, as it does of gas pressure. As van 't Hoff says,47 "if the osmotic pressure follows Gay-Lussac's law and is proportional to the absolute temperature, then, like gas pressure, it will become zero at 0° absolute temperature and will vanish when molecular movements come to rest. It is therefore natural to look for the cause of osmotic pressure in kinetic phenomena and not in attractions."48

CHAPTER IV THE THEORY OF IONIZATION; IONIZATION AND ELECTRICAL CONDUCTIVITY

Of the laws and hypotheses concerning gases, the one that is perhaps of most importance to chemistry is Avogadro's hypothesis. With the aid of this hypothesis, we are able to determine the relative molecular weights49 of such elements and compounds as are gases, or are volatile at higher temperatures. If equal volumes of gases, under the same conditions of temperature and pressure, contain the same number of molecules, then the weights of such equal volumes also represent the relative weights of the molecules composing the gases. As a standard, for expressing the relative molecular weights in definite numbers, the molecular weight of oxygen is taken by convention to be 32, and all other molecular weights are expressed in terms of this standard. The density, or weight of one liter of oxygen at 0° and 760 mm., is 1.429 grams, and the molecular weight expressed in grams (molar weight) of oxygen, 32 grams, occupies, therefore, 32 / 1.429, or 22.4 liters. The weights of this same volume, 22.4 liters, of gases and vapors, calculated for 0° and 760 mm. pressure,50 express then directly, in terms of the oxygen standard, the relative molecular weights of the elements or compounds forming the gases. The weights themselves give us directly their gram-molecular or molar weights.

When molecular weights are determined in this way, with the aid of Avogadro's hypothesis, results are obtained which agree [p034] perfectly with the chemical behavior of the compounds or elements in question. The molecular weights of hydrogen chloride, water, ammonia, and marsh gas, for instance, are found to be 36.5, 18, 17 and 16, respectively, corresponding to the formulæ51 HCl, H₂O, NH₃ and CH₄, and in confirmation of these results we find, by methods used especially in organic chemistry, that these compounds show a chemical behavior agreeing perfectly with the presence of one, two, three and four hydrogen atoms, respectively, in their molecules. Marsh gas, for instance, by treatment with chlorine, yields a monochloride, CH₃Cl, a dichloride, CH₂Cl₂, a trichloride (chloroform), CHCl₃, and a tetrachloride, CCl₄. Water, by proper treatment, may be converted in successive stages into alcohol, (C₂H₅)OH, and then into ether, (C₂H₅)O(C₂H₅), or into sodium hydroxide, NaOH, and sodium oxide, Na₂O.

While the application of Avogadro's hypothesis thus gives results agreeing well with the observed chemical behavior of very many important compounds, observations have been made which, at first sight, do not appear to agree with the requirements of the hypothesis and which seem to raise a doubt as to the universal truth of its fundamental assumption. Thus, if equal volumes of hydrogen chloride and ammonia, of the same temperature and pressure, are brought together, ammonium chloride is formed, both gases being totally consumed. Since, according to the hypothesis, equal volumes, under the conditions obtaining, contain the same numbers of molecules, the formation of ammonium chloride takes place according to the equation NH₃ + HCl → NH₄Cl, and we should anticipate that the molecular weight of [p035] ammonium chloride would be 17 + 36.5 or 53.5. However, when the molecular weight is determined by obtaining the weight of a measured volume of ammonium chloride vapor, at a temperature sufficiently high to vaporize the salt, and the observations are reduced to standard conditions of temperature and pressure, 26.75 grams is found as the calculated weight of 22.4 liters, and this weight, according to this hypothesis, should be the molecular weight of the chloride. This contradiction in two conclusions, each reached by the application of Avogadro's hypothesis to experimental observations, would, at the first glance, make one hesitate to accept the hypothesis as representing a universal truth; it might seem as if in some gases, such as ammonium chloride vapor, there might be only half as many molecules in a given volume as in the same volume of the majority of gases.

The gaseous dissociation of other ammonium salts, of phosphorus pentachloride and pentabromide (PX₅ ⇄ PX₃ + X₂), and of a number of less common compounds, has been demonstrated in similar ways. As a result of the study of each case, the important conclusion has been reached that, as far as our knowledge goes, there are no exceptions to Avogadro's hypothesis, and this hypothesis seems therefore to represent a universal truth.56

The fact that all solvents and all solutes are included in this hypothesis, with the sole limiting condition that the solution must be dilute, is one of great significance and of greatest practical importance, as we may use any suitable solvent for determinations.

When molecular weights are determined in this way, a very large number of compounds give the same molecular weight by the solution method as by the gas method. For instance we have:

Further, the molecular weight of glucose is found in aqueous solutions to be 180, conforming to the formula C₆H₁₂O₆, and agreeing with the molecular weight as obtained by a chemical study of compounds derived from glucose.

