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David L. Byrum Ruamrudee International School Bangkok, Thailand 10510
The Use of Approximations in a High School Chemistry Course Paul S. Matsumoto,* Gary Tong, Stephanie Lee, and Bonita Kam Galileo Academy of Science and Technology, San Francisco, CA 94109; *
[email protected] Scientific literacy is important for all citizens since science has a major impact on society and allows citizens to intelligently participate in debates involving science. An important component of scientific literacy, and a part of the National Science Education Standards (1, 2), is the use of mathematics. A background in mathematics is a good predictor of success in learning chemistry (3). To ease student learning of chemistry, assumptions are used to simplify the content, which leads to an approximation. The use of approximations introduces limitations into a scientific conclusion, which is a valuable concept for citizens to acquire. The public should be skeptical (2) and aware that the use of approximations, for example, in the context of a mathematical model, has limitations. While numerous science courses use mathematics, high school chemistry, high school physics, college chemistry, and college physics courses enroll the largest number of future citizens in science courses that use mathematics. Of the preceding courses, high school chemistry has the largest enrollment (4, 5). Therefore, high school chemistry teachers have the greatest potential to ensure that future citizens understand the use of mathematics, including approximations, in a scientific context. While learning some chemistry topics using approximations, high school students remarked that biology and physics use many fewer approximations than chemistry. Such student remarks reflect students’ misconception or ignorance regarding the use of approximations in science. Thus, we recommend that high school science teachers discuss the use of approximations explicitly, explaining to students both their utility and their limitations. In this article, we will describe high school chemistry topics that use approximations. Dissociation of a Weak Acid
HA H + + A− The information to determine the [H+] may be summarized in a table with the columns containing the chemicals in the reaction and rows containing the initial concentration, the change in concentration, and the equilibrium concentration of the chemicals in the reaction (i.e., ICE table; 6–8)
[change] [equilibrium]
H+
Ka =
A−
e
HA
e
H+
=
HA
e
+x x
0
0
−x
(1)
where the subscript “e” represents equilibrium. Rearrangement gives
x 2 + Ka + H+
x − K a HA
0
0
= 0
The solution to the preceding quadratic equation is
− K a + H+
±
x =
0 2
K a + H+
0
+ 4K a HA
0
+ 4K a HA
0
2
As x > 0, the solution is
− K a + H+
+
x =
0 2
K a + H+
0
(1a)
2 To simplify the calculation, assume that [H+]0 = 0; that is, assume that the [H+] due to the dissociation of water, [H+]0, does not contribute significantly to [H+]e. Thus, H+
=
e
H+
+ x ≈ x
0
Then, eq 1 simplifies to
An example of using approximations is the determination of the hydrogen ion concentration, [H+], owing to the dissociation of a weak acid, HA
[initial]
and
[HA]
[H+]
[A–]
[HA]0
+
0
[H ]0
–x
+x
+x
[HA]0 – x
[H+]0 + x
x
Ka =
x2 HA
− x
0
(2)
which produces the approximation
H+
e
≈
−K a +
K a 2 + 4 K a HA 2
0
(2a)
To further simplify the solution, an additional assumption is made that the dissociation of a weak acid is negligible, that is, [HA]0 ≈ [HA]0 − x . Thus eq 2 further simplifies to
Ka =
H+ HA
2 e
(3)
0
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 7 July 2009 • Journal of Chemical Education
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In the Classroom
which produces a new approximation
H+
e
=
K a HA 0
(3a)
to determine the [H+] due to the dissociation of a weak acid. The preceding assumptions simplify the calculation and lead to the approximation expressed by eq 3a. Introductory chemistry textbooks use eq 2a or 3a to solve such problems—there is less than a 5% difference between these equations,1 when Ka
