Chapter 1: Measurements The metric system Becoming more common in our country. The SI system Developed in 1960, also called the international system of units. Set of seven standard quantities of which all others can be derived. Common Metric Units Length Metric and SI unit is the Meter (m) 1 m = 39.4 inches 1 inch = 2.54 centimeters (cm) Volume Amount of space occupied by a substance Metric unit = Liter (L), SI unit = m3 1 L = 1.06 quarts 946 milliliters = 1 quart Common Metric Units Mass Metric unit is the gram (g), SI unit is the kilogram (kg) 1 kg = 2.2 pounds 454 g = 1 pound Time Metric and SI unit is the second (s) Common Metric Units Temperature Indicates how hot or cold an object is Metric unit is Celsius (C) and SI unit is Kelvin (K) These will be discussed in more detail in Chapter 2. Scientific Notation Used when numbers are really small or really large. General format: A x 10n A is a number between 1 and 10. n is the exponent on 10 (called the power) and is equal to the number of places that the decimal place is moved. Scientific Notation Examples 0.000058 = 5.8 x 10-5 • The decimal is moved five places to the right, hence the –5 power. Note that small numbers (0< x <1) will always have a negative exponent. 58,000 = 5.8 x 104 • The decimal is moved four places to the left, hence the power of 4 is used. Note that large numbers (>1) will always have a positive exponent. Scientific Notation On your calculator, use the EE or EXP key when entering these numbers. Example: (1.8 x 105) x (3.2 x 10-3) = 576 or 5.76 x 102 Measured & Exact Numbers Measured numbers are those which come from using any kind of a measuring device. Examples are: ruler, graduated cylinder, thermometer, scale, etc. Include both certain digits and one uncertain digit. Certain digits are digits that all would agree on. Uncertain digit is the last digit that requires YOU to make a “best guess.” Measured & Exact Numbers “Best Guess” depends on the smallest increments on the measuring device. A ruler may have markings for every 0.1cm. Your “best guess” might be to 0.05cm or even 0.02cm. Measured & Exact Numbers A graduated cylinder – depends on size. A 10-mL has markings every 0.2mL. A 50-mL has markings every 1-mL. Measured & Exact Numbers A Thermometer has markings every degree. Measured & Exact Numbers If a measurement results where it is exactly lined up with a mark, then a zero at the end may be needed. Not everyone will agree on the “best guess,” which means that all measured numbers have some uncertainty. Digital devices – the last digit in display is uncertain. Measured & Exact Numbers An exact number is a either a counted number or an established relationship between two units. Examples are: 12 computers in the room 15 apples in the bag 1 foot = 12 inches 1 pound = 16 ounces Learning Check Are the following numbers exact or measured? There are 50 pages in a book A coin weighs 2.87 grams There are 100 centimeters in one meter A graduated cylinder contains 40.0mL of water Significant Figures Goal: Report answers from calculations by rounding the answer to the correct number of significant figures. Accuracy refers to measurement(s) that are close to the true (accepted) measurement. Precision refers to the agreement of values obtained while repeating the same measurement. In the lab, we desire both of these. Significant Figures Rules are: 1. A number is significant if it is: a. b. c. d. 2. A non-zero digit A zero between two non-zero digits A zero at the end of a decimal number Any digit in the coefficient of a number written in scientific notation A number is not significant if it is: a. b. A zero at the beginning of a decimal number A zero used as the placeholder in a large number without a decimal point. Learning Check How many significant digits do each of the following measured numbers have? 1.02 0.0045 91,000 2.50 x 10-8 0.0150 250. Significant Figures in Calculations A calculation (using measured numbers) is only as accurate as the number that had the least number of significance. Calculators almost NEVER provide answers with the proper significance. Thus, answers will need to be rounded (or possibly require additional zero’s). Rounding Numbers Rules: 1. 