Measurements and Calculations Chapter 5 Measurements and Calculations

Goals of Chapter 5: Measurement & Calculations Express numbers in scientific notation Learn English, metric, & SI system of measurement Use metric system to measure length, volume, and mass Significant digits Dimensional Analysis Temperature Scales Density/Specific Gravity Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Scientific Notation Used to express very large or very small numbers A number is expressed in scientific notation when it is in the form a x 10n where a is between 1 and 10 and n is an integer use positive power for large numbers use negative number for small numbers (decimals) Copyright © Houghton Mifflin Company

Scientific Notation: Large Numbers 210,000,000,000,000,000,000,000 Where is the decimal point? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023

Scientific Notation: Small Numbers 0.0000000902 Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8

Copyright © Houghton Mifflin Company Learning Check Convert the following to scientific notation 100,000 b. 0.0056 c. 45,000,000 d. 0.0000101000 e. 9006 f. 0.00000002 Convert to standard notation g. 8.9 x 108 h. 7.7 x 10-4 i. 7.003 x 106 j. 1.001 x 10-8 Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Units Tell us what scale or standard is being used to represent measurement Scientists need common units to represent quantities like mass, length, time, and temperature If everyone had own set of units – chaos would result US uses English system, most of world (& scientists) use metric system, also SI system Copyright © Houghton Mifflin Company

Units of Measurement

Figure 5.1: Comparison of English and metric units. Copyright © Houghton Mifflin Company

Figure 5.2: Cube representations. Copyright © Houghton Mifflin Company

Figure 5.3: A 100 mL graduated cylinder. 1 mL = 1 cm3 1 milliliter = 1 cubic centimeter 100 mL = 100 cm3 Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Move decimal point left Move decimal point right Power of 10 between each increment Copyright © Houghton Mifflin Company

Chapter 2 SI Conversions

Copyright © Houghton Mifflin Company Learning Check 1000 meters = ___ km (decimal moved ___ places left) 1 meter = ___ cm (decimal moved ___ places right) 10 mm = ___ cm (decimal moved ___ place left) 1 Liter = ___ mL (decimal moved ___ places to right) 600 grams = ___ kg (decimal moved ___ places left) Copyright © Houghton Mifflin Company

Derived Units

Uncertainty of Measurement When using an instrument to measure (such as a ruler or graduated cylinder), we visualize divisions between markings and estimate When making measurement, record all certain numbers and first uncertain number Copyright © Houghton Mifflin Company

Figure 5.5: Measuring a pin. Reading is between 2.8 cm & 2.9 cm These divisions were visualized 2.85 cm is measurement “5” is uncertain Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Significant Figures Includes all numbers recorded in a measurement For pin, length = 2.85 cm: 3 significant figures All certain numbers plus first uncertain Assume to be accurate to ± 1 in last # Pin length is 2.85 ± 0.01 cm Pin is somewhere between 2.84 & 2.86 cm Copyright © Houghton Mifflin Company

Rules for Counting Significant Figures All non-zero digits ARE significant - 125.55 contains FIVE significant figures Leading Zeros ARE NOT significant - 0.00055 contains TWO significant figures 3. Middle Zeros ARE significant - 505 contains THREE significant figures 4. Ending Zeros ARE significant IF there is a decimal present - 500.00 contains FIVE significant figures 5. Ending Zeros ARE NOT significant IF there is no decimal present - 500 contains ONE significant figure

Rules for counting significant figures (continued) Significant Figures only apply to measurements and calculations involving those measurements Any number written in scientific notation is considered significant An ending zero can be expressed as significant by placing a line above it Remember! Just because a number is not significant, does not mean it’s not important! A non-significant zero acts as a placeholder to show the magnitude of the number!

Copyright © Houghton Mifflin Company Learning Check How many significant figures are in the following measurements? 0.0108 0.0050060 110 5.030 x 103 0.00100 480 500 Copyright © Houghton Mifflin Company

To give answer with correct number of significant figures – round off Look at number to right of last s.f. If number is 4 or below, let it go! If number is 5 or above, give it a shove! Do not round off until end of calculations

Rules for Significant Figures in Calculations Multiplication & Division Answer should have same number of significant figures as measurement with smallest number of significant figures Example: 4.56 x 1.4 = 6.384 → 6.4 (3 s.f.) (2 s.f.)* (2 s.f.)*

Rules for Significant Figures in Calculations Addition & Subtraction Limited by smallest number of decimal places Example: 12.11 (2 decimal places) 18.0 (1 decimal place)* + 1.013 (3 decimal places) 31.123 → 31.1 (1 decimal place)* Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Learning Check Perform the following operations and round to the correct number of significant figures a. 5.19 +1.9 + 0.842 b. 1081 – 7.25 c. 2.3 x 3.14 d. 25.36/2.5 Copyright © Houghton Mifflin Company

Dimensional Analysis Dimensional Analysis is a method of problem solving that allows for the conversion of one unit to another by way of a conversion factor. A unit conversion factor is a fraction whose numerator and denominator are equivalent measures. Some common unit conversion factors are given below. You can also use the reciprocal of these. Ask for some reciprocals so you know they understand.

