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1 1.2 Measurement & Scientific Notation

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2 Measurement Measurement You make a measurement every time you Measure your height. Measure your height. Read your watch. Read your watch. Take your temperature. Take your temperature. Weigh a cantaloupe. Weigh a cantaloupe.

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3 Measurement in Science Measurement in Science In science and allied health we Measure quantities. Do experiments. Calculate results. Use numbers to report measurements. Compare results to standards.

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4 Standards of Measurement When we measure, we use a measuring tool to compare some dimension of an object to a standard. When we measure, we use a measuring tool to compare some dimension of an object to a standard. Calipers are used to measure the thickness of the skin fold at the waist. Calipers are used to measure the thickness of the skin fold at the waist.

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5 Stating a Measurement In every measurement, a number is followed by a unit. Observe the following examples of measurements: number + unit 35 m 35 m 0.25 L 0.25 L 225 lb 225 lb 3.4 hr 3.4 hr

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6 The Metric System (SI) The metric system is A decimal system based on 10. Used in most of the world. Used by scientists and in hospitals.

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7 Units in the Metric System In the metric and SI systems, a basic unit identifies each type of measurement:

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8 Length Measurement In the metric system, length is measured in meters using a meter stick. The metric unit for length is the meter (m).

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9 Volume Measurement Volume is the space occupied by a substance. Volume is the space occupied by a substance. The metric unit of volume is the liter (L). The metric unit of volume is the liter (L). The liter is slightly bigger than a quart. The liter is slightly bigger than a quart. A graduated cylinder is used to measure the volume of a liquid. A graduated cylinder is used to measure the volume of a liquid.

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10 Mass Measurement The mass of an object is the quantity of material it contains. The mass of an object is the quantity of material it contains. A balance is used to measure mass. A balance is used to measure mass. The metric unit for mass is the gram (g). The metric unit for mass is the gram (g).

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11 Temperature Measurement The temperature of a substances indicates how hot or cold it is. The temperature of a substances indicates how hot or cold it is. In the metric system, temperature is measured on the Celsius scale. In the metric system, temperature is measured on the Celsius scale. On this thermometer, On this thermometer, the temperature is the temperature is 19ºC or 66ºF. 19ºC or 66ºF.

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12 Scientific Notation A number in scientific notation contains a coefficient and a power of 10. A number in scientific notation contains a coefficient and a power of 10. coefficient power of ten coefficient power of ten coefficient power of ten coefficient power of ten 1.5 x 10 2 7.35 x 10 -4 1.5 x 10 2 7.35 x 10 -4 Place the decimal point after the first digit. Indicate the spaces moved as a power of ten. Place the decimal point after the first digit. Indicate the spaces moved as a power of ten. 52 000 = 5.2 x 10 4 0.00378 = 3.78 x 10 -3 52 000 = 5.2 x 10 4 0.00378 = 3.78 x 10 -3 4 spaces left 3 spaces right 4 spaces left 3 spaces right

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13 1.3 Measured and Exact Numbers

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14 Measured Numbers You use a measuring tool to determine a quantity such as your height or the mass of an object. You use a measuring tool to determine a quantity such as your height or the mass of an object. The numbers you obtain are called measured numbers. The numbers you obtain are called measured numbers.

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15. l 2.... l.... l 3.... l.... l 4.. cm To measure the length of the blue line, we read the markings on the meter stick. To measure the length of the blue line, we read the markings on the meter stick. The first digit 2 plus the second digit 2.7 Estimating the third digit between 2.7–2.8 Estimating the third digit between 2.7–2.8 gives a final length reported as 2.75 cm 2.75 cm or 2.76 cm or 2.76 cm Reading a Meter Stick

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16 Known + Estimated Digits In the length measurement of 2.76 cm, the digits 2 and 7 are certain (known). the third digit 5(or 6) is estimated (uncertain). all three digits (2.76) are significant including the estimated digit.

