An Efficient Method for the Treatment of Weak Acid/Base Equilibria James H. Burness Pennsylvania State University, York Campus, York, PA 17403 The treatment of aqueous acidhase equilibria usually makes up a significant part of the second half of a typical freshman chemistry course. Students often have difficulty with this material because there are many possible equilibria that must be considered before the dominant equilibrium is identified. Furthermore, students must learn to recognize that of several nossible sources of a eiven snecies in solution. some sources contrihute appreciable amounts of the species while others contrihute negligible amounts. Often, justifiable approximations and simplifications are made so that students are not burdened with complex mathematical solutions. In the majority of such courses, the difference between concentration and activity is either assumed to he insignificant or is simply ignored. Likewise, the contribution of water to the hydronium ion concentration in a solution of an acid can be ignored unless the acid is ultraweak or the solution is extremely dilute. A factor that often cannot be ignored, however, is the percentage ionization of a weak acid or base. This is discussed in most. if not all. of the current freshman chemistrv texts. The funhamental'question of whether the amount "1 weak electrolvte that ionizes can be ianored (relative to the stoichiornetri; concentration of the wiak electrolj.tej is usually answered hv the "5% rule": if the percentage ionization is less than or equal to 5%, then the equilibrium concentration of the weak acid or base is essentially the same as the starting (stoichiometric) concentration. This so-called "5% rule" is the topic of this paper. Textbook Treatments For an equilibrium system of the form,
where C = stoichiometric concentration of weak electrolyte and K = K. or Kh. when can the quadratic formula he avoided by assuming that the denominator C - x = C? An examination of a handful of current freshman chemistw texts reveals a variety of approaches to this question. One text ( I ) suggests that the approximation be made and the value of x he determined. Then, if C - x is the same as C to the appropriate number of significant figures, the approximation was valid, and the calculation is complete. If the two values differ in the last significant figure (presumably by more than f1 ) then the nroblem must be reworkedusine the quadratic formula. ~ e v e r aother l texts ( 2 4 ) explicitly mention the "5% rule" and sueeest that the student make the approximation, calculate x , then determine the percentage ionization, uu
my students, this approach is much faster and less prone to error than setting up and solving the quadratic formula. Other texts use related, but slightly different, approaches. One of these (51 advises the student that the approximation is valid i f C > 100K. which imolies a oercent ionization of -.-~ -10% rather than 5%: Another (6) &ply advises using the MSA annroach for all calculations. In an annroach closest to the ~ n ~ d e s c r i b in e dthis paper, Ebbing (n'droposes that the quantity KIC he determined. If this is greater than 0.001 (corresponding to -3% ionization), then he advises the student to use either the quadratic formula or the MSA. None of these approaches is wrong. They are all variations of the same idea, and it is unlikely that anyone is prepared to argue whether the cutoff point should be 3%, 5%, or even 10% ionization. All of the above approaches do, however, share a common disadvantage: they-require that a calculation be performed to determine whether the approximation is valid, and if the approximation is not valid, the time spent reaching that conclusion was wasted. A more efficient approach is to make a quick calculation to see whether the approximation is valid, then to use the results of this calculation to arrive a t the answer to the problem.
.~~ ~~
~~~~
~
The lonlzatlon Ratlo The approach I have been using for the past six or seven years relies on the introduction of a new quantity called the ionization ratio, I:
The ionization ratio is defined in this way because i t is a physically meaningful quantity. As I increases, the degree of ionization (a)increases. The mathematical relationship between a and I is
From this equation i t should be evident that at some point the value of I is small enough to justify the approximation
pxi=Jz Under these conditions, eq 1 reduces to
If the valueofI is reduced further (by reducingKor increasing C), eventually 112 becomes negligible compared to and eq 2 reduces to
x 7%ionization = - X 100
a-p
Two of these texts (3.4) , . soare . the student the trouble of using the quadratic formula by suggesting that if the percentage ionization exceeds 5% then the method of successive approximations (MSA) be used instead of the quadratic formula. Based on my experience and on the comments of
Equations 2 and 3 are approximations to the "exact" solution of eo 1. When can these annroximations he made? The .. figure s h o w a plot of percent dissocintion versus log I. When 1 exceeds 1. the useof ea 3 sueeests more than 1004 dissociation, and the use of eq 5 suggests that the percent dissociation decreases with increasing I. Both results clearly show
C
224
Journal of Chemical Education
~~~
~
~
~~
(3)
Percent Dissociation
vs.
Log Ill
Examples Some examples will illustrate how easily these calculations can be performed. For the sake of brevity, chemical reactions and equilibrium constant expressions have been omitted.
Example 1: Calculate the pH of 0.0500 M acetic acid (Ka = 1.8 X 1016\ Lu
,.
Solution: 1. I = KIC = 1.8 X 10~5/0.0500 = 3.6 X lo-' 2. Since 1< 2 X 3. [Ht] = (C)(a) = C $I=9.5 X lo-' 4. pH = 3.02 Example 2: Calculate the pH of 0.010 M ethylamine (Kh = 5.6 X 10-4~ . . ,.
