proceed to calculate their results on their own effort and record these on their data sheets. When all students have entered their data into the file. the data are processed by a FORTRAN program written for that particular experiment. When a student turns in his report with his completed data sheet a t the end of the period, he can quickly compare the results of his own calculations with the computer printout and can spot any errors made in arithmetic or logic while the exueriment is fresh in mind. he data sheet fo; each experiment is arranged so the data appears on the upper part of the page, followed hy lines of successive steps in the calculation of results. The right edge of the page has a vertical blank area about two inches wide. The printed output of computer results match line-by-line those items on the student's data sheet. After class, the wrintout is cut into vertical strius and are affixed hv ~" transb e n t tape in the blank area of the student's data sheets. This side-by-side comparison of student's results with computer results is not only a convenience to the grader in evaluating the quality of the results and in checking each step of the computation but is especially a n aid to the student in studying his returned laboratorv"rewort. . Student acceutance of. and enthusiasm for, this approach has been almost unanimous. I n even the more involved experiments there has always been sufficient time for students to enter their data a t the terminal and for the instructor to receive the printout well before the end of the period (2 hr 50 min). The following experiments are handled in this manner during the first semester of general chemistry: reaction of copper with sulfur; analysis of KC103-KC1mixture by heating; formulas of hydrates by heating; analysis of soluble sulfate by Bas04 precipitation; molar volume of oxygen a t STP; molecular weight of a volatile liquid; equivalent weight of an active metal by hydrogen replacement. Persons interested in our approach to freshman use of the computer in the laboratory may obtain gratis from the author a packet containing a printout of FORTRAN programs for entering data into a file and for calculations on one experiment, as well as the data sheet with its stripped-on printout. For copies of programs for all seven.experiments and their relevant instruction and data sheets, include check or money order for $3.00 drawn to Galveston College. The language used in these programs is DEC PDP-11 FORTRAN IV. The DEC-PDP-11170 comuuter and weriuheral euuinment ~
~
A Tutorial Program for pH Calculation Glenn E. Palmer University of Prince Edward Island Charlottetown. PEI, Canada CIA 4P3 One of the perennial problems encountered in our freshman course lies in the calculation of pH. The wide-spread availability of electronic calculators has, to a great extent, solved the problem of handling logarithms, isolating the real difficulty that students have with the basic concepts involved. Because these concepts are encountered again . in suhseuuent courses in biology and chemistry, we feel that it is important that an understanding of them be gained in the first vear. Last year, a t the suggestion of some freshman students, we experimented with a computer program designed to generate typical pH problems for consideration. Reaction to the program was quite positive and the results in terms of student performance, although not conclusive, were encouraging. At the time that this program is described in class, students are made aware that they should bring writing materials and a calculator or logarithm table to the session. The program itself begins with an introductory section that outlines its organization and the form in which answers are to be entered.
I t then proceeds, operating in the following format. The user first chooses among four types of pH problem. These are 1) solutions of strone acids and hases 2) solutions of weakacids and bases 3) solutions of salts 4) buffer solutions
When one of these has been selected. a brief introduction. including the general equations for the equilibria involved in the tvwe, is uresented. A tvwical numerical nroblem is then generated and the user is asked to provide an answer to it. If this answer is correct (vide i n f r a ) the program selects and prints one of five positive responses. At that point the user may choose to trv another examule of the same tvoe, ". . or to work on one of the other types listed. If the answer presented is incorrect, then the wrozram al. lows the user to enter an alternatwe answer or to request help with the prohlem a t hand. For the simple cases of strong acids and bases, only one such 'help' is provided. However, for the more complex types, a sequence of 'helps' is available. These are presented in an order that allows the user to become aware of the aspects of the problem that must be considered to reach a solution and the route to that solution. After each 'help,' the user is given the option of presenting a recalculated answer or requesting more aid. When the supply of 'helps'has been exhausted, the program provides the correct answer. The user is then free to try another problem of the same type, one of another variety, or terminate the session. The uropram is able to wrovide a verv. laree number of problems f i r the student toattempt since, except in the case of buffers, both the concentration and the identity of the substrate are randomly selected. In the buffer solution problems, the concentrations of the comwonents are selected
-
(i.e., non-quadratic) mathematical methods will yield acceptable answers for problems involving weak acids and hases and salt solutions. The program allows for some discrepancy between the user's answer and that calculated internally. These allowed differences range from 0.02 pH units in the case of strong acids and bases to 0.10 pH units in problems dealing with salt hydrolyses. The program, written in VAX-11 BASIC V1.2-1, is being executed on a VAX 111750. usineu Decwriter I1 and VT-100 terminals. It contains approximately 350 statements and 25 comments. Copies of the listing and sample executions are available from the author a t the above address for a fee of $5 to cover costs of handling and mailine. should be hv u Pavment " check, made payable to the University of Prince Edwari Island, Account # 1672-382. The author wishes to acknowledge the enthusiastic cooperation of the Chemistry llOA class of 1981-82.
