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ISSN 1430-4171

THE CHEMICAL EDUCATOR

©

http://journals.springer-ny.com/chedr

1996 SPRINGER-VERLAG NEW YORK, INC.

10.1007/s00897960020a

In the Classroom

Mathematically Modeling Dilution STEPHEN DEMEO Hunter College of the City University of New York New York, NY 10021 [email protected]

This article describes a

A

lesson is described in which students interactively derive the dilution equation, CiVi = C f V f . Qualitative description and mathematical modeling are utilized to promote students’ conceptual understanding of this equation.

lesson for allowing high school students to participate in the construction of an equation that describes dilution of liquids in the laboratory.

Introduction

If understood fully a scientific equation can symbolically convey a great quantity of information about scientific concepts. But, unfortunately, to many students equations serve another purpose. They are often thought of as things which must be memorized in order to produce the “right answer”, regardless of the conceptual meaning they connote. Teachers who do not demand such understanding nor test for it are implicitly telling their students to use equations as algorithms, as black boxes which produce answers with a turn of a pen or push of a calculator. Researchers in science education have found that allowing students to use equations solely as algorithmic devices does not necessitate that the students understand the meaning of

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1996 SPRINGER-VERLAG NEW YORK, INC.

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the equation, the problem, or the solution which was generated [1, 2]. Some alternative methods which science researchers claim can promote student understanding of conceptual knowledge, are (1) to qualitatively describe an equation before using quantitative values and (2) produce equations by mathematically modeling physical phenomena [3, 4]. This article describes a lesson for allowing high school students to participate in the construction of an equation that describes dilution of liquids in the laboratory. Students participate in the mathematical modeling of the dilution of orange juice and in so doing develop the equation CiVi = C f V f based on their observations. Where C is the concentration and V is the volume. The subscripts i and f designate the initial and final values. The dilution equation is important for students to understand, because laboratory technicians and scientists from many fields use this formula to prepare chemical reagents from stock solutions. Varying the concentration of a solution by dilution is relevant for experiments that are designed to determine how different concentrations of a solution affect, for example, pH, boiling point, or absorbance.

Qualifying the Concept of Dilution

The lesson is begun by asking students what they think “dilution” means. Their responses are not corrected, but rather explored through a demonstration involving a red dye solution that is consecutively diluted with water in a series of beakers. The students are asked to qualitatively determine the relationship between the concentration of the dye, indicated by color, and the amount of water used to dilute it. (The concept of concentration has been studied extensively in previous classes). It is easy for them to see that as more water is used the concentration of the dye decreases (the color fades).

Quantifying the Concept of Dilution

Students frequently encounter the need to quantitatively dilute a liquid at home when preparing food in their kitchens; thus, many students are already able to solve dilution problems containing simple numbers. With these facts in mind, a second demonstration is initiated by handing a student a can of concentrated orange juice and asking him or her to identify the volume and concentration of its contents. By reading the description on the side of the can, the student reports that the can contains “12 ounces of 100% orange juice”. Orange juice is then made for the class. Another student instructs the class in

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ISSN 1430-4171

THE CHEMICAL EDUCATOR

©

http://journals.springer-ny.com/chedr

1996 SPRINGER-VERLAG NEW YORK, INC.

S 1430-4171 (96) 02020-1

how to prepare it. The can is opened and emptied into a large container, calibrated in ounces. Following the directions on the label, 12 oz can is filled three times with water, each time it is emptied into the large plastic container, which is capped and briefly shaken. As these instructions are followed, count aloud as each 12 ounce can of liquid (the one can of concentrate and the three cans of water) is poured into the plastic container. After counting to four, the class is asked to determine the new concentration of orange juice in the plastic container. Most students will correctly answer 25%. For those who don’t, ask how they would make up a 50% solution, then extrapolate to a 25% solution. Next, the class should be asked how they arrived at the answer of 25%, in other words, what did they do mathematically? Most students simply say that they “divided 100% by 4 getting 25%.” This is written on the board as equation 1 100% = 25% 4

(1)

