#set document(title: "9.5 Measuring Temperature", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.5#h(0.6em)Measuring Temperature #figure(figph[A close-up view of a thermometer shows 19 degrees Celsius and 66 degrees Fahrenheit. The thermometer is positioned near a plant.], alt: "A close-up view of a thermometer shows 19 degrees Celsius and 66 degrees Fahrenheit. The thermometer is positioned near a plant.", caption: [A thermometer that measures temperature in both customary and metric units.]) === 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? #strong[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). #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[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.] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Misconceptions_About_Temperature")[Misconceptions About Temperature] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[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? 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. #figure(figph[An illustration of a thermometer shows temperature in both Fahrenheit and Celsius.], alt: "An illustration of a thermometer shows temperature in both Fahrenheit and Celsius.", caption: [Common Temperatures]) #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The formulas used to convert temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in . #figure(table( columns: 2, align: left, inset: 6pt, table.header([Fahrenheit to Celsius], [Celsius to Fahrenheit]), [#math.equation(block: false, alt: "C equals the fraction 5 over 9 open parenthesis F minus 32 close parenthesis")[$C = frac(5, 9) ( F − 32 )$]], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$]], )) ] #examplebox("Example 1")[Converting Temperatures from Fahrenheit to Celsius][ A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius? #solutionbox[ Use the formula in to convert from Fahrenheit to Celsius. #math.equation(block: true, alt: "C, equals, the fraction 5 over 9 open parenthesis F minus 32 close parenthesis; C, equals, the fraction 5 over 9 open parenthesis 392 minus 32 close parenthesis; C, equals, the fraction 5 over 9 open parenthesis 360 close parenthesis; C, equals, 200")[$C & = & frac(5, 9) ( F − 32 ) \ C & = & frac(5, 9) ( 392 − 32 ) \ C & = & frac(5, 9) ( 360 ) \ C & = & 200$] So, 392 °F is equivalent to 200 °C. ] ] #examplebox("Example 2")[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? #solutionbox[ Use the formula in to convert from Celsius to Fahrenheit. #math.equation(block: true, alt: "F, equals, the fraction 9 over 5 C plus 32; F, equals, the fraction 9 over 5 open parenthesis 20 close parenthesis plus 32; F, equals, 36 plus 32; F, equals, 68")[$F & = & frac(9, 5) C + 32 \ F & = & frac(9, 5) ( 20 ) + 32 \ F & = & 36 + 32 \ F & = & 68$] The temperature was 68 °F. ] ] #examplebox("Example 3")[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? #solutionbox[ Use the formula in to convert from Celsius to Fahrenheit. #math.equation(block: true, alt: "F, equals, the fraction 9 over 5 C plus 32; F, equals, the fraction 9 over 5 open parenthesis 3 close parenthesis plus 32; F, equals, 5.4 plus 32; F, equals, 37.4")[$F & = & frac(9, 5) C + 32 \ F & = & frac(9, 5) ( 3 ) + 32 \ F & = & 5.4 + 32 \ F & = & 37.4$] Then, compare the temperatures. #math.equation(block: true, alt: "36 F ° less than 37.4 F ° less than 46 F °")[$36 #h(0.28em) attach(upright(F), tl: °) < 37.4 #h(0.28em) attach(upright(F), tl: °) < 46 #h(0.28em) attach(upright(F), tl: °)$] Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator. ] ] === 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 . #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The formulas used to estimate temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in . #figure(table( columns: 2, align: left, inset: 6pt, table.header([Fahrenheit to Celsius], [Celsius to Fahrenheit]), [#math.equation(block: false, alt: "C equals the fraction F minus 30 over 2")[$C = frac(F − 30, 2)$]], [#math.equation(block: false, alt: "F equals 2 C plus 30")[$F = 2 C + 30$]], )) ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Temperature_Conversion_Trick")[Temperature Conversion Trick] ] #examplebox("Example 4")[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? #solutionbox[ 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. ] ] #examplebox("Example 5")[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? #solutionbox[ To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, and divide the result by 2. #math.equation(block: true, alt: "the fraction open parenthesis 98.6 minus 30 close parenthesis over 2 equals the fraction 68.6 over 2 equals 34.3")[$frac(( 98.6 − 30 ), 2) = frac(68.6, 2) = 34.3$] A reasonable value for average body temperature is 34.3 °C. ] ] #examplebox("Example 6")[Using Conversion to Determine