A Pocket Calculator Program for the Solution of pH Problems via the Method of Successive Approximations Wayne C. Guida Eckerd College St. Petersburg, FL 33733 Chemistry students generally encounter pH problems in several of their courses beginning with general chemistry. Calculation of the pH of a solution prepared by dissolving a weak polyprotic acid or any of its salts in water is usually treated in a rigorous fashion in the analytical chemistry course. Since many polyprotic acids occur naturally in the form of amino acids, polycarhoxylic acids, and inorganic acids like phosphoric acid, this type of calculation is also encountered by students of biochemistry. In our analytical chemistry and biochemistry courses our students solve such problems hy Volume 60
Number 2
February 1983
101
employing a method that utilizes successive approximations. This method, which has been nsed extensively by Professor John Walters in his courses at the University of WisconsinMadison, is based upon techniques described by Laitinen (9) and Butler (10) and has recently been described in depth by Brewer i l l ). The method relies on the use of a charge balance equation (or an amnholvte for anv of the amuhoteric suecies) . " eauation . in conjunction with distribution functions known as u-fractions (9).Thus, for a polyprotic acid (H,A) one can write H,
ASH+ + H,-~A-~+H++H,-~A-~=H+ k%
Km
+ H,-*A-"'
.
K"
H+ + A-"
(1)
and charge balance requires that [Ht] = [H,-lA-']
+ 2[H,-zA-2] + ~[H,-zA-~] K, + . . . n[A-"1 + -
(2)
IHt1 If the formal concentration ( C J is known, the concentration of [H+] and the ionization constants of the acid (K,). ~ h u s , each term in eqn. (Z), including the last term for the ionization of water, can be calculated if Cf, K , and [H+] are known. In other words, the calculation of [H+]requires a knowledge of [H+]! However, the value of [H+] can be estimated by assuming that only the first ionization of H,A is important and solving as one would for a weak monoprotic acid. The a-fractions can then he calculated from this valne of [H+]as can the term for the ionization of water. Now, eqn. (2) can be nsed to obtain a better approximation for [H+]. A series of successive approximations is then repeated until convergence is achieved. (1 5 m < n),an equation For the ampholytes H,-,A-m such as equation 3 is used.
was introduced into solution (i.e.. . . CG). . One can then obtain an ~~~
~
~
approximate value of [H+]from eqn. (31, which in conjunction with the appropriate a-fraction yields a better estimate of [H,-,A-"']. Again successive approximations are repeated until convergence is achieved. The nature of the a-fractions is such that they can he readily committed to memory by students (12). Therefore, students generally do not experience serious difficulty performing the calculations described above. They can usually calculate pH to two decimal places with the aid of a hand-held calculator in a reasonable time. Convergence is achieved rapidly when the ionization constants are separated from one another by several orders of magnitude. Furthermore, the student obtains the true value of [H+] without having to rely on gross approximations. The method of successive approximations also affords u-fractions (and hence the concentration of each species in solution) simultaneously with [H+]. Of course. activitv corrections have been nealected however, ionic strengths can be estimated from the results obtained in these calculations which can then be used for activity corrections. Although the exact solution to the problems described above can be obtained by solving polynomial expressions in [H+] by numerical methods (13). The method of successive auuroximations is readily amenable to computation with hand hkld non-programmable calculators forsystems that are not too complex. In addition, the student can observe various phenomena involving solutions of polyprotic acids. For example, the effect of ignoring all hut the first ionization 102
Journal of Chemical Education
of a polyprotic acid or the effect of assuming that the formal concentration of an ampholyte is equal to its equilibrium concentration can easily be seen. Since successive approximations are at the heart of more sophisticated calculations such as quantum mechanical SCF calculations, this procedure exnoses students to an imnortant comnutational method. ;hove. The user supplies the ionization constants for the acid, the formal concentration, and information concerning the nature of the species of interest. The valne of [H+]is displayed after each iteration. The valne of each a-fraction can be recalled from an appropriate memory register after each iteration and the program terminates when convergence is achieved. Di- and triprotic acids can be handled with the T I 58C, and polyprotic acids as large as hexaprotic can be handled with the T I 59. The tolerance for convereence can be snecified by the user. After students have nerformed several calculations hv hand. they are allowed to utilize the programs to check their results. Since some students nossess one of these two calculators. thev can utilize the prog~amswith their own calculators. 0the"r students utilize calculators available within the department. The programs are invaluable (to the instructor as well as the student) for calculations involving polyprotic acids with ionization constants that are not widely separated such as citric acid. For such species, convergence is usually achieved only after many iterations are The programs, therefore, allow the requisite calculations to be done in a reasonable time period. Listings of the programs are available upon request.
Balancing complex Chemical Equations Using a Hand-Held Calculator Robert A. Alberty
Massachusetts Institute of Technology Cambridge. MA 02139 The process of balancing a chemical equation is equivalent to solvine for each " a set of simultaneous linear equations-one element in the chemical species involved. Since the solution of simultaneous linear equations is the central prohlem of linear algebra, the fundamentals of chemical stoichiometry can he expressed in the most general way in terms of matrices. The basic theory has heen very clearly described by Smith and Missen (14) who have considered the following questions: If a closed system contains certain specified chemical species, how many independent chemical eqnations are required to represent chemical changes in the system? What is a permissible set of chemical equations? Smith and Missen (14) give references to the extensive literature on this subject since it was introduced by Gibbs in 1878. This article is nrimarilv concerned with a more limited question: If certain specified chemical species are involved in a reaction. what are the stoichiometric coefficients? We will return to the more general questions a t the end. The reasons for discussine the balancing of chemical equations now is the increasedavailability of hand-held calculators with a capacity for matrix multiplication and matrix inversion. Several hand-held programmable calculators have built-in programs for the solution of simultaneous linear equations. This means that complex chemical eqnations can be quickly balanced with the same calculators that students are using to solve other numerical problems. Balancing a chemical equation is an application of the element-balance equations ~
~
where Ah, is the subscript of the kth element in the molecular formula of species i, x, is the number of moles of species i, and
