In the Classroom
An Interactive Graphical Approach to Temperature Conversions Jonathan Mitschele Chemistry Department, Saint Joseph’s College, Standish, ME 04084;
[email protected] The relationship of temperature readings on different temperature scales is a common topic in introductory chemistry texts (1-4), and the difficulties students have with temperature scale conversions have been discussed several times over the past 70 years in this Journal (5-11). A variety of formulas for converting temperatures on one scale to those on another are offered by textbooks, usually with little more elaboration than the inclusion of a picture of two thermometers using two different scales, and often with little attention given to the units involved. Examples of such equations for converting from Celsius temperatures to Fahrenheit temperatures include the following:
Table 1. Celsius and Fahrenheit Temperatures as a Function of the Length of the Column of Liquid in a Thermometer
9 Temperature °F = Temperature °C + 32 5
L/ cm
T/ °C
T/ °F
0.00
-20.0
––––
2.00
-8.1
18.7
4.00
3.9
38.8
6.00
15.3
59.5
8.00
27.0
80.3
9 °F °C + 32 °F ? °F = 5 °C 100
t°F = 1.8 t°C + 32 9 °F (t°C ) + 32 °F 5 °C
My experience is that students, confronted with such recipes in a textbook, simply memorize them. Given that this is done by rote, with no understanding of what is involved in the derivation of these formulas, it is not surprising that retention is brief and misunderstanding all too common. For several years I have used a simple graphical approach in my classes to explain temperature conversions. Unlike the usual textbook approach, the meaning of conversion formulas is made clear and units are used in a consistent manner. I begin the exercise by passing out dual temperature thermometers (Fisher-EMD alcohol-in-glass thermometers, model S41576G) and 15-cm rulers to pairs of students. I ask each student pair to find the Celsius and Fahrenheit temperatures at 2-cm intervals along the length of the thermometer, beginning at the bottom of the Celsius scale (᎑20 °C on our thermometers). I then use student values to construct a threecolumn table and plot a graph; Table 1 and Figure 1 show typical results for the first five measurements. The linear relationships between temperatures measured with either Celsius or Fahrenheit temperature scales and the length of the column of liquid in the thermometer is clearly evident as shown in Figure 1. Implicit in Table 1 is a relationship between Celsius and Fahrenheit temperatures. After discussing these points, I plot corresponding pairs of Celsius and Fahrenheit temperatures to determine the relationship between them. One can use the Celsius and Fahrenheit values collected as a function of column length for this purpose. Alternatively, one can have students find the Fahrenheit temperatures that correspond to the set of Cel-
80
Temperature / (°C or °F)
t °F =
60
40
20
0
-20
-40 0
2
4
6
8
10
Column Length / cm Figure 1. Temperature versus column length of a thermometer. Circles represent Celsius temperature values; squares represent Fahrenheit temperature values.
sius values, 0 °C to 100 °C in 10 °C increments; the corresponding Fahrenheit temperatures have whole number values and can be read accurately on the thermometer. There are some advantages to this data set: there is less scatter in the data students collect this way and the significance of the ratio 180 °F/100 °C is made clearer. Table 2 lists pairs of temperature values for the second choice of data sets and a plot of the data is shown in Figure 2. The relationship between Fahrenheit and Celsius temperatures is obviously linear, with a slope of 180 °F/100 °C and an intercept of 32 °F. Therefore, the equation used to
JChemEd.chem.wisc.edu • Vol. 79 No. 10 October 2002 • Journal of Chemical Education
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In the Classroom 250
Table 2. Fahrenheit Temperatures as a Function of the Celsius Temperatures T/ °F
0.0
32.0
10.0
50.0
20.0
68.0
212 200
Temperature / °F
T/ °C
150
30.0
86.0
40.0
104.0
50.0
122.0
60.0
140.0
50
70.0
158.0
32
80.0
176.0
90.0
194.0
100.0
212.0
rise = 180.0
100
run = 100.0
0 -20
0
20
40
60
80
100
120
Temperature / °C Figure 2. Fahrenheit temperatures versus Celsius temperatures.
convert Celsius temperatures into Fahrenheit temperatures can be written as: 180 °F Temperature °F = Temperature °C + 32 °F 100 °C where Temperature °C represents the Celsius temperature with units °C and Temperature °F the Fahrenheit temperature with units °F. The significance and units of the multiplicative factor 180 °F/100 °C and the additive term 32 °F are obvious, and the conversion equation, thus understood in terms of a graphical representation, is not so likely to be easily forgotten by students. The relationship between the Celsius and Kelvin scales can be discussed at this point. It is easy to show that the equation used to convert Celsius temperatures into Kelvin temperatures can be written as: 1K Temperature K = Temperature °C + 273 K 1 °C where Temperature °C represents the Celsius temperature
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with units °C and Temperature K the Kelvin temperature with units K. Literature Cited 1. Chang, R. Chemistry, 6th ed.; McGraw–Hill: New York, 1998; p 18. 2. Jones, L.; Atkins, P. Chemistry: Molecules, Matter, and Change, 4th ed.; W. H. Freeman: New York, 1999; p 57. 3. Masterton, W. L.; Hurley, C. N. Chemistry: Principles and Reactions, 4th ed.; Harcourt: Philadelphia, 2001; p 9. 4. Oxtoby, D. W.; Gillis, H. P.; Nachtrieb, N. H. Principles of Modern Chemistry, 4th ed.; Saunders: Philadelphia, 1999; p A11. 5. Blann, J. G. J. Chem. Educ. 1930, 7, 2946. 6. Midgeley, C. P. J. Chem. Educ. 1965, 42, 322 (see responses to this article, p 646). 7. Ander, P. J. Chem. Educ. 1971, 48, 325. 8. Estok, G. K. J. Chem. Educ. 1973, 50, 495. 9. Gorin, G. J. Chem. Educ. 1980, 57, 350. 10. Nordstrom, B. H. J. Chem. Educ. 1993, 70, 827. 11. Rudman, R. J. Chem. Educ. 1998, 75, 1646.
Journal of Chemical Education • Vol. 79 No. 10 October 2002 • JChemEd.chem.wisc.edu
