II
This second action would mean that we could obtain reactions of silver ions, such as the precipitation of silver sulphide, without the intermediate formation of the free ions themselves.
The consequences of the first, the ordinary, conception of such actions as direct actions of silver ions, have been analyzed by Haber[465] from the point of view of the velocities of the actions, by which the complex must be formed from its components and be decomposed into them, in order to satisfy the facts concerning the precipitations. [p233]
We may consider, with Haber, a liter of a 0.05 molar solution of K2Ag(CN)3, containing an excess[466] of 0.95 mole potassium cyanide. In such a solution, the concentration of silver-ion is reduced to 8E−24 gram-ion per liter. Now, according to the best determinations of the ultimate dimensions of molecules, about 1024 molecules are estimated to be contained in a mole (gram-molecule), and 1024 ions, therefore, in a gram-ion (e.g. in 108 grams of silver-ion there would be 1024 individual silver ions). Then a liter of the solution we are considering would contain, at any moment, only eight individual silver ions, which are different ones from moment to moment, since the reversible reactions Ag+ + 3 CN− ⇄ Ag(CN)32−, are going on continually. Thus 100 c.c. of the solution would not contain even one silver ion all the time, but the requirements of the equilibrium conditions could be met[467] by silver ions "flashing up and disappearing" in such a way, that the required average concentration in unit time is maintained. There is nothing irrational in such a conception.
One may ask, however, what must be the velocities, with which the complex is formed from the components, and is resolved into them, in order to satisfy an instability constant[466] 10−22 and still enable us to obtain a practically instantaneous precipitation, say of silver sulphide, the action being analyzed on the basis of the ordinary conception that only the silver-ion itself, and not the complex ion, is directly active in the formation of the silver sulphide. A condition of equilibrium, in a reversible action, implies that the velocities of the two continuous, opposed reactions are equal (p. [94]). For the action Ag+ + 3 CN− → Ag(CN)32− the velocity of formation of the complex is proportional to a characteristic constant, KFormation, to the concentration, [Ag+], of the silver-ion, and to the third power (see p. [94]) of the concentration, [CN−], of the cyanide-ion. The velocity of the opposed reaction of decomposition of the complex is proportional to another characteristic constant, KDecomposition, and to the concentration, [Ag(CN)32−], of the complex ion. For the condition of equilibrium, the velocities of the opposed reactions are equal, and we derive the relation:
| [Ag+] × [CN−]3 | = | KDecomposition | = | 1 | . |
| [Ag(CN)32−] | KFormation | 1022 |
The equilibrium constant of the complex ion is, then, the ratio of the velocity constants of its decomposition and formation (see p. [94]). Now, the velocity constant, KFormation, represents the concentration, in moles, of the complex ion [Ag(CN)32−],that is formed in unit time from unit concentrations of its components Ag+ and CN−, and it may be considered as the reciprocal of a time constant, TFormation, the time required to form unit concentration of the complex ion, while the components are maintained at unit concentration. The analogous reciprocal relation holds for the velocity constant, KDecomposition, and a time constant, TDecomposition. The equilibrium equation, therefore, expresses also the following relations:[468] [p234]
| [Ag+] × [CN−]3 | = | TFormation | = | 1 | . |
| [Ag(CN)32−] | TDecomposition | 1022 |
In words, the time required for the spontaneous decomposition of one mole of the complex is 1022 times as long as the time required to form one mole of the complex, from uniformly unit concentrations of the components. If the concentration of silver-ion is reduced to 1 / 1022 and the concentrations of the cyanide-ion and the complex ion are maintained at 1, the formation of the complex takes place 1022 times as slowly as when [Ag+] = 1, and a condition of equilibrium is produced, the time required to decompose and to form the same amount of the complex being now equal.
This relation of time constants may be used to obtain some idea of the consequences of assuming certain limiting values for one or the other, the ratio being maintained at the value 1 / 1022. If the time constant TFormation for the formation of the complex be taken as one ten-thousandth of a second,[469] then, according to Haber, a molar solution of potassium argenticyanide would not be able to form in thousands of years sufficient silver ions to be discovered by any direct test, a result which is not compatible with the precipitation of silver sulphide and of metallic silver in a few minutes, since silver ions could not be supplied rapidly enough. It is evident, thus, that the ratio 1 / 1022 must indicate an exceedingly small value for TFormation, if only silver ions form silver sulphide and silver.