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17. l 3.... l.... l 4.... l.... l 5.. cm The first and second digits are 4.5. The first and second digits are 4.5. In this example, the line ends on a mark. In this example, the line ends on a mark. Then the estimated digit for the hundredths place is 0. Then the estimated digit for the hundredths place is 0. We would report this measurement as 4.50 cm. We would report this measurement as 4.50 cm. Zero as a Measured Number

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18 Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects 2 soccer balls 4 pizzas Defined relationships Defined relationships 1 foot = 12 inches 1 meter = 100 cm An exact number is not obtained with a measuring tool.

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19 1.4 Significant Figures in Calculations

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20 Significant Figures in Measurement The numbers reported in a measurement depend on the measuring tool. Measurements are not exact; they have uncertainty. The significant figures for a measurement include all of the known digits plus one estimated digit.

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21 All non-zero numbers in a measured number are significant. MeasurementNumber of Significant Figures 38.15 cm4 5.6 ft2 65.6 lb3 122.55 m5 Counting Significant Figures

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22 Leading zeros precede non-zero digits in a decimal number. Leading zeros in decimal numbers are not significant. Measurement Number of Significant Figures 0.008 mm1 0.008 mm1 0.0156 oz3 0.0156 oz3 0.0042 lb2 0.0042 lb2 0.000262 mL 3 0.000262 mL 3 Leading Zeros

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23 Sandwiched zeros occur between nonzero numbers. Sandwiched zeros are significant. MeasurementNumber of Significant Figures MeasurementNumber of Significant Figures 50.8 mm3 2001 min4 2001 min4 0.0702 lb3 0.0702 lb3 0.40505 m 5 0.40505 m 5 Sandwiched Zeros

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24 In numbers without decimal points, trailing zeros follow non-zero numbers. Trailing zeros are usually place holders and not significant. Measurement Number of Significant Figures 25 000 cm 2 200 kg1 200 kg1 48 600 mL3 48 600 mL3 25 005 000 g 5 25 005 000 g 5 Trailing Zeros

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25 Significant Figures in Scientific Notation All digits including zeros that appear in the coefficient of a number written in scientific notation are significant. All digits including zeros that appear in the coefficient of a number written in scientific notation are significant. Scientific NotationNumber of Significant Figures 8 x 10 4 m1 8.0 x 10 4 m2 8.00 x 10 4 m3

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26 A calculated answer must relate to the measured values used in the calculation. In calculations involving addition or subtraction, the number of decimal places are counted. In calculations involving multiplication or division, significant figures are counted to determine final answers. Significant Numbers in Calculations

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27 Rules for Rounding Off Calculated Answers To obtain the correct number of significant figures, an answer may be rounded off. To obtain the correct number of significant figures, an answer may be rounded off. When digits of 4 or less are dropped, the rest of the numbers are the same. For example, rounding 45.832 to 3 significant figures gives When digits of 4 or less are dropped, the rest of the numbers are the same. For example, rounding 45.832 to 3 significant figures gives 45.832 rounds to 45.8 (3 SF) When digits of 5 or greater dropped, the last retained digit is increased by 1. For example, rounding 2.4884 to 2 significant figures gives When digits of 5 or greater dropped, the last retained digit is increased by 1. For example, rounding 2.4884 to 2 significant figures gives 2.4884 rounds to 2.5 (2 SF)

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28 Adding Significant Zeros When a calculated answer requires more significant digits, zeros are added. When a calculated answer requires more significant digits, zeros are added. Calculated answerZeros added to give 3 significant figures 44.00 1.51.50 0.20.200

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29 An answer obtained by adding or subtracting has the same number of decimal places as the measurement with the fewest decimal places. An answer obtained by adding or subtracting has the same number of decimal places as the measurement with the fewest decimal places. Proper rules of rounding are used to adjust the number of digits in the answer. Proper rules of rounding are used to adjust the number of digits in the answer. 25.2 one decimal place 25.2 one decimal place + 1.34 two decimal places + 1.34 two decimal places 26.54calculated answer 26.54calculated answer 26.5 answer with one decimal place 26.5 answer with one decimal place Adding and Subtracting

