Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is the degree Kelvin and is sometimes called absolute temperature. We will express temperature using the Kelvin scale. Although the Kelvin scale is the unit of temperature for the metric system we will most often encounter another temperature scale. It is the Celsius or Centigrade scale. The Centigrade scale is related to the Kelvin scale by a simple formula. This is the scale most frequently used by chemists. The temperature scale the we are most familiar with is the Fahrenheit scale. It is the scale of the English system of measurement. But which ever of the three scales that we use will be attempting by comparison to some standards to record the temperature (hotness or coldness) of a substance. OBJECTIVES 1. The student will identify the three temperature scales and list or identify the freezing point and boiling point of water as well as normal body temperature using the three scales. 2. The student will convert between the three scales given the formulas to within the accuracy of the data provided. 3. The student will use graphical analysis to determine the relationship between the Centigrade and Fahrenheit scales. DISCUSSION A. Temperature Scales Compared In comparing temperature scales it is necessary to identify some physical phenomena that always takes place at a particular temperature. We will use three such temperatures, the freezing point of water, the boiling point of water and normal body temperature. The following table includes the temperature of the three scales at these points. Included also is the common abbreviation for the term.
Scale Abbreviation Water Freezes Water Boils Normal Body Temperature Fahrenheit F 32 212 8.6 Centigrade C 0 100 37 Kelvin K 237 373 310 The scale that we are accustomed to is the Fahrenheit scale. The numbers associated with the three standard points have meaning to as with respect to hot and cold. We have a feeling for temperature when it is expressed using this scale. On the other hand, the Kelvin and Centigrade scales are just numbers at this time. Whether fifty degrees Centigrade (50 ) represents something that is hot or cold is difficult for us to determine. It will be our task in the next parts of this unit to develop formulas that will allow us to convert between the different scales, and in time we will develop a feeling for temperature no matter which scale is used. B. Kelvin Centigrade Conversations The connection between the Centigrade and Kelvin scales is the easiest to identify and work with. Let us take a look again at the data in the previous table with respect to these two scales. Scale Water Freezes Water Boils Normal Body Temperature C 0 100 37 K 273 373 310 The algebraic difference between the values of temperature at the three points is always the same. At the freezing point of water 273 K - 0 C =273 At the boiling point of water 373 K - 100 = 273 At Normal Body Temperature 310 K - 37 C = 273 The difference between the Kelvin scale and the Centigrade is always 273 or put another way the degrees Kelvin are always 273 plus the degrees Centigrade.
Following is a formula that describes this constant relationship K = C + 273 Example Problem (1) Change 50 C to K Using the formula substitute the value of 50 for the C in the equation. a) K = C + 273 b) K = 50 + 273 c) K = 323 K Note that 50 C and 323 K are the same temperature physically, just represented on two different scales. Example Problem (2) Change 847 K to C. Write the equation, substitute in the K, rearrange to solve for C and perform the calculation. a) K = C = 273 Formula b) 847 = C + 273 847 for K c) 847-273 = C + 273-273 Rearrangement d) 847 273 = C Calculation 574 = C Note that in step c) 273 was subtracted from both sides of the equation. This was done to clear one side of the equation of everything except the unknown quantity. This operation will work as long as you add or subtract the quantity from both sides of the equation.
C. Fahrenheit Centigrade Conversions The conversion between Centigrade and Kelvin was a simple addition or subtraction. Unfortunately the relationship between Centigrade and Fahrenheit is not so convenient. Let us compare the two scales and approach the problem in the same way as we did for the Centigrade to Kelvin conversion. Scale Water Freezes Water Boils Normal Body Temperature F 32 212 8.6 C 0 100 37 The algebraic difference between the points is not always the same. At the freezing point of water 32 F - 0 = 32 At the boiling point of water 212 F - 100 C = 112 At Normal Body Temperature 8.6 F - 37 C = 61.6 It will be necessary for us to find some other relationship between the two scales. This can be done by investigating the possibility of a linear (straight line) relationship between the Fahrenheit and Centigrade scale. To discover whether there is such a relationship will require us to make a graph and to plot on this graph temperatures in the Fahrenheit scale versus those of the Centigrade. We already have three sets of data points, that is those temperatures on the two scales that are physically equivalent, we need only to plot them and see what relationship arises. For this plot, let the y-axis be degrees Fahrenheit and the x-axis be the degrees Centigrade. Plotting the three points with respect to their Fahrenheit and Centigrade values would look something like the following:
