Experimenting with Temperature Conversions

Lisa Goldsmith

Math 3190 A

Abstract: After a brief description on thermal energy and how it is measured, the student will discover how information from both the Fahrenheit and Celsius temperature scales can derive an equation of a line relating the two. The student will graph the line on their graphing calculator and use the value function of the calculator to quickly convert between degrees Fahrenheit and degrees Celsius. After closely following directions given for this lesson, the student will be asked to repeat the same steps in creating a line and graphing to relate the Kelvin and Celsius temperature scales. The student will think critically and describe similarities between the equation of lines they have derived and the formulas that convert between the different scales.

This lesson is designed for an upper middle grades student who has/is taken/taking Algebra 1 or a freshman/sophomore in high school. The lesson would be great for a student in a physical science course.

Difficulty level: Medium

Prerequisites:

· Knowledge and moderate efficiency in deriving an equation of a line given two points.

· General Algebra skills.

· Adequate knowledge of a graphing calculator including how to enter an equation of a line, graphing the line, changing the viewing window for the graph.

· Background knowledge on potential and kinetic energy.
Experimenting with Temperature Conversions

Have you ever thought about why things become hot? When you place your Pizza Rolls in the microwave they become hot. But makes them hot? As the microwave cooks the Pizza Rolls the molecules in the food move faster – gaining kinetic energy.

Ø Remember that kinetic energy is the energy of motion.

Ø You could say that the warmer an object becomes the greater kinetic energy it has. You can also say that the warmer an object becomes the more TERMAL ENEGRY it contains.

Ø Thermal energy is the TOTAL energy (Kinetic and Potential) the atoms and molecules of a substance contain.

When you measure temperature, you are essentially measuring the amount of thermal energy a substance contains. If you are measuring the temperature of ice cream you’ll find that it’s much lower than the temperature of a cup of hot chocolate. Molecules and atoms move much slower in substance that we describe as “cold” and much faster in hot chocolate, which we describe as “hot”.

There are different temperature scales that are used as standards, or for reference, when measuring the temperature of a substance. Two scales that you may be most familiar with are the Fahrenheit and Celsius scales.

Ø In the United States the Fahrenheit scale is commonly used when referring to everyday things.

o The temperature you see on television in the morning that tells you how hot or cold it is outside is in degrees Fahrenheit.

o When you go to the doctor and the nurse takes your temperature she is measuring it in degrees Fahrenheit.

Ø The Fahrenheit scale was originated by a German physicist, G.D. Fahrenheit.

Ø On the Fahrenheit scale, 32° is assigned to the temperate at which water freezes and 212° is assigned to the temperature at which water boils.

Ø The Celsius scale is the scale that is used most frequently around the world and in the science laboratory.

Ø Andres Celsius, a Swedish astronomer, first suggested the use of this scale.

Ø On the Celsius scale, 0° is given to the temperature at which water freezes and 100°is the temperature at which water boils.

o There is a 100-degree difference between the freezing and boiling point of water on the Celsius scale.

§ What is the difference between the freezing and boiling point of the Fahrenheit scale? ________

As a scientist, sometimes we are given information in the Fahrenheit scale and we need to have it in degrees Celsius for our experiment, or vise versa.

So what do we do? How do we take the degrees Fahrenheit and change it into degrees Celsius so we can use it?

This is when our math skills come in!! Arithmetic formulas have been created to covert between the two scales.

Ø Fahrenheit to Celsius °C= (ºFahrenheit –32) x (5/9)

Ø Celsius to Fahrenheit °F= (ºCelsius x 9/5) + 32

Try a few!

45°F = ________°C

85°C = ________°F

There is still another way to convert between the Fahrenheit and Celsius scale that you will discover momentarily. Using your graphing calculator, you will graph a line that relates both the Fahrenheit and Celsius scales and then you can convert between the two with just a push of a button! Let’s get started!!

In order to graph a line we need two points to determine the slope.

32° Fahrenheit = 0° Celsius (FREEZING)

212° Fahrenheit = 100° Celsius (BOILING)

With this information you can translate it into X and Y coordinates; X corresponding to Fahrenheit and Y corresponding to Celsius.

( ____________°F, ____________°C)

Now that you have two points you can find the slope of the points.

(Remember Y2-Y1 )

X2 –X1

Your slope should be in fraction form. If it’s not, go to the Math button on the calculator and select 1: Frac and then hit Enter and it will change the decimal answer into a fraction.

Using the slope-intercept form of a line, Y = Mx + B, where M is the slope and B is the y-intercept, we must now find B by plugging in one of the points into our equation with the slope.

Once you have the y-intercept, B, you have the equation for the line that will convert between the Fahrenheit and Celsius scales.

The next step is to enter the equation into your graphing calculator.

At the Y= menu type in your equation. Remembering to use parentheses () around the slope.

After you’ve entered in the equation press the graph button to view the graph.

OOPS! The graph’s not there!! We need to adjust the viewing window so that we can see the graph of our line.

Enter these values in the Window for the graph:

Xmin= -100

Xmax=220

Ymin=-100

Ymax=220

Now that you can see the graph, use the trace feature to first trace along the line. Notice that the X value represents °F and the Y value represents °C … this is much easier than using the formulas isn’t it!