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30 An answer obtained by multiplying or dividing has the same number of significant figures as the measurement with the fewest significant figures. An answer obtained by multiplying or dividing has the same number of significant figures as the measurement with the fewest significant figures. Use rounding to limit the number of digits in the answer. Use rounding to limit the number of digits in the answer. 110.5 x 0.048 = 5.304 (calculator) 110.5 x 0.048 = 5.304 (calculator) 4 SF 2 SF 4 SF 2 SF The final answer is rounded off to give 2 significant figures = 5.3 (2 SF) The final answer is rounded off to give 2 significant figures = 5.3 (2 SF) Multiplying and Dividing

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31 1.6 SI and Metric Prefixes

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32 Prefixes A prefix in front of a unit increases or decreases the size of that unit. The new units are larger or smaller that the initial unit by one or more factors of 10. A prefix indicates a numerical value. prefix=value 1 kilometer=1000 meters 1 kilogram=1000 grams

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33 Metric and SI Prefixes

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34 An equality states the same measurement in two different units. An equality states the same measurement in two different units. Equalities are written using the relationships between two metric units. Equalities are written using the relationships between two metric units. For example, 1 meter can be expressed as 100 cm or as 1000 mm. For example, 1 meter can be expressed as 100 cm or as 1000 mm. 1 m=100 cm 1 m=100 cm 1 m=1000 mm Metric Equalities

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35 Metric Equalities for Length

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36 Metric Equalities for Volume

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37 Metric Equalities for Mass Several equalities can be written for mass in the metric system Several equalities can be written for mass in the metric system 1 kg=1000 g 1 g=1000 mg 1 mg= 0.001 g 1 mg=1000 µg

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38 1.6 Writing Conversion Factors

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39 The quantities in an equality use two different units to describe the same measured amount. The quantities in an equality use two different units to describe the same measured amount. Equalities are written for relationships between units of the metric system, U.S. units or between metric and U.S. units. For example, Equalities are written for relationships between units of the metric system, U.S. units or between metric and U.S. units. For example, 1 m = 1000 mm 1 lb = 16 oz 2.20 lb = 1 kg Equalities

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40 Exact and Measured Numbers in Equalities Equalities written between units of the same system are definitions; they are exact numbers. Equalities written between units of the same system are definitions; they are exact numbers. Equalities written between metric-U.S. units, which are in different systems, represent measured numbers and must be counted as significant figures. Equalities written between metric-U.S. units, which are in different systems, represent measured numbers and must be counted as significant figures.

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41 Some Common Equalities

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42 Equalities on Food Labels The contents of packaged foods in the U.S. are listed as both metric and U.S. units. The contents of packaged foods in the U.S. are listed as both metric and U.S. units. The content values indicate the same amount of substance in two different units. The content values indicate the same amount of substance in two different units.

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43 A conversion factor is a fraction in which the quantities in an equality are written as the numerator and denominator. A conversion factor is a fraction in which the quantities in an equality are written as the numerator and denominator. Equality: 1 in. = 2.54 cm Each unit can be written as the numerator or denominator. Thus, two conversion factors are possible for every equality. Each unit can be written as the numerator or denominator. Thus, two conversion factors are possible for every equality. 1 in. and 2.54 cm 2.54 cm 1 in. Conversion Factors

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44 A word problem may contain information that can be used to write conversion factors. Example 1: At the store, the price of one pound of red peppers is $2.39. 1 lb red peppers $2.39 $2.391 lb red peppers Example 2: At the gas station, one gallon of gas is $1.34. 1 gallon of gas $1.34 $1.341 gallon of gas Conversion Factors in a Problem

