Solving equilibrium problems is a standard part of the chemistry curriculum. For example, the equilibrium expression

(1)

gives rise to the rational polynomial

(2)

where d is the initial concentration of N₂, f is the intial concentration of H₂, and x is the variable related to the change in the concentrations of the reactants and products. Modern freshman general chemistry texts and upper-level physical chemistry texts use either the quadratic method (see ref 1) or the method of (successive) approximations (see ref 2) to solve numerical equilibrium problems such as eq 2. Both methods represent no significant advances in teaching and learning methodology for over sixty years and both methods present difficulties, but especially to students who are weak in mathematics. Furthermore, since mathematical operations are time-consuming, generally educators are forced to set numerical assessment tasks that foster low-level thinking skills. There has been some use of spreadsheets for multiple equilibria problems (3) or use of Newton’s method to solve single-equilibrium problems (4). Donato has proposed using graphics calculators to graph a rearranged version of eq 2 (5). All of these approaches require some degree of mathematical manipulation.

Graphics calculators and spreadsheets can be used to directly graph or to tabulate (6–9) the unarranged left-hand side of eq 2, giving rise to several methods of solving equilibrium problems. Essentially one selects that value of x, for which the numerical value of the unarranged left-hand side of eq 2 equals the numerical value of the right-hand side: these methods are detailed in the online material. The use of such technology in this and other areas of chemistry (10–16), empowers novice students, especially those who lack confidence in their mathematical ability.

The main feature of the tabular and graphical approaches is that students perform no numerical calculations and no mathematical manipulation, thus avoiding the learning difficulties experienced by students who are weak in mathematics and enabling students of all abilities to achieve at least some success (17). The approaches are fast and can be generalized to equilibria of any molecularity. It has been noted elsewhere (10–16) that the use of technology in chemistry enables students to do numerical experiments, thus moving equilibrium calculations from tedious mathematical exercises into the realm of experimental science. The graphical and tabular approaches allow students to explore the equilibrium concept, which is an important part of discovery and learning (6).

Literature Cited

1. Lehfeldt, R. A. A Text-Book of Physical Chemistry; E. Arnold: London, 1899; p viii.
2. Lowry, T. M.; Cavell, A. C. Intermediate Chemistry, 5th ed.; MacMillan: London, 1947.
3. Carter, D. R.; Frye, M. S.; Mattson, W. A. J. Chem. Educ. 1993, 70, 67.
4. Joshi, B. D. J. Chem. Educ. 1994, 71, 551.
5. Donato, H., Jr. J. Chem. Educ. 1999, 76, 632.
6. Pólya, G. How To Solve It; A New Aspect of Mathematical Method, 2nd ed.; Princeton University Press: Princeton, 1971.
7. Pólya, G. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving; Wiley: New York, 1962.
8. Goldenberg, P. Multiple Representations: A Vehicle for Understanding Understandings. In Software Goes to School; Perkins, D. N., Schwartz, J. L., West, M. M., Wiske, M. S., Eds.; Oxford University Press: New York, 1995; pp 155–171.
9. Kaput, J. Technology and Mathematics Education. In A Handbook of Research on Mathematics Teaching and Learning; Grouws, D., Ed.; Macmillan: New York, 1992.
10. Lim, K. F. J. Computer Chem. 2006, 5, 139 (accessed May 2008).
11. Lim, K. F.; Coleman, W. F. J. Chem. Educ. 2005, 82, 1263.
12. Lim, K. F. Aust. J. Educ. Chem. 2004, 64, 24.
13. Lim, K. F. J. Chem. Educ. 2005, 82, 145.
14. Lim, K. F. Using Spreadsheets in Chemical Education To Avoid Symbolic Mathematics. In CCCE Newsletter: Using Computers in Chemical Education; Spring 2003; Paper 5 (accessed May 2008).
15. Lim, K. F. Using Spreadsheets To Teach Quantum Theory to Students with Weak Calculus Backgrounds. In Maths for Engineering and Science; Hirst, C. Ed.; LTSN MathsTEAM: Edgbaston, U.K., 2003; p 24.
16. Lim, K. F. New Directions in the Teaching of Physical Sciences 2003, 1, 16.
17. Riddell, S.; Tinklin, T.; Wilson, A. Disabled Students in Higher Education: Perspectives on Widening Access and Changing Policy; Routledge: Abingdon, U.K., 2005.