«The Solubility-Product in Volumetric Analysis.»—A final instance of the application of the solubility-product principle to the ordinary methods of analytical chemistry, may be taken from the field of quantitative, volumetric analysis. A particularly accurate method of determining silver consists in precipitating it as silver chloride by means of a standardized solution of sodium chloride. The aim of the method is to recognize, as exactly as [p150] possible, the point where the action AgNO_{3} + NaCl → AgCl ↓ + NaNO_{3} has just completed itself, ‹i.e.› where one equivalent of sodium chloride has been added for just one equivalent of silver nitrate present. A very sensitive method[316] depends on the fact that silver chloride can be made to coagulate by the vigorous shaking of its suspensions, and that the coagulated chloride settles rapidly, leaving a clear supernatant liquid, in which the appearance of the faintest turbidity may be recognized, when the sodium chloride solution is carefully added to the silver nitrate solution under investigation. Now, when sodium chloride solution is added in this way to silver nitrate, a point is reached (called the "neutral point"),[317] where the addition of a further drop or two of sodium chloride solution will still produce a precipitate, and where one would be inclined to decide that too little of the chloride had been used to complete the action. But, at the same time, the addition of a few drops of silver nitrate solution to the solution at the "neutral point" also produces a precipitate of silver chloride, seemingly indicating that an excess of sodium chloride had been used, and apparently contradicting the previous result. As a matter of fact, such a behavior is exactly what is to be expected from a solution, when exactly equivalent quantities of silver nitrate and sodium chloride have been brought together in solution. The solution is then saturated with silver chloride, K_{S.P.} = [Ag^{+}] × [Cl^{−}] and [Ag^{+}] = [Cl^{−}], and the further addition ‹either› of the silver-ion (silver nitrate) ‹or› of the chloride-ion (sodium chloride) should produce a precipitate, according to the principle of the solubility-product. The correct end-point in the determination is, thus, the "neutral point," for at that point the quantity of silver present is equivalent to the quantity of sodium chloride added.