J. E. House, Jr. and R. C. Reiter Illinois State University Normal, 61761
I
Errors in Calculating Hydrogen Ion Concentration
I n calculating the pH of solutions of acids and bases in general chemistry, one is always faced with calculating the hydrogen ion concentration. The general equation which must be solved for a monoprotic acid or base is
xa
+ K z - Kc = 0
Percent Error Introduced by Avoiding the Full Quadratic Form
(2)
where K is the dissociation constant of the acid or hase, c is the total acid or base concentration, and x is the concentration of the ionized acid or base. Application of the quadratic formula gives The approximate solution, x = (Kc)'/,
(4)
is obtained by dropping x from the denominator of eqn. (1). I n both cases, the negative roots have no physical significance. There have been several discussions of the errors introduced by avoiding the full quadratic form in solving such problems.' Various "rules" have been proposed to guide the student in deciding when the abbreviated calculation is adequate. I n some general chemistry courses, we prefer to treat this problem by the following procedure. We have carried out both the full quadratic and abbreviated calculations for a range of concentrations and dissociation constants using an IBM 1620 computer with a program written in FORTRAN 11. These results are compared and the percent error calculated, the "exact" result always being the smaller. These results for the most appropriate ranges of concentrations and dissociation constants are shown in the table. Dissociation constants smaller than 10-"nd larger than 10-2 are not included for obvious reasons. A few minutes study of this table is all that is required to lead students to the following conclusions.
The extent of acid or base dissociation (which gives rise to the error) depends greatly on the acid or bsse concentration and increases rapidly as the solution becomes more dilute.
It is particularly revealing to display these results graphically as shown in the figure. From this figure, students can quickly see which combinations of concentration and dissociation constant lead to approximate answers outside the desired limit of accuracy, regardless of what limit is placed on the accuracy of the result. Such exercises are valuable for student assignments, especially if an introduction to computer programming
The possibility of avoiding the full quadratic crtlculation depends on both the dissociation constant and the concentration of the acid or hase. The abbreviated calculation may not be used for dilute solutions of acids or bases with dissociation constants as large as lo-* and may alwaw be used when the dissociation constant is lo-' or smaller. If one can tolerate s n error of say 5%, the abbreviated cdclcolation may be used for acids or bases with dissociation constants if the e a e m t r a t ~is suficimtly high. as large as
' MEEHS,F. R., J. CHEM.EDUC.,42, 609 (1965). KING,E. L., J. CHEM. EDUC.,31, 183 (19.54). NIGHTINGALE, E. R., J. CHEW. Eouc., 34, 277 (1957).
0.2
0.4
0.6
0.8
1.0
CONCENTRATION Percent error venvr concentrotion for several rolues of K.
Volume 45, Number
10, October 1968
/ 679
is ~rovided in the course. Smaller tables can be generated quickly without the use of a computer, however. Many numerical problems are suggested by the results shown in the table and figure.
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Journol o f Chemicol Educotion
Acknowledgment .
The authors would like t o acknowledge the cooperation of the Division of Computer Services of Illinois State University.