Connecting the three points with a straight line shows us that there is indeed a linear relationship between the two scales. The exact nature of the relationship can be discovered by using the general equation for a straight line and applying it to this situation. This general equation is: y = mx + b The variables in this equation have the following meanings: y = any value of y that is on the line x = any value of x that is on the line m = the slope of the line which is the change in y divided by the corresponding change in x. Δy = ( Δ = change) Δx b = the y-intercept or the point at which the line crosses the y-axis. For our specific case in which we have plotted the degrees Fahrenheit versus the degrees Centigrade the equation could be written in the following fashion F = m C + b The variables in the specific equation have the following meanings F = any temperature in Fahrenheit that lies on the line, in our case 32, 8.6 and 212. C = any temperature in Centigrade that lies on the line, in our case 0, 37, and 100. m = = the slope of the line which is the change in the value of the Fahrenheit temperature divided by the corresponding change in the value of the Centigrade values. Δ F Δ C
b = the point at which the lines crosses the axis labeled with degrees Fahrenheit. To find the relationship between the Fahrenheit and Centigrade scales we must determine the slope of the line as well as the y-intercept. The y-intercept being the point at which the line crosses the y-axis is easily determined by inspecting the completed graph, it is 32. The slope is determined by picking any two points on the line (two points will determine a straight line) and dividing the change in the degrees Fahrenheit by the change in the degrees Centigrade. Let us pick the two points, one at the freezing point of water, the other at its boiling point. These points are: Freezing point Boiling point (0 C, 32 F) (100 C, 212 F) We use them in the following fashion to determine the slope. slope = m Δ F Δ C slope = 212 32 100 0 = Δ F Δ C 180 slope = 100 F slope = 1.8 C That is for every 1.8 degree change on the Fahrenheit scale there is a 1 degree change on the Centigrade. Substituting the values for the slope and the y-intercept into our original formula gives us the relationship between Fahrenheit and Centigrade. F = 1.8 C + 32 Looking at the graph again let us see how this equation was arrived at.
This derivation results in the equation: F = 1.8 C+ 32 and can be used to change from degrees Centigrade to degrees Fahrenheit as well as from degrees Fahrenheit to degrees Centigrade. A more familiar form of this equation is F = 5 x C + 32 where the fraction 5 is equal to 1.8.
D. Centigrade to Fahrenheit Conversions Centigrade to Fahrenheit conversions are carried out conveniently with the equation as it is written. Example Problem (3) Convert 20 C to F. Substitute the values in the formula and perform the conversion. a) F = 5 x C + 32 The formula b) F = 5 x 20 C + 32 Substitution of C made c) F = 180 + 32 5 F = 36 + 32 Calculations F = 68 A day that is 20 C would be a pleasant day of 68 F! Example Problem (4) Convert -25 C to F. a) F = 5 x C + 32 The formula b) F = 5 x (-25 ) + 32 Substitution of C made *NOTE* The negative sign must be carried with the number c) F = 225 + 32 Calculations 5 F = -45 + 32 F = -13
E. Fahrenheit to Centigrade Conversions Fahrenheit to Centigrade conversions are made with the same formula but by rearranging it. The following algebraic steps are used to perform the rearrangement. F = C + 32 5 a) Add a 32 to both sides F 32 = C + 32-32 5 Which is equivalent to * F 32 = C * 5 b) Multiply both sides by 5/. 5 5 ( F 32 ) = x C 5 Which is equivalent to 5 ( F 32 ) = C The algebraic manipulations can be made correctly as long as both sides of the equation are treated the same. The formula in its more familiar form is: C = 5 ( F 32 ) But remember this is simply our original formula in a different form.
Example Problem (5) Convert 100 F to C. a) C = 5 ( F 32 ) The formula b) C = 5 (100-32 ) Substitution of the F. c) C = 5 (68) Example Problem (6) C = 340/ Note that the operation within the parenthesis is done first C = 37.8 Convert 32 F to C. a) C = 5 ( F 32 ) The formula b) C = 5 (-32-32 ) Substitution of F. c) C = 5 (-64) C = 320 C = -35.5
PROBLEMS 1. Make a table and compare the Fahrenheit, Centigrade and Kelvin scales. Include on the table the abbreviation used for each as well as the temperature at the freezing point of water, boiling point of water and normal body temperature. 2. Convert the following temperatures from degrees Centigrade to Kelvin. a. -210 C b. -100 C c. 0 C d. +55 C e. 1055 C 3. Convert the following from Centigrade to Fahrenheit. a. -210 C b. -40 C c. 0 C d. 80 C e. 10,500 C 4. Convert the following from Fahrenheit to Centigrade. a. 8,500 F b. 0 F c. -55 F d. 35 F e. -40 F 5. Following is a list of equivalent Centigrade and Fahrenheit temperatures. On a piece of graph paper plot the F vs C, determine the slope and y-intercept as well as the equation that results. F C 50 10 68 20 86 30 104 40 122 50