Another Method for Solving Equations in Acid-Base Calculations To the Editor:
From equations (I),(21, (31, we get: [OH-]
THISJOURNALpublished a method for solving equations in acid-base calculations in September 1980 [57 [9],6201. I would like to describe another method which involves only four iterations to arrive at the correct answer and can he performed with a simple programmable calculator. If we consider the problem presented in that issue, a solution containingsodium benzoate 2 X 10W5F;Kb = 1.59.10-lo, we can write the following equations: CB = 2 X 10W5 = [HB] + [B-]
(1)
=
[HB] + [H30+]
(5)
This equation tells us that [OH-] is the result of two contributions: the hydrolysis of B- and the dissociation of water. Which is more important? I t depends on the concentration of NaB. Suppose the first reaction predominates in the solution: B- t H z O f H B + O H -
then [OH-] would be (Kb X C~)1'2= 5.63 X 10-8. Now suppose the second reaction predominates: H20 t Hz0 f H30+ + OH-
it would give [OH-] = (K,)'/2 = 1 X 10-7 to the solution. It is evident that the main contribution of OH- comes from water and not from B-. This reasoning allows us to determine which [OH-] will he used as a first approximation in the iterative program. I t is based on eqns. ( 1 )and (51, written as follows:
Volume 60 Number 10 October i983
911
[OH-]
= (K,
[B-] =
+ Kb X [B-])lI2
(6)
Cs X [OH-] [OH-]
+ Kb
The first approximation, [OH-] = 1 X 10-7 is introduced into eqn. (7) to obtain an approximation of [B-1, which is then introduced into eqn. (61,and the process is repeated until the desired accuracy is reached. The results are presented in the following table: [OH-]. 10' 1.0000000 1.1478219 1.1478502 1,1478502
[B-1. 10" 1.9968250 1.9972334 1.9972334
Then, [OH-] = 1.1478502 X 10-7 and pOH = 6.9401148. I t does not agree with the result given by the authors in their article, because there was a little mistake in eqn. (3) written there, which should he: [OH-l3
+ Kb X [OH-l2 - ( K , + K b X CB)[OH-] - Kb X Kw = 0
The method can he applied to very difficult problems, like the hydrolysis of (NHJHzPOa, the formation of complex ions, the solubility of salts under different conditions (pH, complexation, common ions), and redox reactions. Alelandro C. Olivieri Brown 2666 2000 Rosaria RepObliCa, Argentma
To the Editor: Jim Carr and I think that Dr. Olivieri's letter contains valid points, and the idea of deciding which contribution to [OH-] is larger is pedagogically useful. However, our method requires no precalculation. Paul 6. Kelter and Jim Carr University of Wisconsin-Eau Claire Eau Claire. WI 64701
Dilemma Over Extinction Coefficient Units To the Editor: I would like to comment on the six options described in the recent note by Donald C. Wigfield, "The Extinction Coefficient: SI and the Dilemma of its Units-Six Options.'' [J. CHEM.EDUC.59,27 (1982).] In offering six options, it appears that the author presents us with a hexalemma rather than a dilemma. However, since only four out of the six options are legitimate and of these four, three are equivalent, it may he that it is a dilemma after all. Suppose we examine the options in detail. The extinction coefficient (molar absorption coefficient) c. can be written as c = Alcl. where A is the absorbance (a Dure numhtrl, < is the cunrt.ntri#rlt,n i i t h t . .~ll~st.tnct.it~~d 1 i~ the I ~ I lrnrrh. I C'lrd\.. the dimel#ii(~n* d e are 1encrl1.1am~~unt bf suhs&nce)-1. he author presents six po&bilities for tabulating values of c. 0 ~ t i o n ~ Tabulate 1. a numher with no units attached. The numbers customarilv tahulated are based on mol/L as a concentration unit a n d cm for the unit of length. Those who perpetuate this method no doubt do so in the belief tbat the user of the table will have a divine revelation while contem912
Journal of Chemical Education
author would not have mentioned it were it not f i r the fact tbat it is commonly done. (Unfortunately, this lamentable practice is not confined to the extinction coefficient.) Option 2. Take the numher in Option 1 and restore its correct unit: namely, L/(mol em). After a few moments thought, we see that the concentration is in m o l L and the length is in cm. This is an infinite improvement over Option 1. [For that matter, tabulating the value in hogsheads per (stonemole furlong) would be an infinite improvement over Option 1. At least it tells us that the concentration is in stonemoles per hogshead and the length is in furlongs.] Options 3 and 4. Take L/(mol em), convert liters to cm3 or m3, combine the volume and length units, then diddle with the SI prefixes until the factors of 10 are eliminated. This doesn't change the unit, but it does combine the powers of length, which is laudable; neatnesswise, that is. The author seems to despise Optidn 3 inthe form cm2/mmol. ". . . a rather poor, hut amusing, option .'. ."What is poor about mmol/cm3 = (mol/L) for concentration and cm for length? As for amusement, I've gotten bigger laughs out of the table of logarithms. Looking at Option 4, m2/damol, I Fgree. It's probably a loser. I haven't found anyone yet who can use dekamoles with conviction. Option 6. Tabulate the same numher as all the above options but invent a new unit, the "extinction coefficient unit" (ecu?) as a way to legitimize present practice. This option only requires that divine revelation tell us what an ecu is. Some years ago I read a research paper on calorimetry by a distinguished thermodynamicist who suggested that the unit for heat capacity and entropy, cal/K, he named to honor Gibhs and proceeded to tahulate his data in gbdmol; 1ghs = 1cal/K. In a misguided effort to advance this noble cause, I formulated and subsequently published a textbook problem using the gibbs as a unit of heat capacity. This invention produced a large increase in entropy, but did little to honor Gihbs, who has probably reached the asyrntotic limit on that kind of thing anyway. I suggest we leave the invention of new units to the appropriate.internationa1committees. With any measure of good luck, they won't get around to it. That leaves only Option 5 , the SI derived unit m2/mol. The author objects to this on the grounds that the tabulated value will differ from,those currently tahulated by a factor of ten, and seems to say that the results of using SI units in tahulations ". . . are a woefully unnecessary extra complication to the practice of science." What advantage is gained by changing from L/(mol cm) to m2/mol? The advantage of the SI unit is simply that it is the systemati'c,unit. Since only SI base units appear in this unit, it is immediately obvious to the user of the table that it is the systematic unit. The user then knows that the systematic units for concentration (mol/m") and for length (m) are the appropriate ones to use with the tahulated value. No logical process is required for this. It is all contained by definition. On the contrary, in the case of L/(mol em), a few moments of logical thought are required to conclude tbat the concentration should be in molL and the length in cm. (Not many moments in this simple case.) Finally, if we use the systematic units we do not have to specify the units when we write about the equation, and the beginner does not have to remember, "Oh yes, in this equation I must use concentration in m o l L and length in em." Having to rememher that, or havingto rememher that this is a special case so that you must look up the equation to decide which units are appropriate, I regard as a "woefully unnecessary extra complication" in the learning and practice of science. Gilbert W. Castellan University of Maryland College Park, MD 20742