To find out exact X and Y values, press the 2nd button and then the Trace button. This brings up the Calculate menu or Calc menu for short. We want to find values so press the number one, or hit enter.

It will ask you for the X value (Fahrenheit) and it will give you the corresponding Y value (Celsius). Try it!!

Here are a few conversions for you to evaluate using the value feature on your graphing calculator. *Round to two decimal places.

1. 98.6°F(normal temperature of the human body) = ____________ °C

2. 84°F (the temperature of a warm spring day) = ______________ °C

3. 134°F(highest temperature observed in US, Greenland Ranch, CA 1913) =____________°C

4. 32 °F (water freezes) = _____________ ºC

5. 212ºF (water boils)= ______________ºC

Try some on your own fill in the blanks for the numbers you chose and write the corresponding temperature in degrees Celsius. *Note: If you receive an error message when you type in a value you should change your viewing window to include the number you are trying to convert.

6. °F= ______________ºC

7. °F= ______________ºC

8. °F= ______________ºC

9. °F= ______________ºC

10. °F= ______________ºC

Temperature has no upper limit, in other words, a substance will continue to become hotter as the molecules and atoms increase their speed. However, there is a place where all molecular motion comes to a stop, this is Absolute Zero.

Ø Absolute Zero is the temperate at which molecules have lost all available kinetic energy.

Absolute zero is referred to on a temperature scale that many scientists use called the Kelvin scale. The Kelvin scale is part of the International System of Measurements (SI). This scale is named after a British mathematician and physicist, William Thomson, First Baron Kelvin.

Ø 0K (absolute zero) is the temperature at which all motion stops

Ø 273K is the melting point of ice

Ø 373K is the boiling point of water

The Kelvin scale is similar to the Celsius scale in that the divisions between melting and boiling are the same (373 – 273 = 100 and 100 – 0 = 100), which makes it easy to convert between the two scales.

Ø K = ºC + 273

Ø ºC = K - 273

Following the same procedure as before with the Fahrenheit and Celsius temperature scales, use the information provided below to find the equation for the line relating the Kelvin and Celsius scales and graph the line in your graphing calculator.

273K = 0ºCelsius (FREEZING)

373K = 100ºCelsius (BOILING)

Remember you will need to change your window to be able to accurately see the graph of the line.

Answer the following questions by using the value function on your calculator.

1. 5K = ______________ºC

2. 0K = ______________ºC

3. 115K = _____________ºC

4. 275K = _____________ºC

5. 350K = _____________ºC

6. 273K = _____________ºC

7. 373K = _____________ºC

Try you own now!

8. ____________K = _____________ºC

9. ____________K = _____________ºC

10. ____________K = _____________ºC

Compare your equation of the line relating both Fahrenheit/Celsius and Kelvin/Celsius to the formula that converts between the two given in the handout. Do you see any similarities? What conclusions can you draw from this?

Answer Key

Ø What is the difference between the freezing and boiling point of the Fahrenheit scale? ___180°_____

Try a few!

45°F = __7.22______°C

85°C = ___185_____°F

With this information you can translate it into X and Y coordinates; X corresponding to Fahrenheit and Y corresponding to Celsius.

( _____32_______°F, _____0_______°C)

( _____212_______°F, _____100_______°C)

Now that you have two points you can find the slope of the points.

(Remember Y2-Y1 )

X2 –X1

M= (100-0) ¸ (212-32) = .5556 = 5/9

Using the slope-intercept form of a line, Y = Mx + B, where M is the slope and B is the y-intercept, we must now find B by plugging in one of the points into our equation with the slope.

Y= (5/9)X + B

0= (5/9)(32) + B

B= -17.78

Y= (5/9)X – 17.78

1. 98.6°F(normal temperature of the human body) = ___37.0_________ °C

2. 84°F (the temperature of a warm spring day) = ______28.87________ °C

3. 134°F(highest temperature observed in US, Greenland Ranch, CA 1913)=____56.66____°C

4. 32 °F (water freezes) = ____0_________ ºC

5. 212ºF (water boils)= _____100_________ºC

6 – 10 Answers will vary depending what they choose.

273K = 0ºCelsius (FREEZING)

373K = 100ºCelsius (BOILING)

M = (100-0) ¸ (373-273) = 1

Y= 1X +B

0= 273 + B

B= -273

Y= X – 273

Window

Xmin=-100

Xmax=373

Ymin=-200

Ymax=250

1. 5K = _____-268_________ºC

2. 0K = _____-273_________ºC

3. 115K = _____-158________ºC

4. 275K = ______2_______ºC

5. 350K = ______77_______ºC

6. 273K = ______0_______ºC

7. 373K = _____100________ºC

8-10 Answers will vary.

°C= (ºFahrenheit –32) x (5/9) and Y= (5/9)X – 17.78

You can see that the slope of the line and the multiplying factor in the formula are the same.

ºC = K – 273 and Y= X – 273

Both the equation of the line and the conversion formula from Kelvin to Celsius are the same.

The conversion formulas that are used today were probably derived using the same steps you used during this activity.