potential ambiguity (i.e., two unequal but physically meaningful roots) never arises. When an acid, HA, is dissolved in water, the concentration of A- a t equilibrium can he obtained from the definition of K,:
Further, if HA is the only source df H+ and A-, then (1) [A-I.,' + Ka [A-leq - K,, lH.410 = 0 where [HA10is the initial concentration of the acid. Wemust prove that (1) the quadratic equation always yields a value of [A-I,, such that 0 < [A-Ieq < [HAIo;and (2) of the two roots of the auadratic eauation.. onlv . one satisfies the above inequality. Proof: The solutions of eqn. (1) are
[A-I,
But, KO
=
-K, i d K G Z+ 4Ko[HA10 2
(2)
+ 4K0[HA10 < (Ka + 2[HA]o)2.So, it follows that d K a P+ dK.[HA]a
< K,
+ 2[HA]o
(3)
Substitution of eqn. (3) into eqn. (2), and utilization of 4K, [HA10 > 0, yields: 0 < [A-I,,
< [HA10
for that solution of eqn. (2) which corresponds to the positive square root. On the other hand, the solution of eqn. (2) which corresponds to the negative square root must be a negatiue real number because both KO and {Ka2+ 4K,[HA]ol are positive. The proof is now complete.
Computer Quiz on the Periodic Table J. S. Wasson Dalhousie Uniuersity Halifax, N.S. B3H 2S7 Some asoects of the ~eriodictable are best learned bv practice which we offer students in the form of an interactive computer program PERIOD. The introductory section of PERIOD reviews general concepts such as whether rows or columns corresoond to periods. Before this introductory section a periodic t a b l e k printed but with three atomic symbols and numbers (chosen at random) missing. The student must identify each of the unknowns given only the atomic symbols. The student determines the atomic number, most stable electronic configuration, typical oxidation state, and name. The program supplies information values, some help for inappropriate responses, and additional descriptive information. The wording of the questions on the quantum numbers n and 1 was influenced by Pilar's recent article (J. CHEM. EDUC., 55, 2 (1978). The information values are calculated following Bonchev and Kamenska (Croat. Chem. Acta, 51,19 (1978). This calculation can easily he omitted. Scores on all questions are printed a t the end of the program. This Fortran program has run successfully on a CDC6400 for two years and has been submitted to the Central Program Exchange, Wulfruna St., Wolverhampton, WV1 ILY.
Volume 56. Number 8. August 1979 / 527