2. 3. If the first digit to be dropped is a 4 or less, it and all other following digits can be dropped. If the first digit to be dropped is a 5+, then the last retained digit is increased by one. If the digit to be dropped is exactly 5 (nothing after it), then round up if it makes the digit even or down if that makes the digit even. Learning Check 3.46, round to two s.f.’s. 54.48, round to two s.f.’s. 135.51, round to three s.f.’s. 8.74528, round to three s.f.’s. Multiplication & Division The final answer is rounded to have the same number of significant digits as the measurement with the fewest s.f.’s. Example: 24.65 x 0.67 = 16.5155 (calc.) Example: (2.85 x 67.4) / 4.39 = 43.756264 (calc.) Example: (8.00) / (0.250) = 32 (calc.) Addition & Subtraction The final answer is rounded to have the same number of decimal places as the measured number with the fewest decimal places. Example: 2.045 + 34.1 = 36.145 (calc.) Example: 255 – 175.65 = 79.35 (calc.) Example: 89.15 – 82.95 = 6.2 (calc.) Mixed Calculations Apply the rules for each type of calculation. Don’t round intermediate answers – round only at the very end. Example: (23.8 + 4.25) / 67.85 = 0.413411938 Example: (17.92 – 16.82) x 0.01957 = 0.021527 SI & Metric Prefixes To increase or decrease metric units, prefixes are used as a multiplier. Prefixes that increase the size: G = giga, M = mega, 106 or 1,000,000 k = kilo, 103 or 1,000 Prefixes that decrease the size: d = deci, 10-1 or 0.1 c = centi, 10-2 or 0.01 m = milli, 10-3 or 0.001 m = micro, 10-6 or 0.000001 Some equalities 1 m = 100 cm 1 kg = 1000 g 1 L = 1,000,000 mL 1000 m = 1 km 10 dL = 1 L Note: Volume – a milliliter is equivalent to a cube that measures 1 cm x 1cm x 1cm or 1 cm3 (also referred to as a cubic centimeter – cc). Conversion Factors To change from one unit to another, you will multiply by the appropriate conversion factor. Conversion factors are written in the form of a fraction. Example: 100 cm = 1 m can be written as: 100 cm 1m or 1m 100 cm Using Conversion Factors / Dimensional Analysis Fence-post method uses number + unit. When conversion factor is multiplied correctly, then all of the units – except those desired in the answer – will cancel out. Dimensional Analysis Ex) Convert 35 inches to centimeters Ex) Convert 160 pounds to kilograms Dimensional Analysis For problems with more than one step, they can be done either as a series of steps or as one continuous problem. Note that you will only round answers at the very END! Dimensional Analysis Convert 25,000 feet to km As one continuous conversion… Dimensional Analysis Problems with units in both the numerator and denominator can create problems. Will see some problems involving “clinical” calculations. Dimensional Analysis Ex) Convert 35 miles per hour to meters per second Dimensional Analysis Ex) A certain medicine requires that 250mg per kilogram of body mass is to be given. What dose should be given to a child that weighs 48lbs? Dimensional Analysis Area and volume conversions can also be easily missed. Ex) A concrete footing measuring 3.0 feet by 2.0 feet by 1.5 feet is poured. What volume of concrete is needed, in cm3? Note: 1 ft3 12 in3, Rather: 1 ft3 = (12)3 in3 = 1728 in3 ! Dimensional Analysis Volume of concrete = 3.0ft x 2.0ft x 1.5ft = 9.0ft3 Alternatively, EACH dimension could be converted to cm first, then multiplied out to yield the volume. Density The density of an object is equal to its mass divided by its volume. m D V You will be asked to solve for any one of the D, m, or V’s. Density Problems Ex) An object has a mass of 35.0g and occupies 5.2mL. What is its density? Density Problems Ex) The mass of an iron bar is 1500g. What volume does it occupy? The density of iron is 7.9g/mL. Can rearrange formula or use Density as conversion factor.