### Learning Objectives

After completing this section, you should be able to:

- Convert between Fahrenheit and Celsius.
- Identify reasonable values for temperature applications.
- Solve application problems involving temperature.

When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature.

In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor).

### Checkpoint

*When reading temperatures, it’s important to look beyond the degree symbol to determine which temperature scale the units express. For example, 13 °C reads “13 degrees Celsius,” indicating that the temperature is expressed using the Celsius scale, while 13 °F reads “13 degrees Fahrenheit,” indicating that the temperature is expressed using the Fahrenheit scale.*

### Who Knew?

#### How Many Temperature Scales Are There?

Did you know that in addition to Fahrenheit and Celsius, there is a third temperature scale widely used throughout the world? The Kelvin scale starts at absolute zero, the lowest possible temperature at which there is no heat energy present at all. It is primarily used by scientists to measure very high or very low temperatures when water is not involved.

### Converting Between Fahrenheit and Celsius Temperatures

Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember.

But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at?

Figure 9.16 lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression.

### FORMULA

The formulas used to convert temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in Table 9.3.

Fahrenheit to Celsius | Celsius to Fahrenheit |
---|---|

$C=\frac{5}{9}(F-32)$ | $F=\frac{9}{5}C+32$ |

### Example 9.40

#### Converting Temperatures from Fahrenheit to Celsius

A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius?

#### Solution

Use the formula in Table 9.3 to convert from Fahrenheit to Celsius.

So, 392 °F is equivalent to 200 °C.

### Your Turn 9.40

### Example 9.41

#### Converting Temperatures from Celsius to Fahrenheit

On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit?

#### Solution

Use the formula in Table 9.3 to convert from Celsius to Fahrenheit.

The temperature was 68 °F.

### Your Turn 9.41

### Example 9.42

#### Comparing Temperatures in Celsius and Fahrenheit

A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator?

#### Solution

Use the formula in Table 9.3 to convert from Celsius to Fahrenheit.

Then, compare the temperatures.

Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator.

### Your Turn 9.42

### Reasonable Values for Temperature

While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in Table 9.4.

### FORMULA

The formulas used to estimate temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in Table 9.4.

Fahrenheit to Celsius | Celsius to Fahrenheit |
---|---|

$C=\frac{F-30}{2}$ | $F=2C+30$ |

### Example 9.43

#### Using Benchmark Temperatures to Determine Reasonable Values for Temperatures

Which is the more reasonable value for the temperature of a freezer?

- 5 °C or
- –5 °C?

#### Solution

We know that water freezes at 0 °C. So, the more reasonable value for the temperature of a freezer is −5 °C, which is below 0 °C. At temperature of 5 °C is above freezing.

### Your Turn 9.43

- 48 °C or
- 148 °C?

### Example 9.44

#### Using Estimation to Determine Reasonable Values for Temperatures

The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body temperature in degrees Celsius:

- 98.6 °C,
- 64.3 °C, or
- 34.3 °C?

#### Solution

To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, and divide the result by 2.

A reasonable value for average body temperature is 34.3 °C.

### Your Turn 9.44

- 102 °C,
- 37 °C, or
- 20 °C?

### Example 9.45

#### Using Conversion to Determine Reasonable Values for Temperatures

Which is a reasonable temperature for storing chocolate:

- 28 °C,
- 18 °C, or
- 2 °C?

#### Solution

Use the formula in Table 9.3 to determine the temperature in degrees Fahrenheit.

A temperature of 82.4 °F would be too hot, causing the chocolate to melt. A temperature of 35.6 °F is very close to freezing, which would affect the look and feel of the chocolate. So, a reasonable temperature for storing chocolate is 18 °C, or 64.4 °F.

### Your Turn 9.45

### Solving Application Problems Involving Temperature

Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems.

### Example 9.46

#### Using Subtraction to Solve Temperature Problems

The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the freezer?

#### Solution

Use subtraction to find the difference.

So, the temperature in the freezer is −17 °C.

### Your Turn 9.46

### Example 9.47

#### Using Addition to Solve Temperature Problems

A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was the temperature after the scientist heated it?

#### Solution

Use addition to find the new temperature.

The temperature of the liquid was 41 °C after the scientist heated it.

### Your Turn 9.47

### Example 9.48

#### Solving Complex Temperature Problems

The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C every 15 minutes. How long until the compound reaches its optimum temperature?

#### Solution

**Step 1:** Determine the optimum temperature in degrees Celsius using the formula in Table 9.3.

**Step 2:** Subtract the starting temperature.

**Step 3:** Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature.

**Step 4:** Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes.

So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature.

### Your Turn 9.48

### Check Your Understanding

### Section 9.5 Exercises

37 °C or

37 °F

32 °C or

0 °C

212 °C

or 212 °F

8 °C,

23 °C, or

75 °C

33 °C,

30 °C, or

3 °C

700 °C,

70 °C, or

7 °C

- 60 °C,
- 40 °C, or
- 20 °C?