While there are, then, very many agreements in the molecular weights determined by the solution and by the older methods, it was recognized, at the outset,58 that there is also a large number of apparently abnormal cases, in which, in particular, much lower molecular weights are obtained by the solution methods than by the gas method,—lower even than the weights consistent with the accepted atomic weights of the elements in the compounds in question.59 For instance, we find 36.5 to be the molecular weight of hydrogen chloride in the gas form, but in aqueous solution its apparent molecular weight, as determined on the basis of van 't Hoff's hypothesis, is not even a constant; it is found to be less than 36.5 and approaches the limit 18.25, the more dilute the solution, [p038] the lower being the apparent molecular weight.60 For sodium chloride, the formula weight, corresponding to the formula NaCl, is 58.5. This would also represent its smallest molecular weight in gas form, consistent with the accepted atomic weights for sodium and chlorine. In aqueous solution, again, the apparent molecular weight of sodium chloride is found to be less than 58.5, and more than 29.25, the value found depending on the concentration of the solution used. For zinc chloride we have, likewise, in aqueous solution values much less than 136 and tending toward the limit 45, whereas the formula weight for ZnCl₂ is 136.

These are instances of a very large class of apparent gross discrepancies between the requirements of the Avogadro-van 't Hoff principle and the generally accepted molecular weights of common compounds. There are three ways, in particular, in which one might be inclined to regard such results: in the first place, one might be tempted to consider that van 't Hoff's extension of Avogadro's hypothesis to solutions is justified in a considerable number of cases, but not as a universal expression, applicable to all dilute solutions. This seems, indeed, to have been van 't Hoff's own attitude originally. Such a view, since it does not throw new light on the matter, but simply shelves the question of the source of the discrepancy, would be tenable only after all other explanations had been found unsatisfactory.

In the second place, we might be inclined to consider whether a molecule like hydrogen chloride is not dissociated in aqueous solution into two smaller molecules, hcl, in which hydrogen and chlorine would appear as atoms with the weights h = 0.5 and cl = 17.75, which are half as large as the atomic weights determined from a study of volatile compounds of hydrogen and chlorine. If we remember that our atomic weights are confessedly maximum weights, and not minimum weights—although they are almost certainly also the true atomic weights—such a view would be, at least, worthy of some consideration. But, in the first place, it would be extraordinary that we should never have found, in the thousands of [p039] hydrogen derivatives that have been investigated, any compound, the molecule of which, in the gaseous condition, contained a single such atom of hydrogen, with the weight 0.5, or an uneven multiple of it: that only even multiples or pairs h₂, corresponding to the atom H, should always have been found. In the second place, such an explanation of the results of the molecular weight determinations in aqueous solutions given above, would soon lead to difficulties, which make the view altogether untenable. For instance, the molecule of zinc chloride, according to the data given, would have to break down into three molecules and, if these were of uniform composition, we would have to assume chlorine atoms two-thirds or one-third as large as Cl. Since a moment ago we had to assume chlorine atoms one-half as large as Cl, we would have to conclude that the atomic weight of chlorine could be, at most, Cl / 6, which is the largest common divisor of Cl / 2 and Cl / 3. No chemist would seriously consider an atomic weight for chlorine one-sixth as large as the accepted weight, for that would mean that, in all the chlorine compounds investigated in the condition of gases, we have always at least six such atoms occurring together, and otherwise always multiples of six. Consequently such an interpretation of the so-called "abnormal" behavior of solutions of hydrogen chloride, sodium and zinc chlorides, etc., although at one time advanced by some chemists, must be considered as altogether untenable.

A third explanation of the "abnormally" low molecular weights, which certain substances in aqueous solutions possess, is, that the molecules of these compounds are capable of dissociation into smaller molecules of unlike composition, somewhat like ammonium chloride when it is heated, and that the substances in question are dissociated more or less considerably in this fashion in the solutions under consideration. Hydrogen chloride, for instance, besides existing as such (as HCl), in aqueous solutions, might be capable of dissociating, and actually be dissociated, to a considerable degree into molecules containing either only hydrogen or only chlorine (HCl ⇄ H + Cl); the average of the weights of the molecules in a mixture of molecules, HCl, H, and Cl, would be less than 36.5, and, according to the proportion of dissociated and undissociated molecules of hydrogen chloride, the average would lie between the limits 36.5 and (1 + 35.5) / 2, or 18.25. Such an [p040] explanation,61 made with certain additions and restrictions, was advanced in 1885 by Arrhenius, a Swedish chemist and physicist, when he learned of the exceptional behavior of these solutions, as noted by van 't Hoff. Although at first this interpretation occasioned considerable criticism, it has maintained itself successfully for twenty years, on the basis of a wide range of accumulated facts, and it has been of remarkable value and benefit in the development of all branches of chemistry and the allied sciences.

