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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

Qualitative vs. Quantitative Qualitative: appearance. the substance was a white powder Quantitative: Measurement The substance had a mass of 5.6 grams Copyright © Houghton Mifflin Company

Why are measurements important? Pay for gasoline by the gallon Construction: must have accurate measurements Welding: measurements must be accurate Automobiles: Most items on cars have a measurement Landscaping: Measurements important for spatial relationships Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Measurements: 2 Parts Number + Unit Number alone is meaningless If I tell you a 2 x 4 is 5 long – what does this mean: 5 feet, 5 inches, 5 meters? Unit alone is meaningless If I tell you the 2 x 4 is feet long – what does this mean: 2 ft, 4 ft, 8 ft? ALWAYS INCLUDE BOTH NUMBER AND UNIT! Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Scientific Notation Used to express very large or very small numbers Express using number between 1 & 10 multiplied by a power of 10 10 is to positive power for large numbers 10 is to a negative number for small numbers (decimals) Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Large Numbers Decimal point moved to left Power of 10 = # places moved = + number 125 = 1.25 x 100 = 1.25 x 102 Decimal point moved 2 places left 1700 = 1.7 x 1000 = 1.7 x 103 Decimal point moved 3 places left 93,000,000 = 9.3 x 107 Decimal point moved 7 places left Copyright © Houghton Mifflin Company

Small Numbers (Decimals < 1.0) Decimal point moved to the right Power of 10 = # places moved = negative number 0.010 = 1.0 x 0.010 = 1.0 x 10-2 Decimal point moved right 2 places 0.000167 = 1.67 x 10-4 Decimal point moved right 4 places 0.089 = 8.9 x 10-2 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

Copyright © Houghton Mifflin Company BASIC UNITS Physical Quantity English System Metric System Mass Pound or slug Gram or kilogram Length inch, foot, yard Centimeter or meter Volume Quart or gallon Liter, milliliter, or cm3 Time Hours, minutes, seconds Seconds Temperature Degrees Fahrenheit °C or Kelvin Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Metric Prefixes Kilo- hecto- deka- UNIT deci- centi- milli- k h dk d c m Move decimal point left Move decimal point right Power of 10 between each increment Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Examples 1000 meters = 1 km (decimal moved 3 places left) 1 meter = 100 cm (decimal moved 2 places right) 10 mm = 1 cm (decimal moved 1 place left) 1 Liter = 1000 mL (decimal moved 3 places to right) 600 grams = 0.6 kg (decimal moved 3 places left) Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Table 5.2 Copyright © Houghton Mifflin Company

Copyright © Houghton Mifflin Company Table 5.3 Copyright © Houghton Mifflin Company

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 Dimensional Analysis Copyright © Houghton Mifflin Company

Uncertainty of Measurement When using 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

Significant Figures Ex. 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 Captive zeros (between nonzero digits) Example: 1.008 (4 s.f. - all numbers are significant) Trailing zeros (right end of number) Significant if bar is placed over zero or over zero to right Example: 200 (only #2 is significant = 1 s.f.) 2Ō0 (2 s.f. - 2 & Ō both significant) 20Ō (3 s.f. - 2 and both zeros are significant)

Copyright © Houghton Mifflin Company Following decimal to right of nonzero digit = significant 5.00 = 3 s.f. Copyright © Houghton Mifflin Company

Non-significant zeros Leading zeros, precede non-zero digits Example: 0.0025 = 2 s.f. - zeros are non-significant Both: some numbers are sig. some are not Copyright © Houghton Mifflin Company

Rules for counting significant figures (continued) All nonzero integers are significant Example: 1457 has 4 s.f. Exact numbers have unlimited number of s.f. Determined by counting 8 apples, 21 students Not obtained from measuring devices From definitions 1 inch is exactly 2.54 cm Will not limit numbers in calculations

Copyright © Houghton Mifflin Company Use same rules for scientific notation (10x not s.f.) 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 <5 round down If number is ≥5 round up Do not round off until end of calculations

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

Copyright © Houghton Mifflin Company 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

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

Copyright © Houghton Mifflin Company 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

Figure 5.7: Converting 70 degrees Celsius to Kelvin units. Copyright © Houghton Mifflin Company

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

Figure 5.8: Comparison of the Celsius and Fahrenheit scales. Copyright © Houghton Mifflin Company

Density: amount of matter in a given volume of substance Density = mass / volume Determine mass using a balance Determine volume by calculations, graduated cylinder, or water displacement Units are in g/cm3, g/mL, kg/L, lb/gal

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