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45 To start solving a problem, it is important to identify the initial and final units. A person has a height of 2.0 meters. What is that height in inches? To start solving a problem, it is important to identify the initial and final units. A person has a height of 2.0 meters. What is that height in inches? The initial unit is the unit of the given height. The final unit is the unit needed for the answer. The initial unit is the unit of the given height. The final unit is the unit needed for the answer. Initial unit = meters (m) Final unit = inches (in.) Initial and Final Units

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46 In working a problem, start with the initial unit. In working a problem, start with the initial unit. Write a unit plan that converts the initial unit to the final unit. Write a unit plan that converts the initial unit to the final unit. Unit 1 Unit 2 Select conversion factors that cancel the initial unit and give the final unit. Select conversion factors that cancel the initial unit and give the final unit. Initial x Conversion= Final unit factor unit unit factor unit Unit 1 x Unit 2=Unit 2 Unit 1 Problem Setup

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47 Setting up a Problem How many minutes are 2.5 hours? Solution: Initial unit = 2.5 hr Final unit=? min Unit Plan=hr min Setup problem to cancel hours (hr). Inital Conversion Final unit factor unit unit factor unit 2.5 hr x 60 min = 150 min (2 SF ) 1 hr 1 hr

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48 Often, two or more conversion factors are required to obtain the unit of the answer. Unit 1 Unit 2Unit 3 Additional conversion factors are placed in the setup to cancel the preceding unit Initial unit x factor 1 x factor 2 = Final unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit 2 Unit 1 Unit 2 Using Two or More Factors

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49 How many minutes are in 1.4 days? Initial unit: 1.4 days Unit plan: days hr min Set up problem: 1.4 days x 24 hr x 60 min = 2.0 x 10 3 min 1.4 days x 24 hr x 60 min = 2.0 x 10 3 min 1 day 1 hr 1 day 1 hr 2 SF Exact Exact = 2 SF 2 SF Exact Exact = 2 SF Example: Problem Solving

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50 Be sure to check your unit cancellation in the setup. What is wrong with the following setup? 1.4 day x 1 day x 1 hr 24 hr 60 min 24 hr 60 min Units = day 2 /min is Not the final unit needed Units don’t cancel properly. The units in the conversion factors must cancel to give the correct unit for the answer. Check the Unit Cancellation

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51 Identify the initial and final units. Write out a unit plan. Select appropriate conversion factors. Convert the initial unit to the final unit. Cancel the units and check the final unit. Do the math on a calculator. Give an answer using significant figures. Typical Steps in Problem Solving

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52 Clinical Factors Conversion factors are also possible when working with medications. Conversion factors are also possible when working with medications. A drug dosage such as 20 mg Prednisone per tablet can be written as A drug dosage such as 20 mg Prednisone per tablet can be written as 20 mg Prednisone and 1 tablet 1 tablet 20 mg Prednisone 20 mg Prednisone and 1 tablet 1 tablet 20 mg Prednisone

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53 A percent refers to a ratio of the parts to the whole. % = Parts x 100 % = Parts x 100 Whole Whole A percent factor is written by choosing a unit to express the percent. Write 100 of the same unit in the denominator. The second factor is the inverse of the first. For example, a food contains 30% (by mass) fat. 30 g fat and100 g of food 100 g of food30 g fat Percent as a Conversion Factor

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54 1.8 Density

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55 Density compares the mass of an object to its volume. Density compares the mass of an object to its volume. In the density expression, the mass of an object or substance is written in the numerator and its volume in the denominator. In the density expression, the mass of an object or substance is written in the numerator and its volume in the denominator. D = mass = g or g = g/cm 3 volume mL cm 3 volume mL cm 3 Note: 1 mL = 1 cm 3 Density

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56 Volume by Displacement A solid displaces its volume of water when the solid is placed in water. A solid displaces its volume of water when the solid is placed in water. Therefore the volume of the solid is calculated from the volume difference. Therefore the volume of the solid is calculated from the volume difference.