Solution: 1. I = KIC = 5.6 X 10~410.010 = 0.056 2. Since I > 0.002, the 5% approximation is not valid and MSA should be used: 3. [OH-] 1CJI 2.4 X 4. 5.6 X lo-' = 1.210.010- 2.4 X So r = 2.1 X 10W Repeating this procedure for one more iteration gives the same value, so [OH-] = 2.1 X 10P (pH = 11.32). Note: Another solution would have been to recognize that I < 0.10,s~the "25% approximation" is valid and eq 2 can be
-
Percent dissociation of a weak electrolyte calculated from the ionization ratio using eq 1(quadraticCurve).eq 2 (25% apprax. curve).and eq 3 (5% approx. curve).The figurewas drawn using LoNs 1-2-3.
used:
that a t some noint the annroximations lead not onlv to -incorrect answers, but to illogical answers as well. The curves plotted using eqs 1 and 2 diverge a t a point that corresponds to approximately 25% dissociation. Thus, eq 2 could be used as long as the acid is not more than 25% dissociated (in other words, as long as I is less than or equal to -0.10). The use of eq 2 could be referred to as the "25% approximation". The use of eq 3 is equivalent to assuming that C x is eoual t o C (i.e.. ea 3 renresents the "5% anoroximation"). .. he valueof the ionizatibn ratiocorresponding to5% ionization is 0.0025. 1 usuallv tell mv students that the 5% rule is valid if I is less than br equai to 0.002 simply because the latter value is easier to remember. An ionization ratio of 0.002 actually corresponds to 4.4% ionization. There are two distinct advantages to usinn the ionization ratio concept in acidbase equilibGa: (1)a simple division of K by C allows the student to compare I to 0.002 and decide whether the "5% approximation" will be valid, and (2) if the approximation is valid, all the student needs to do is press the "sauare root" kev on the calculator to determine the degreeif ionization (see eq 3). If the concentration of H+ or OH- is desired. the student simnlv needs to multiolv a (which is already showing on the caiculator display) 6y-the stoichiometric concentration. The use of the ionization ratio to calculate x can be used for any equilibrium expression of the form
-
This means that it can be used for weak acid, weak base, and hydrolysis calculations. The typical procedure is (1) calculate I (2) if I 5 0.002, then the 5%approximation is valid, and (3) a = a III x = [H*]or [OH-]= ( c ) ( o ) = c,i (51 if1> 0.002, calculate x using steps S end 4, then use the method 01succensive appnximations.
a = g-112 0.209 (20.9%ionized) x = [OH-] = (C)(a) = (0.010)(0.209)= 2.1 X
Example 3: In a particular solution of henzoic acid (K. = 6.4 X 3.5%of the acid is ionized. What is the concentration of the acid? Solution: Since less than 5%of the acid is ionized, a = v% so I = uZ= (0.035)z = 1.22 X Thus, I=KIC-C=KII = 6.4 X 10-5/1.22 X C = 0.052 M Conclusions The use of the ionization ratio concent enables students to perform many of the calculations f i r aqueous acidbase eauilibria more auicklv and efficientlv. Just as students can compare an equijibriim con.stant with the numher 1 to decide whether the position of eouilibrium liesfar to the left or right, they can compare the ionization ratio with 0.002 to decide whether they are permitted to neglect the ionization of the weak electrolyte. Furthermore, the value for I (which is already showing on the calculator display) can easily be used to determine directlv the concentration of H+ or OH-. Even if the method of successive approximations is used, the value of the ionization ratio will quickly provide a first approximation for subsequent iterations. The feedback of students concerning this approach has been very positive. They report that the concept of the ionization ratio is easily understood and that its use simplifies subsequent calculations. Uterature Clted 1. Martimer, C. E. Chernutry,6th ad.: Wadaworth: Bclmoat, CA. 1986:Chapter 17. 2. Brow, T. L.:Le May, H. E. Chemistry: Tho Cenfrd Science. 4th ad.; Prentice-Hall: Englewiwd Cliffs, NJ, 1988:Chapter 17. 3. Zumdahl, S.C h e m ~ t r u2nd , 4.; Heath: Lexington, MA, 1989:Chapter 14. 4. Whitten, K. W ; Oailey, K. D.: Davis, R E.&nard Chemistry. 3rdad.; Ssunders: New York.1988:Chapter 18. 5. Kotz. J. C.: Purceil, K. F. Chemistry ond Chemicol Rrarfiuily: Saundera: New York, 1987:Chepter 15. 6. McBusrrie, D. A,; Rack. P.A . Gonard Chemistry. 2nd ed.:W. H.Freeman: New York, 1987:Chapter 15. 7. Ebbing, D. D.Ganaroi Chemistry:Hought~nMifflin: Boatan, 1984:Chapter 18.
Volume 67
Number 3
March 1990
225
