CHAPTER 2: SCIENTIFIC MEASUREMENTS Problems: 1-26, 37-76, 80-84, 89-93 2.1 UNCERTAINTY IN MEASUREMENTS measurement: a number with attached units To measure, one uses instruments = tools such as a ruler, balance, etc. All instruments have one thing in common: UNCERTAINTY! INSTRUMENTS CAN NEVER GIVE EXACT MEASUREMENTS! mass: a measure of the amount of matter an object possesses measured with a balance and NOT AFFECTED by gravity usually reported in grams or kilograms weight: a measure of the force of gravity usually reported in pounds (abbreviated lbs) MASS WEIGHT EARTH MOON SPACE mass = 68 kg mass = 68 kg mass = 68 kg weight = 150 lbs weight = 25 lbs weight = 0 lbs volume: Amount of space occupied by a solid, gas, or liquid. measured using graduated cylinder, a buret, a pipet, a volumetric flask, etc. generally in units of liters (L), milliliters (ml), or cubic centimeters (cm 3 ) 1 ml 1 cm 3 Note: When the relationship between two units or items is exact, we use the to mean is exactly equal to rather than using the usual = sign. also know the following equivalents for the English system 1 gallon 4 quarts 1 quart 2 pints 1 pint 2 cups CHM130 Chapter 2 Notes page 1 of 9
2.2 SIGNIFICANT DIGITS (also called Significant Figures or Sig Figs ) When a measurement is recorded, all the numbers known with certainty are given along with the last number, which is estimated. All the digits are significant because removing any of the digits changes the measurement's uncertainty. Ruler A 0 1 2 3 4 5 Ruler B 0 1 2 3 4 5 Ruler C 4.1 4.2 4.3 4.4 Ruler Measurement/quantity # of sig figs A B C Which ruler above gives the most accurate measurement? Guidelines for Sig Figs (if measurement is given): Count the number of digits in a measurement from left to right: 1. When a decimal point is present: For measurements 1, count all the digits (even zeros). 60.2 cm has 3 sig figs, 5.0 m has 2 sig figs, 186.00 g has 5 s.f. For measurements less than 1, start with the first nonzero digit and count all digits (even zeros) after it. 0.011 ml and 0.00022 kg each have 2 sig figs 2. When there is no decimal point: Count all non-zero digits and zeros between non-zero digits 125 g has 3 sig figs, 107 ml has 3 sig figs Placeholder zeros may or may not be significant 1000 may have 1, 2, 3 or 4 sig figs CHM130 Chapter 2 Notes page 2 of 9
Example: How many significant digits do the following numbers have? # of sig figs # of sig figs # of sig figs a. 165.3 c. 90.40 e. 0.19600 b. 105 d. 100.00 f. 0.0050 2.6 EXPONENTIAL NUMBERS 2.7 SCIENTIFIC NOTATION Some numbers are very large or very small difficult to express. Avogadro s number = 602,000,000,000,000,000,000,000 an electron s mass = 0.000 000 000 000 000 000 000 000 000 91 kg To handle such numbers, we use a system called scientific notation. Regardless of their magnitude, all numbers can be expressed in the form where N 10 n N =digit term= a number between 1 and 10, so there can only be one number to the left of the decimal point: #.#### n = an exponent = a positive or a negative integer (whole #). To express a number in scientific notation: Count the number of places you must move the decimal point to get N between 1 and 10. Moving decimal point to the right (if # < 1) negative exponent. Moving decimal point to the left (if # > 1) positive exponent. Example: Express the following numbers in scientific notation (to 3 sig figs): 555,000 0.000888 602,000,000,000,000,000,000,000 CHM130 Chapter 2 Notes page 3 of 9
Also, in some cases the number of sig figs in a measurement may be unclear: For example, Ordinary form Scientific Notation Express 100.0 g to 3 sig figs: Express 100.0 g to 2 sig figs: Express 100.0 g to 1 sig fig: Thus, some measurements usually those expressing large amounts must be expressed in scientific notation to accurately convey the number of sig figs. 2.3 ROUNDING OFF NONSIGNIFICANT DIGITS How do we eliminate nonsignificant digits? If first nonsignificant digit < 5, just drop ALL nonsignificant digits If first nonsignificant digit 5, raise the last sig digit by 1 then drop ALL nonsignificant digits last significant digit 72.58643 g first nonsignificant digit For example, express 72.58643 with 3 sig figs: 72.58643 to 3 sig figs Example: Express each of the following with the number of sig figs indicated: a. 376.276 to 3 sig figs b. 500.072 to 4 sig figs c. 0.00654321 to 3 sig figs d. 1,234,567 to 5 sig figs e. 2,975 to 2 sig figs Be sure to express measurements in scientific notation when necessary to make it clear how many sig figs there are in the measurement. CHM130 Chapter 2 Notes page 4 of 9