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57 Density Using Volume Displacement The volume of zinc is calculated from the displaced volume 45.0 mL - 35.5 mL = 9.5 mL = 9.5 cm 3 45.0 mL - 35.5 mL = 9.5 mL = 9.5 cm 3 Density zinc = mass = 68.60 g = 7.2 g/cm 3 volume 9.5 cm 3

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58 Sink or Float Ice floats in water because the density of ice is less than the density of water. Aluminum sinks because it has a density greater that the density of water.

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59 Density represents an equality for a substance. The mass in grams is for 1 mL. For a substance with a density of 3.8 g/mL, the equality is: Density represents an equality for a substance. The mass in grams is for 1 mL. For a substance with a density of 3.8 g/mL, the equality is: 3.8 g = 1 mL For this equality, we can write two conversion factors. For this equality, we can write two conversion factors. Conversion 3.8 g and 1 mL factors1 mL 3.8 g Density as a Conversion Factor

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60 Specific Gravity Specific gravity compares the density of a substance to the density of water (1.00 g/mL). Specific gravity compares the density of a substance to the density of water (1.00 g/mL). Specific gravity = density of substance density of water For example, the density of mercury is For example, the density of mercury is 13.6 g/mL. Specific gravity = 13.6 g/mL = 13.6 1.00 g/mL The units cancel. Thus specific gravity has no units.

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61 1.9Temperature

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62 Temperature is a measure of how hot or cold an object is. Temperature is determined by using a thermometer. Some thermometers contain a liquid that expands with heat and contracts with cooling. Other types of thermometers are electronic. Temperature

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63 Temperature Scales Temperature is measured using the Fahrenheit, Celsius, and Kelvin temperature scales. The reference points are the boiling and freezing points of water.

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64 On the Fahrenheit scale, there are are 180°F between the freezing and boiling points and on the Celsius scale, there are 100 °C. On the Fahrenheit scale, there are are 180°F between the freezing and boiling points and on the Celsius scale, there are 100 °C. 180°F = 9°F =1.8°F 100°C 5°C 1°C In the formula for Fahrenheit, the value of 32 adjusts the zero point of water from 0°C to 32°F. In the formula for Fahrenheit, the value of 32 adjusts the zero point of water from 0°C to 32°F. °F = 9/5 T°C + 32 or°F = 1.8 T°C + 32 or°F = 1.8 T°C + 32 Fahrenheit Formula

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65 The equation for Fahrenheit is rearranged to calculate T°C. The equation for Fahrenheit is rearranged to calculate T°C. °F = 1.8 T°C + 32 Subtract 32 from both sides and divide by 1.8. Subtract 32 from both sides and divide by 1.8. °F - 32 = 1.8 T°C ( +32 - 32) °F - 32 = 1.8 T°C 1.8 1.8 1.8 1.8 °F - 32 = T°C 1.8 1.8 Celsius Formula

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66 A person with hypothermia has a body temperature of 34.8°C. What is that temperature in °F? °F = 1.8 (34.8°C) + 32 exact tenth's exact exact tenth's exact = 62.6 + 32 = 94.6°F tenth’s Solving A Temperature Problem

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67 On the Kelvin Scale, the lowest possible temperature is 0 K called absolute zero. On the Kelvin Scale, the lowest possible temperature is 0 K called absolute zero. 0 K = –273 °C On both K and °C scales, there are 100 units between freezing and boiling. On both K and °C scales, there are 100 units between freezing and boiling. 100 K = 100°Cor 1 K = 1 °C The Kelvin temperature is obtained by adding 273 to the Celsius temperature. The Kelvin temperature is obtained by adding 273 to the Celsius temperature. K = °C + 273 Kelvin Temperature Scale

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68 Some Temperature Comparisons

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