Ask the class if “48 ounces/12 ounces” could be substituted for the value of “4” in the denominator in equation 1. There is no resistance to this since 48 divided by 12 equals 4. Thus, equation 2 is re-written: 100% = 25% 48oz 12oz

(2)

Equation 2 is rearranged to give equations 3 and 4. 100% ×

12oz = 25% 48oz

100% × 12oz = 25% × 48oz

(3) (4)

Next ask, “What do each of the numerical values represent?” The answer is that 100% is the initial concentration of orange juice in the can; 12 oz is the initial volume of orange juice in the can; 25% is the final concentration of orange juice in the plastic container; and 48 oz is the final volume of orange juice in the plastic container. These responses are substituted into the equation, producing equation 5. The equations 6,7, and 8 exemplify the process of shortening the written notation, which many students rarely see performed.

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1996 SPRINGER-VERLAG NEW YORK, INC.

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(initial concentration of orange juice in can) × (initial volume of orange juice in can) = (final concentration of orange juice in container) × (final volume of orange juice in container)

(5)

(Conc. initial)(Vol. initial) = (Conc. final)(Vol. final)

(6)

(Ci )(Vi ) = ( C f )(V f )

(7)

CiVi = C f V f

(8)

Finally.

Equation 8 is our mathematical model, in terms of concentration and volume, of orange juice being diluted with water. Once this mathematical model has been derived, the students are given the opportunity to change their responses to the earlier question about the meaning of dilution. From both the qualitative demonstration and from just having seen that adding water to orange juice decreases the concentration of the orange juice, they know that the concept of dilution concerns two variables. While discussing some of the newer responses, illustrate the scientists’ notion of solute and solvent can help clarify the meaning of dilution. Write the scientists’ meaning of dilution on the blackboard. From this point, different forms which express concentration in units of volume, namely molarity, M and if desirable, normality, N can be substituted in the formula. Point out that Ci and C f in the dilution formula can be expressed in any of these forms, but, stipulate that both variables must be expressed in the same form. Concentration expressed per unit mass (i.e.; mass percent, molality, parts per thousand, parts per million, parts per billion) cannot be used in this equation since the density of the liquid must be considered. Exploration of this concept can further lead to an interesting discussion with the students. Using their equation students can solve other problems that involve chemicals found in the laboratory as opposed to the kitchen. To demonstrate the equation’s usefulness, these examples might include numerical values that are not commonly used by students and cannot be determined intuitively. Further problems require solving for other variables in the equation, not only C f as originally demonstrated. Ultimately, the students are able to write instructions for preparing solutions from stock solutions based on their mathematical model. For instance, they might be asked how to prepare a 2 M aqueous sulfuric acid solution from an 18 M commercial sulfuric acid stock solution.

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1996 SPRINGER-VERLAG NEW YORK, INC.

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They become aware that a volume variable, Vi or V f , must be supplied in order to use their dilution equation and solve the problem. Any quantitatively solved problems can be related to the students’ qualitative description. Qualifying equations and mathematical modeling are two nonalgorithmic methods of teaching equations. These methods help students to understand the scientific concept as well as generate the correct answer. Using equations in these ways • enables students to learn concepts meaningfully • increases motivation and memory retention by connecting equations to concrete materials in their world • increases a student’s ability to deal with variations in problem type • enables students to understand the significance of their resolution of the problem • enables students to understand when to apply and not apply an equation While the theory presented in this article is fairly simplistic, the scientific reasoning is complex. Students must pass from empirical to numerical to symbolic understanding and then apply and generalize what they have learned in order to solve new problems. This activity attempts to facilitate a transition between these various levels of understanding. Giving students the equations directly is far less time consuming, but a student’s conceptual understanding is more important than saving time or just the “right answer.” REFERENCES

1. Resnick, L. B. Science 1983, 220, 477. 2. Lythcott, J. J. Chem. Educ. 1990, 67, 248. 3. Heller, J. I.; Reif, F. Cognition and Instruction 1984, 1, 177. 4. Sawrey, B. A. J. Chem. Educ. 1990, 67, 253.