Research: Science and Education
Temperature Scale Conversion as a Linear Equation: True Unit Conversion vs Zero-Offset Correction Reuben Rudman* Department of Chemistry, Adelphi University, Garden City, NY 11530
The rarity of addition/subtraction operations in unit conversions is readily perceived from the Guide for the Use of the International System of Units (SI) (1). Of the more than 500 factors given for converting unit quantities to acceptable SI units, the only conversion factors that include addition/ subtraction operations are those relating to the temperature scales. The proportionality constant (9/5) used to convert any temperature interval on the Celsius scale to the corresponding temperature interval on the Fahrenheit scale is the same at any point on the thermometric scale. However, if one wishes to convert a temperature reading from one scale to another, then the proportion of 9/5 is not of itself sufficient because the scale offset, due to the arbitrary selection of the zero point in each scale, must also be taken into account. The resulting difficulties students have in grasping the whys and wherefores of temperature scale manipulations have been discussed in these pages in the past (2–6 ), most recently by Nordstrom (7) . Most of the discussions (2, 3, 5, 6 ) were devoted to the techniques and methods used to convert between the Celsius (C) and Fahrenheit (F) scales. However, none of these previous articles presented the main points of this present discussion: 1. The general equation for a straight line (y = ax + b) is the paradigm for many of the calculations taught in introductory chemistry. 2. Most unit conversions are special cases of this equation where a ≠ 1 and b = 0 (slope not equal to 1 and y-intercept equal to zero). 3. Many instrumental corrections are special cases of this equation where a = 1 and b ≠ 0. 4. Many of the conversions between the various common temperature scales are general cases of this equation (a ≠ 1 and b ≠ 0) wherein the two operations used in effecting this conversion are (i) a zero-point offset correction and (ii) a normal conversion-factor procedure.
Since most students taking introductory chemistry, at either the high school or college level, are already familiar with the general equation for a straight line, it is possible to drastically improve their comprehension, retention, and selfconfidence in the early stages of the course by relating these “new” fundamental concepts and quantitative manipulations to a subject they have already learned. Nordstrom (7 ) focused his discussion on the conversion of Celsius to Kelvin. He introduced concepts used by statisticians (8, 9) in describing general measurement scales: nominal (assignment to categories only), ordinal (qualitative ranking within categories), interval (quantitative, with equal intervals between adjacent measurements, but no absolute zero-point), and ratio (same as interval, with an absolute zeropoint). Therefore, only ratio scales allow for proportional *Email:
[email protected].
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calculations (see point 2, above); that is, “we can multiply or divide numbers only on the ratio scale (7).” Also, to quote one statistics introductory textbook, “a 20 degree temperature relative to a 40 degree temperature is not in the same ratio as 60 degrees is to 120 degrees” (8) where the unstated implication is that an absolute scale is not being used. However, it is clear that the Kelvin1 (K) and Rankine (R) scales, which are based on absolute zero, are ratio scales and one can convert between them without the need for addition or subtraction. In general, variables that are related to each other by an equation for a “straight line with a nonzero intercept” correspond to the statisticians’ interval scale, while a “straight line with a zero intercept” relationship describes their ratio scale. In order to be able to convert from one scale to another using a simple proportionality constant (as is the case for most of the 500+ conversion factors listed by NIST [1 ]), it is necessary for there to be a straight-line relationship with a zero intercept. Unfortunately, by concentrating on the differences between unit and scale conversions, Nordstrom has not fully explained the subject of temperature-scale conversion. Furthermore, he incorrectly states that temperature scales are the only examples of interval scales that are of interest to chemists. It is true that temperature conversions are the only unit conversions in which the nonzero intercept is incorporated into the standard conversion formula. However, it is not the only instance of nonzero intercepts encountered in experimental measurements. Any data set in which the instrumental zero point is not a true zero and which requires a zero-offset correction falls into the category of the statisticians’ interval scale.2 In order to convert any of these interval scale measurements to a true ratio scale, it is necessary to correct for the true zero point. Thus, it is important to realize that, contrary to what is found in statistics texts (usually written by individuals with no training in chemistry or physics) and as brought to the attention of the readers of this Journal (7), interval scales are very commonly encountered in experimental chemistry and physics. However, they are usually not treated as actual scales but rather as “correction terms” to be applied to the data before doing the “actual” calculations. Ideally, the same treatment should be used in F ↔ C conversions. Instead we blend both steps into one equation and call it a scale conversion, thereby confusing our students (and ourselves?). Temperature Scale Conversion Procedures Understanding that the C ↔ F interconversion formula [°F = (9/5) (°C) + 32] is a straight-line equation helps the student follow the steps involved in the temperature conversion. Since each scale has its own arbitrary, though universally accepted, zero point, conversion from one temperature scale to another generally requires two operations: adjustment of the zero point (by addition or subtraction) and/or conversion of the temperature intervals (by multiplication or division).