This simple fact, that the very solutions which give abnormally low molecular weights for the dissolved compounds are also good conductors of electricity, was explained by a theory of electrolytic dissociation or of ionization, which Arrhenius had developed62 from a study of the conductivity of electrolytes. The same fact has aided in establishing this theory which has led to the elucidation of vital problems of electrical conductivity and to a successful [p041] explanation of the problem of the apparently abnormal osmotic pressures (and molecular weights) of electrolyte solutes. It has thus removed the last difficulty in the way of accepting the van 't Hoff-Avogadro Hypothesis (p. 15) as true for all dilute solutions, exactly as the discovery of gaseous dissociation made it possible to recognize in the original Avogadro Hypothesis a universal truth (p. 36) about gases. And to these results was added, chiefly as the fruit of the work of Ostwald, with the aid of the theory of Arrhenius, the most successful and accurate formulation of the problem of the chemical activity of electrolytes, known in the history of chemistry.

When a current is passed through the solution of an ionogen, the electrified particles carry their charges to the electrodes (see [p042] below). They are called the ions64 of the electrolyte; the positively charged ions are distinguished as cations from the negatively charged anions, and the electrode toward which the cations move is called the cathode (negative electrode), and the electrode to which the anions move is called the anode (positive electrode).

The dissociation of hydrogen chloride may be expressed, in the terms of the assumptions made, in the following equation: HCl ⇄ H⁺ + Cl⁻; that is, hydrogen chloride is dissociated, to a greater or smaller extent and in reversible fashion, into positively charged hydrogen ions H⁺, and negatively charged chloride ions Cl⁻, and the charge on each chloride ion is equal in quantity to the positive charge on each hydrogen ion. Zinc chloride is dissociated according to the equation ZnCl₂ ⇄ Zn²⁺ + 2 Cl⁻, and, according to (2), the charge on each zinc ion is twice as great in quantity as the charge on each chloride ion, and therefore twice as great also as the charge on each hydrogen ion (see below, p. 58). It is practically certain, according to more recent results, that the ions are combined with water to form hydrates, such as H⁺(H₂O)_x and Cl⁻(H₂O)_y.65 This does not modify, essentially, the fundamental assumptions of the theory, but contributes rather to a satisfactory explanation of the rôle of water as an ionizing agent, a question to which we shall return later.

One of the most fundamental and most characteristic properties of elements is considered to be the affinity which their atoms show for electrons; thus, the atoms of metals like sodium and potassium, which are generally called "electropositive" elements,68 show an enormous tendency to lose one electron each and to form positively charged particles69 Na^-ε (= Na⁺) and K^-ε (= K⁺).70 The atoms of strongly electronegative elements, like chlorine, have a tremendous tendency for gaining and holding electrons beyond the number originally in such atoms. Thus, chlorine atoms tend to assume an electron each; they thereby become negatively charged particles, Cl^+ε ( = Cl⁻).

On the basis of these views, we have in sodium chloride NaCl a substance, whose molecules contain an atom, Na, with a tremendous tendency to lose an electron, and an atom, Cl, which has a tremendous affinity for an electron. It is natural to suppose, then, that both tendencies will be satisfied by the passage of an electron from the sodium to the chlorine atom, NaCl → Na^-εCl^+ε. Or, if we use the sign + to designate the positive charge produced on an atom by the loss of an electron and the sign − to indicate the charge gained through the assumption of an electron, we have71: NaCl is Na⁺Cl⁻. Similarly we have in hydrogen chloride H^-εCl^+ε or H⁺Cl⁻. It is altogether likely, therefore, that the atoms in a molecule of sodium chloride or of hydrogen chloride already possess electric charges,72 so that, even while combined, [p044] their tendencies to lose or gain electrons are satisfied. It is also possible that the atoms are held together in the molecule by the electrical attraction of the opposite charges.73 The force with which opposite electrical charges attract each other depends, as is well known, on the nature of the surrounding medium. Now, when molecular sodium chloride or hydrogen chloride is dissolved in water (a favorable medium), a decided decrease in the attraction (see p. 62), between the charged atoms within the molecules is brought about, and a process of ionization results: H⁺Cl⁻ ⇄ H⁺ + Cl⁻. The charged particles are called ions only after they have separated from one another and have become independent molecules, capable, for example, of moving in opposite directions.

While the atoms of some metallic elements tend to lose a single electron and form ions Me⁺ (e.g. Na⁺, K⁺), the atoms of other elements tend to lose two or more electrons, forming bivalent ions, Me²⁺ (e.g. Zn²⁺, Fe²⁺, etc.), or trivalent ions, Me³⁺ (e.g. Bi³⁺, Fe³⁺), and so forth. Similarly, atoms of the so-called negative elements may assume two or more electrons, forming bivalent ions, X²⁻ (e.g. S²⁻), and so forth.