Reasonable Values for Temperatures][ Which is a reasonable temperature for storing chocolate: - 28 °C, - 18 °C, or - 2 °C? #solutionbox[ Use the formula in to determine the temperature in degrees Fahrenheit. #math.equation(block: true, alt: "F, equals, the fraction 9 over 5 C plus 32; F, equals, the fraction 9 over 5 open parenthesis 28 close parenthesis plus 32 equals 82.4; F, equals, the fraction 9 over 5 open parenthesis 18 close parenthesis plus 32 equals 64.4; F, equals, the fraction 9 over 5 open parenthesis 2 close parenthesis plus 32 equals 35.6")[$F & = & frac(9, 5) C + 32 \ F & = & frac(9, 5) ( 28 ) + 32 = 82.4 \ F & = & frac(9, 5) ( 18 ) + 32 = 64.4 \ F & = & frac(9, 5) ( 2 ) + 32 = 35.6$] 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. ] ] === 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. #examplebox("Example 7")[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? #solutionbox[ Use subtraction to find the difference. #math.equation(block: true, alt: "4 minus 21 equals minus 17")[$4 − 21 = − 17$] So, the temperature in the freezer is −17 °C. ] ] #examplebox("Example 8")[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? #solutionbox[ Use addition to find the new temperature. #math.equation(block: true, alt: "35 plus 6 equals 41")[$35 + 6 = 41$] The temperature of the liquid was 41 °C after the scientist heated it. ] ] #examplebox("Example 9")[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? #solutionbox[ #strong[Step 1:] Determine the optimum temperature in degrees Celsius using the formula in . #math.equation(block: true, alt: "C, equals, the fraction 5 over 9 open parenthesis F minus 32 close parenthesis; C, equals, the fraction 5 over 9 open parenthesis 392 minus 32 close parenthesis; C, equals, the fraction 5 over 9 open parenthesis 360 close parenthesis; C, equals, 200")[$C & = & frac(5, 9) ( F − 32 ) \ C & = & frac(5, 9) ( 392 − 32 ) \ C & = & frac(5, 9) ( 360 ) \ C & = & 200$] #strong[Step 2:] Subtract the starting temperature. #math.equation(block: true, alt: "200 C ° minus 20 C ° equals 180 C °")[$200 #h(0.28em) attach(C, tl: °) − 20 #h(0.28em) attach(C, tl: °) = 180 #h(0.28em) attach(C, tl: °)$] #strong[Step 3:] Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature. #math.equation(block: true, alt: "180 divided by 9 equals 20")[$180 ÷ 9 = 20$] #strong[Step 4:] Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes. #math.equation(block: true, alt: "15 ⁢ minutes times 9 equals 135 ⁢ minutes; 135 ⁢ minutes equals 2 ⁢ hours ⁢ 15 ⁢ minutes")[$15 "minutes" × 9 = 135 "minutes" \ 135 "minutes" = 2 "hours" 15 "minutes"$] So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Metric_System_in_5_Minutes")[Learn the Metric System in 5 Minutes] ] === Key Terms - temperature === Key Concepts - 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. - 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). === Videos - #link("https://openstax.org/r/Misconceptions_About_Temperature")[Misconceptions About Temperature] - #link("https://openstax.org/r/Temperature_Conversion_Trick")[Temperature Conversion Trick] - #link("https://openstax.org/r/Metric_System_in_5_Minutes")[Learn the Metric System in 5 Minutes] === Formulas To convert temperature from Fahrenheit to Celsius: #math.equation(block: true, alt: "C equals the fraction 5 over 9 open parenthesis F minus 32 close parenthesis")[$C = frac(5, 9) ( F − 32 )$] To convert temperature from Celsius to Fahrenheit: #math.equation(block: true, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$] To estimate temperature from Fahrenheit to Celsius: #math.equation(block: true, alt: "C equals the fraction F minus 30 over 2")[$C = frac(F − 30, 2)$] To estimate temperature from Celsius to Fahrenheit: #math.equation(block: true, alt: "F equals 2 C plus 30")[$F = 2 C + 30$] === Projects ==== Cooking + Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the metric system. + Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary System of Measurement, using cups, tablespoons, or teaspoons as required. + What did you observe? Was it easier to convert from one system to another? Which system allows for more precise measurements? What kitchen tools would you need in your kitchen if you used the metric system? ==== Shopping + Compare the average gas price in California to the average gas price in Puerto Rico. + What conversions did you need to make to do the comparison? + Do you think that the price of the gasoline is affected by the units in which it is sold? ==== Sports + What system of measurement is used for track and field events? Why do you think this system is used? + What system of measurement is used for football? Why do you think this system is used? + Research various sports records. Which units of measurement is used? What do you think influenced the unit of measure used?