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
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F
In all these cases, it is necessary to correct for the zero point. This explains why calculations not involving a temperature interval require the use of the Kelvin scale (in the metric system).
C
5 1
2 6
R
3
K
Figure 1. The six possible temperature-scale conversions. The six equations are R = F + 459.67 (1) K = C + 273.15
(2)
R = (9/5)K
(3)
F = (9/5)C + 32
(4)
F = (9/5)K – 459.67
(5)
R = (9/5)C + 491.67 (6) F ↔ R and C ↔ K are changes in zero-point only. K ↔ R is a change in degree-size only (since 0 °K = 0 °R). F ↔ C, C ↔ R, and F ↔ K entail both zero-point and degree-size changes. Each conversion can proceed in either direction, so there are actually 12 possible equations.
The steps to be used and the order in which they are applied depend on the particular conversion that is being performed. The four temperature scales (C, K, F, R) result in six types of temperature conversions, as shown in Figure 1. The six corresponding equations are general or special cases of the straight-line equation y = ax + b, where y and x are the temperature values, a is the degree conversion factor, and b is the zero-point “correction” term. Of the six, C ↔ K and F ↔ C are the most commonly encountered conversions in introductory chemistry.
Temperature Interval Conversion To convert between the metric and English systems simply apply the scale conversion multiplier ∆ tE = 1.8 ∆ tm, where E represents the English system (Fahrenheit or Rankine degrees) and m the metric system (Celsius or Kelvin degrees). As long as a range of temperature is considered, there is no need to use an absolute scale. This concept is applicable, for example, when a temperature range such as the ∆ t used in heat capacity calculations is encountered. Similarly, the common teaching-lab strategy of not relying on the zero-point of a balance and always weighing by difference is the equivalent of using temperature intervals where the actual zero-point is irrelevant.3 To convert between C and K or between F and R, apply the zero-point offset correction of 273.15 or 459.67, respectively. Temperature Scale Conversion In this case, we must consider not only the degree conversion but also the zero-point offset correction (Fig. 1). Note that for many of the equations in which temperature appears, use of the Celsius scale is conceptually equivalent to using a faulty measuring device (e.g., a balance with a zero offset or a meter stick which is missing the first couple of centimeters).
Alternative Scale-Conversion Procedure An alternative systematic method for converting between scales, which is the same for all scales and for changes in either direction (e.g., C → F and F → C), involves three steps: 1. Convert the starting temperature to its absolute scale; that is, adjust for the zero point; and/or 2. Convert by true scale conversion; that is, multiply or divide by 1.8, thereby obtaining the equivalent absolute temperature in the other scale; and/or 3. Adjust to the relative scale in the new system by correcting for its zero point.
For example, to convert 25.0 °C to °F: Step 1: 25.0 + 273.15 = 298.15 C → K Step 2: 298.15 × 1.8 = 536.67 K→R Step 3: 536.67 – 459.67 = 77.0 R → F This procedure makes the conversion process crystal clear and separates the steps of zero-point adjustment and scale conversion. In summary, the temperature-scale conversion formula is an example of the general straight-line equation, whereas most other unit conversion formulas are special cases of this equation. Notes 1. Because several temperature scales are being compared, for the sake of uniformity of expression the older, non-SI form of the Kelvin scale (based on degrees Kelvin) is used. 2. Modern discussions of statistics (8, 9) present the measurement scales discussed by Nordstrom. Here too the only example of an interval scale that is described is the F ↔ C conversion. 3. For example, during a laboratory session where measurements of mass must be taken several hours apart there may be a zero-point drift on the balance or another student may have changed the zero point. In order to eliminate this source of error, students are instructed to make all their measurements independent of the zero reading of the balance, by determining each mass using a tare.
Literature Cited 1. Taylor, B. N. Guide for the Use of the International System of Units (SI); NIST Special Publication 811; U.S. Department of Commerce, National Institute of Standards and Technology: Gaithersburg, MD, 1995; Appendix B.8, pp 46–68. 2. Blann, J. G. J. Chem. Educ. 1930, 7, 2946. 3. Midgeley, C. P. J. Chem. Educ. 1965, 42, 322 (also, see responses on p 646). 4. Ander, P. J. Chem. Educ. 1971, 48, 325. 5. Estok, G. K. J. Chem. Educ. 1973, 50, 495. 6. Gorin, G. J. Chem. Educ. 1980, 57, 350. 7. Nordstrom, B. H. J. Chem. Educ. 1993, 70, 827. 8. Federer, W. T. Statistics and Society, 2nd ed.; Dekker: New York, 1991; pp 20–21. 9. Olinick, M. An Introduction to Mathematical Models in the Social and Life Sciences; Addison-Wesley: Reading, MA, 1978; pp 224–226.
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