[Note 2:  The ionic changes involved in the precipitation of barium sulphate are very simple: 

Ba^{++} + so_{4}^{—­} —­> [BaSO_{4}]

This case affords one of the best illustrations of the effect of an excess of a precipitant in decreasing the solubility of a precipitate.  If the conditions are considered which exist at the moment when just enough of the Ba^{++} ions have been added to correspond to the so_{4}^{—­} ions in the solution, it will be seen that nearly all of the barium sulphate has been precipitated, and that the small amount which then remains in the solution which is in contact with the precipitate must represent a saturated solution for the existing temperature, and that this solution is comparable with a solution of sugar to which more sugar has been added than will dissolve.  It should be borne in mind that the quantity of barium sulphate in this !saturated solution is a constant quantity! for the existing conditions.  The dissolved barium sulphate, like any electrolyte, is dissociated, and the equilibrium conditions may be expressed thus: 

(!Conc’n Ba^{++} x Conc’n so_{4}^{—­})/(Conc’n BaSO_{4}) = Const.!,

and since !Conc’n BaSO_{4}! for the saturated solution has a constant value (which is very small), it may be eliminated, when the expression becomes !Conc’n Ba^{++} x Conc’n so_{4}^{—­} = Const.!, which is the “solubility product” of BaSO_{4}.  If, now, an excess of the precipitant, a soluble barium salt, is added in the form of a relatively concentrated solution (the slight change of volume of a few cubic centimeters may be disregarded for the present discussion) the concentration of the Ba^{++} ions is much increased, and as a consequence the !Conc’n so_{4}! must decrease in proportion if the value of the expression is to remain constant, which is a requisite condition if the law of mass action upon which our argument depends holds true.  In other words, so_{4}^{—­} ions must combine with some of the added Ba^{++} ions to form [BaSO_{4}]; but it will be recalled that the solution is already saturated with BaSO_{4}, and this freshly formed quantity must, therefore, separate and add itself to the precipitate.  This is exactly what is desired in order to insure more complete precipitation and greater accuracy, and leads to the conclusion that the larger the excess of the precipitant added the more successful the analysis; but a practical limit is placed upon the quantity of the precipitant which may be properly added by other conditions, as stated in the following note.]