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
N is the numher of species. The total numher of moles of element k in the system is represented by R k , and M is the numher of elements. Equation (4) can he used to balance a chemical equation involving N species and M elements in the following way. We will use the M simultaneous equations to calculate the numbers of moles x i of N - 1 species required to make one mole of the remaining species. We can do this only if N = M 1. I n order to illustrate this process we will consider a specific example. Doris Kolh (15) has recently discussed halancing complex redox equations by inspection. Her most complicated redox equation provides a good example for use of a hand-held calculator. Without coefficients the reaction is
To put the equation in more familiar form we multiply through by five to clear of fractions and put the species with the negative roefficients on the right-hand side.
+
There are 5 elements and so there are 5 element-halance equations of the form of eqn. (4). To calculate the numbers of moles of Ph(N312, C r ( M n 0 4 ) ~Cr20s, , MnOa, and NO required to make one mole of Pbj04, we have to solve the simultaneous linear equations
There is a row for each element and a column for each species, except for the species Ph304 which we are using to define the B column. The unknown numher of moles of Ph(Nd2, Cr(MnO&, Cr203, MnOz, and NO are represented by xl, xz, xa, x4, and zs,respectively. These equations may be written more conveniently in matrix notation (16,171.
The first matrix is referred to as the system formula matrix. The matrix on the rieht - is the element-abundance vector written as a column matrix. In order to understand how this problem is solved with a hand-held calculator we need to introduce some linear algebra. Matrices are represented with hold-face roman type, and so eqns. (4) and (7) may be written Ax=B
(8)
The same result is obtained no matter which species is put in the B rolumn. This formulation of the problem makes it easy to see several ways we can get into trouhle in trying to balance a chemical eqnatioh. 1) Too few species. If we eliminate any one of the species, we do not have a square matrix which is required for the inversion. 2) Too many species. If we add NO2 to the list of species, we will have six unknowns but only five linear equations. However. we can substitute NO? for NO in the statement of the problem. 3) Two elements in constant ratio. If, for example, N and O had occurred in these five species only as Nos, the five equations would not have been linearly independent, and we could have calculated the numbers of moles of only four species. The system effectively contains four "elements." 4) Ions. If ions are involvid, electrical charge has to be conserved throuah an additional linear eon. (14). Now let us return to the more general questions, given above, which are addressed hy Smith and Missen (14). If a closed chemical system contains S species and C "elements," the numher R of independent reactions required to represent all possible chemical changes in the system is given by In the example given above R = 1. If we add Pb(NO& to the list of species, R = 2. We can obtain the stoichiometric coefficients for the second chemical reaction by substituting Pb(NO3)2for Pbs04 in the above procedure. If we want to halance the eauation for the formation of PblN01Ig. . ". ", instead of Pb304, we simply multiply A-' given in eqn. (10) by the column matrix for P h i N o ~ ) ~ , . When S > C + 1,we h G e to be careful in selecting the suecies for the system formula matrix. These soecies have to ancing redox equations. They are not necessary for balancing equations but may he useful for other reasons.
Mass Spectral Analysis of Halogen Compounds David K. Holdsworth
If matrix A has an inverse A-I, then x = A-' B
University of Papua New Gu~nea BOX320 University P.O. Papua New Guinea
(9)
The program for the hand-held calculator can obtain the inverse A-' and multiply it hy B to obtain, in this example, the numbers x, of moles of Ph(NaI2, Cr(MnOa)z, Cr203, Mn02, and NO required to make a mole of Ph304. The result, expressed in the form of eqn. (9) is
The inverse of the svstem formula matrix is simwlv . . eiven " for interest; it is an intermediate step in obtaining the solution eiven in the column matrix on the left. The values of x riven Fn the column matrix on the left are the stoichiometriccoefficients in the balanced chemical eauation
Chemistry students can readily note the presence of one chlorine or one or two bromine atoms in an organic molecule by examining a mass spectrum and recognizing the characteristic molecular ion abundance patterns, spaced a t twomass-unit intervals, 3:1,1:1 and 1:2:1. Other halogen combinations give patterns that are more difficult to interpret. The abundances of some peaks may be less than 5% of the base peak and would not normally be plotted on a line diagram. A pocket calculator can be programmed to decide unequivocally and display the halogen combination in a rnolecular-ion or fragmentation-ion cluster. It can be seen from Table 1that it is not possihle to indicate particular halogen atoms in an ion-cluster by examination of the ( M 2)lM or (X 2llX percentage values alone, since certain halogen combinations give practically identical values. However,
+
+
Volume 60
Number 2
February 1983
103