Choose a unit conversion factor that… Introduces the unit you want in the answer Cancels out the original unit so that the one you want is all that is left.

“Canceling” out Words Which conversion factor will convert feet to inches? ? Feet 12 inches 1 foot 1 foot 12 inches You can cancel a unit if it appears in both the numerator and the denominator Feet x 12 inches 1 foot

Learning Check: Choose the appropriate conversion factor. Inches to feet Minutes to hours Meters to centimeters

Copyright © Houghton Mifflin Company Single-Step Problems Choose a conversion factor that will get you from the starting unit to the ending unit Multiply the given number by the conversion factor that will cancel out the original unit Solve the problem by multiplying all number on top of the fraction and divide by the number on the bottom of the fraction Be sure to include a UNIT in your answer! Copyright © Houghton Mifflin Company

Example 1: Convert 8 yards to feet Make a decision: What conversion factor will you use? 1 yard 3 feet 3 feet 1 yard Set up the problem: Multiply the measurement by the conversion factor. 8 yards x 3 feet = 24 feet 1 yard Solve the problem: Perform the multiplication Multiply all numbers on the top Divide by all numbers on the bottom

Example 2: Converts 16 quarts to gallons What are the two conversion factors comparing quarts and gallons? Which one will “cancel” quarts? 16 quarts x 1 gallon = 4 gallons 4 quarts

Copyright © Houghton Mifflin Company Learning Check Water is often bottled in 0.750 L containers. Convert 0.750 L into quarts. 1 L = 1.06 quarts Copyright © Houghton Mifflin Company

Multiple-Step Problems Problem that requires the use of multiple conversion factors Determine which conversion factors are needed to get from starting to ending unit Develop plan to put the required conversion factors in order to cancel them out 3. Solve the problem Multiply all numbers on top of fractions and divide by all numbers on the bottom of fractions Include a UNIT in your answer! Copyright © Houghton Mifflin Company

Example 3 : Convert 26.2 miles to kilometers Determine conversion factors 1 mile = 1760 yards 1 meter = 1.094 yards 1000 meter = 1 kilometer Outline plan Miles  yards  meters  kilometers Solve the Problem 26.2 miles x 1760 yards x 1 meter x 1 kilometer 1 mile 1.094 yards 1000 meters Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Learning Check If a race car travels around a track at the speed of 225 miles/hour, what is the speed of the car in kilometers/hour? Use conversion factors from Example 3 Copyright © Houghton Mifflin Company

Temperature Conversions Copyright © Houghton Mifflin Company

Figure 5.6: The three major temperature scales. Copyright © Houghton Mifflin Company

Converting between Kelvin & Celsius To convert from Kelvin to Celsius: T°C = TK – 273 Liquid Nitrogen boils at 77K, what is this in Celsius? T°C = 77 – 273 = -196 °C To convert from Celsius to Kelvin: TK = T°C + 273 The bp of water on top of Mt. Everest is 70 °C. Convert to K. TK = 70 + 273 = 343 K

Fahrenheit/Celsius Conversions To convert from Celsius to Fahrenheit: T°F = 1.80(T°C) + 32 If the temperature is 28°C, what is this in °F? T°F = 1.80(28) + 32 = 50.4 + 32 = 82°F (2 s.f.) To convert from Fahrenheit to Celsius: T°C = (T°F – 32)/1.80 If you have a temperature of 101°F, what is this in °C? T°C = (101 – 32)/1.80 = 69/1.8 = 38°C Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Learning Check a. Which temperature is colder, 172 K or -75 oC? b. Hot tubs are often maintained at 41 oC, what is the temperature in Fahrenheit? Copyright © Houghton Mifflin Company

Density Units are in g/cm3 or g/mL Defined as the amount of matter present in a given volume of a substance Density = mass / volume Units are in g/cm3 or g/mL Volume is dependent on temperature, so density is as well

Densities of Common Substances @ 20oC Density (g/cm3) Oxygen 0.00133 Aluminum 2.70 Hydrogen 0.000084 Iron 7.87 Ethanol 0.785 Copper 8.96 Benzene 0.880 Silver 10.5 Water 1.000 Lead 11.34 Magnesium 1.74 Mercury 13.6 Salt (sodium chloride) 2.16 Gold 19.32 Copyright © Houghton Mifflin Company

Determining volume by water displacement Place water in graduated cylinder & record level Add object Record volume after addition of object Volume is difference between second volume and first volume Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Figure 5.9: Tank of water. Copyright © Houghton Mifflin Company

Figure 5.9: Person submerged in the tank. Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Learning Check A student wants to identify the main component in a liquid cleaner. He finds that 35.8 mL of the liquid weighs 28.1 g. Which is the main component? a. chloroform 1.483 g/mL b. diethyl ether 0.714 g/mL c. isopropyl alcohol 0.785 g/mL d. toluene 0.867 g/mL Copyright © Houghton Mifflin Company