The SIMPLIFICATION of CERTAIN CALCULATIONS BASED on the MASS LAW JOSEPH D. PARENT kY0llaUniversity of Chicago, Chicago, Illinois
I
N CASES where the utmost accuracy is not necessary in mass law calculations, certain simplifications are in order. Thus, one commonly uses stoichiometric concentrations instead of activities, especially in such instances where activity coefficients are unknown and may not be readily or accurately determined. If this is done, the accuracy of such computations is lowered to an extent which depends on the numerical values of the ignored coefficients and their arrangement in the mass law equation. Granting that circumstances exist, which limit the exactness desired or possible in such calculations, one may, in addition to using concentrations instead of activities, simplify the mathematical equations to a degree consistent with this reduction in accuracy. The equations for dissociation and hydrolysis, in particular, may he easily treated in this fashion. Let us consider the case of a simple dissociation. The mass law expression, based on concentrations, for the reaction AB e A B is:
+
If Co is the original concentration before dissociation occurs, and (Co - x) is th6 equilibrium value and if there is no common material from another source, equation (1) becomes:
On solving by means of the quadratic formula, one obtains as the only physically acceptable solution:
These equations are particularly suitable for dissociations of weak electrolytes in aqueous solution, equation
( 4 ) being given by Kolthoff and Sandell' and others for the dissociation of a weak acid or base. Equation (3) may be simplified in a manner similar to that to be given below for the more common and useful equation (4). The latter may be rearranged as follows:
Now if
Co
2 25, the first term under the radical ob-
A-
viously cannot exceed in value one per cent. of the second term under the radical, and may be dropped, if no greater accuracy than this is required in the final answer. This being so, equation (5) may be written:
This is a very useful equation for solving dissociation problems in elementary quantitative analysis, but cannot be found in the textbooks on that subject. If, Co however, - > 2500, the value of the square root term
K -
in equation (5) becomes 2 100, and the first term may be dropped, as it cannot be greater than one per cent. of this. Equations (5) and (7) may now be reduced to the expression: x
-
-
~
D
(8)
Such an equation is very commonly used for the disso' KOLTEOFEAND SANDELL,"Textbook of quantitative inorganic analysis," The Macmillan Ca., New York City, 1936, p. 37. equations 5 a n d 6 p. 39.
ciation of weak acids or bases in aqueous solution, being (H+) = and (OH-) = respectively, in these two instances. The extent of dissociation of a substance is greater the less the concentration and the greater the dissociation constant, other factors being fixed, and i t should be noted that these things were quantitatively considered in deriving equations (7) and (8). The use of the figure one per cent. above is arbitary, and one should actually use that value which is consistent with the desired or attainable accuracy. The error incurred by using concentrations instead of activities is freauentlv as hieh " as several Der cent. and can be greater as, for example, in concentrated solutions of electrolytes. Hydrolysis equations lead to mass law expressions of the same form as those given above for dissociation, and if the same limitations and simplificationsobtain, equations ( 4 ) , (7), or (8)may be used, provided that K is replaced by K,, the hydrolysis constant. Thus equation (8) would be s Z dKX which conforms to the
~KZ
equation (H+) =
~KZ,
4 5
which is usually given for a
solution of a salt of the "strong acid-weak base" type and (OH-) =
which is commonly used for a
salt of the "weak acid-strong base" type. SUMMARY
Simplified solutions for the mass'law expressions for dissociation and hydrolysis have been derived, some of which are new. Conditions are given which permit one to know when the simplified equations may be used. This had not been done rigorously before.