2.4 ADDITION AND SUBTRACTION OF MEASUREMENTS When adding and subtracting measurements, your final value is limited by measurement with the largest uncertainty i.e. the number with the fewest decimal places. Ex 1: 106.61 + 0.25 + 0.195 = 107.055 107.055 to the correct number of sig figs: Ex 2: 725.50 103 = 622.50 622.50 to the correct # of sig figs: 2.5 MULTIPLICATION AND DIVISION OF MEASUREMENTS When multiplying or dividing measurements, the final value is limited by the measurement with the least number of significant figures. Ex 1: 106.61 0.25 0.195 = 5.1972375 5.1972375 to the correct # of sig figs: Ex 2: 106.61 91.5 = 9754.815 w/ correct sig figs: MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS: When multiplying or dividing measurements with exponents, use the digit term (N in N 10n ) to determine number of sig figs. Ex. 1: (6.02 10 23 )(4.155 10 9 ) = 2.50131 10 33 How do you calculate this using your scientific calculator? Step 1. Enter 6.02 10 23 by pressing: 6.02 then EE or EXP (which corresponds to 10 ) then 23 Your calculator should look similar 6.02 x10 23 Step 2. Multiply by pressing: Step 3. Enter 4.155 10 9 by pressing: 4.155 then EE or EXP (which corresponds to 10 ) then 9 4.155 x10 9 Your calculator should now read to: CHM130 Chapter 2 Notes page 5 of 9
Step 4. Get the answer by pressing: = Your calculator should now read 2.50131 x10 33 The answer with the correct # of sig figs = Be sure you can do exponential calculations with your calculator. Most of the calculations we do in chemistry involve very large and very small numbers with exponential terms. Ex. 2: (3.75 10 15 ) (8.6 10 4 ) = 3.225 10 20 The answer with the correct # of sig figs = Ex. 3: (1.90 10 15 ) (2.500 10 8 ) = 7600000 The answer with the correct # of sig figs = SIGNIFICANT DIGITS AND EXACT NUMBERS Although measurements can never be exact, we can count an exact number of items. For example, we can count exactly how many students are present in a classroom, how many M&Ms are in a bowl, how many apples in a barrel. 2.8 UNIT EQUATIONS AND UNIT FACTORS Unit equation: Simple statement of two equivalent values Conversion factor = unit factor = equivalents: - Ratio of two equivalent quantities Unit equation Unit factor 1 dollar = 10 dimes 1 dollar 10 dimes or 10 dimes 1 dollar Unit factors are exact if we can count the number of units equal to another or if both quantities are in the same system of measurement i.e. both in the metric system (e.g. cm and meters) or in the English system (inches and feet). CHM130 Chapter 2 Notes page 6 of 9
For example, the following unit factors and unit equation are exact: 365.25 days 1 year 7 days 1 week 12 inches 1 foot and 1 yard 3 feet Exact equivalents or unit factors have an infinite number of sig figs never limit number of sig figs! Other equivalents are inexact or approximate because they are measured values or approximate relationships, such as 1.61 km 1 mile 65 mi hour 2.2 lb. kg Approximate equivalents do limit the sig figs for the final answer. 2.9 UNIT (or DIMENSIONAL) ANALYSIS PROBLEM SOLVING 1. Write the units for the answer. 2. Determine what information to start with. 3. Arrange all other unit factors showing them as fractions with correct units in the numerator and denominator, so all units cancel except for the units needed for the final answer. 4. Check for correct units and number of sig figs in the final answer. Example 1: You and a friend decide to drive to San Diego, which is about 375 miles from Glendale. If you average 65 mph with no stops, how many hours does it take to get there? CHM130 Chapter 2 Notes page 7 of 9
Example 2: If a marathon is 26.2 miles, how many yards are in a marathon? (1 mile 5280 feet and 1 yard 3 feet) Example 3: The speed of light is about 3.00 10 8 meters per second. Express this speed in miles per hour. (1.61 km=1 mile, 1000 m 1 km) 2.10 THE PERCENT CONCEPT PERCENTAGES Percent: Ratio of parts per 100 parts 10% is 10 25, 25% is 100 100, etc. To calculate percent, divide one quantity by the total of all quantities in sample: Percentage = one part total sample 100% Ex. 1 What percentage is female in a chemistry class with 25 women and 20 men? What percentage is male? (Express answers to 3 sig figs.) CHM130 Chapter 2 Notes page 8 of 9
Writing out Percentage as Unit Factors Ex. 1: Water is 88.8% oxygen by mass. Write two unit factors using this info. Ex. 2: A 1968 penny was cast from a mixture of 95.0% copper and 5.0% zinc by mass. Write four unit factors using this information. Percentage Practice Problems Ex. 1 An antacid sample was analyzed and found to be 10.0% aspirin by mass. What mass of aspirin is present in a 3.50 g tablet of antacid? Ex. 2 Water is 88.8% oxygen and 11.2% hydrogen by mass. How many grams of hydrogen are present in 500.0 g of water? Ex. 3 Steel is an alloy of iron mixed with other elements like carbon and chromium. If a sample of high carbon steel contains is 1.35% carbon by mass, what mass of the steel contains 3.50 g of carbon? CHM130 Chapter 2 Notes page 9 of 9