Uncertainty in Measurements

Transcription

1 Chapter 2: Measurements Uncertainty in Measurements A measurement is any number with a unit attached. It is not possible to make exact measurements, and all measurements have uncertainty. We will generally use metric system units, these include. the meter, m, for length measurements the gram, g, for mass measurements the liter, L, for volume measurements Chapter 2 2 1

2 Scientific Notation Numbers in science are often very large or very small. To avoid confusion, we use scientific notation. Scientific notation utilizes the numeric digits in a measurement followed by a power of ten. The numeric digits are expressed as a number between 1 and 10. Chapter 2 3 Applying Scientific Notation Re-write these numbers in Scientific Notation: Step 1: Identify the first nonzero digit in the number 345, , Step 2: Place a decimal after the first nonzero digit in the number followed by the remaining numeric digits and x x x 10 Step 3: Indicate how many places the decimal has moved by the power of 10. A positive power of 10 indicates that the decimal moves to the right (Large Number). A negative power of 10 indicates that the decimal moves to the left (Small Number). 345, spots 3 spots 3.45 x x 10-3 Chapter 2 4 2

3 Examples of Scientific Notation There are 26,800,000,000,000,000,000,000 helium atoms in 1.00 L of helium gas. Express the number in scientific notation. Step 1: Identify the first nonzero digit: Here it s a 2 Step 2: Place the decimal after the 2, followed by the other numerical digits and x 10 : 2.68 x 10 Step 3: Count the number of places the decimal has moved and in which direction: 22 places to the right (so a positive power) Add the positive power of 10 to complete the scientific notation atoms Chapter 2 5 Another Example The typical length between two carbon atoms in a molecule of benzene is m. What is the length expressed in scientific notation? Step 1: Identify the first nonzero digit: Here it s a 1 Step 2: Place the decimal after the 1, followed by the other numerical digits and x 10 : 1.40 x 10 (We keep the last zero here!) Step 3: Count the number of places the decimal has moved and in which direction: 7 places to the left (so a negative power) Add the negative power of 10 to complete the scientific notation m Chapter 2 6 3

4 Scientific Notation and Your Calculator During this class, you will have to be able to add, subtract, multiply and divide numbers that are listed in scientific notation. That means you need to be able to input these numbers into your calculators. You DO NOT do this by typing the number then multiplying it by 10 to a power. Rather, you use a special button on your calculator, such as the EE button (shown here) or the EXP button, to type in the entire number plus the 10 to the power. The EE Button For Example: 1.25 x x 10 ^ 12 NO 1.25 EE or EXP 12 YES Chapter 2 7 Dimensional Analysis: Unit Equations Dimensional analysis (DA) uses the canceling of units to set up calculations. A unit equation is a simple statement of two equivalent quantities. For example: 1 hour 60 minutes 1 minute 60 seconds Do you know some others? Chapter 2 8 4

5 Dimensional Analysis: Conversion Factors A conversion factor takes the unit equation and converts it into a ratio. For each unit equation, you can write two conversion factors! For the unit equation 1 hour 60 minutes, we can write two unit factors: 1 hour or 60 minutes 60 minutes 1 hour Numerator Denominator Chapter 2 9 Dimensional Analysis (DA) Problem Solving To use dimensional analysis for problem solving, prepare a plan by following these steps: Step 1: Write down the given value and units (What you have) Step 2: Write down the unit asked for in the answer (What you want) Step 3: Write out a plan that will let you convert from what you have to what you want. Step 4: Determine which conversion factor(s) is needed to convert the unit in the given value to the unit in the answer. Chapter

6 Dimensional Analysis Problem How many days are in 2.5 years? Step 1: Determine what you have: 2.5 years Step 2: Determine what you want:??? days Step 3: Write a plan to convert from what you have to what you want. Step 4: Select conversion factor(s) that allows you to perform your plan: 1 year 365 days 365days 2.5 years 912.5days 910 days 1year Chapter 2 11 Another DA Problem A can of Coca-Cola contains 12 fluid ounces. What is the volume in quarts? (1 qt 32 fl oz) Step 1: Determine what you have: 12 fl oz Step 2: Determine what you want:??? quarts. Step 3: Write out your plan to convert from fl oz to qts Step 4: Select conversion factor(s): 1 qt 32 fl oz 1 qt 12 fl oz qt 0.38 qt 32 fl oz. Chapter

7 Another DA Problem A marathon is 26.2 miles. What is the distance in yards (1 mi 1760 yards)? Step 1: Determine what is given: 26.2 miles Step 2:. Determine what you want:??? yards. Step 3: Write out your plan to convert from miles to yds Step 4: Select conversion factor(s): 1 mi 1760 yards 1760 yd 26.2 mi 46,112 yd 1mi 46,100 yd Chapter 2 13 The Metric System The metric system is used by scientists and health professionals throughout the world for scientific measurements. In 1960, a modification of the metric system, called the International System of Units (SI), was adopted by scientists to provide uniformity of units throughout the world. The metric system offers simplicity over the English system by having a single base unit for each measurement. Chapter

8 Mass Measurements The mass of an object is a measure of the amount of matter it posses. Mass is measured with a balance and is not affected by gravity. Weight is the force exerted by gravity on an object Mass and weight are not interchangeable The SI unit for mass is the kilogram (kg) 1 kg 2.20 lb Chapter 2 15 Length Measurements The metric and SI unit for length is the meter (m) This unit is slightly longer than a yard (the value used in common measurements in the US) 1 m 39.4 in 1 in 2.54 cm Chapter

9 Volume Measurements Volume is the amount of space occupied by matter. The SI unit for volume is the cubic meter (m 3 ) The metric unit (and the more often used unit) is the liter (L) There are several instruments for measuring volume 1 ml 1 cm 3 Chapter 2 17 Temperature and Time Temperature is a measure of how hot or cold an object is. There are three temperature scales: Celsius C (Metric) Fahrenheit F (English) Kelvin K (SI) The SI unit for time is the second (s) Chapter

10 Common Units of Measurement Physical Quantity Length Mass Volume Time Temperature Metric Meter (m) Gram (g) Liter (L) Second (s) Celsius ( C) SI Meter (m) Kilogram(kg) Cubic meter (m 3 ) Second (s) Kelvin (K) Chapter 2 19 Metric System Advantage An advantage of the metric system is that it is a decimal system. It uses prefixes to enlarge or reduce the basic units by factors of 10. For example, the base unit for length is the meter (m): A kilometer is 1000 meters. A centimeter is 1/100 of a meter. Chapter

11 Metric System Prefixes You will need to be able to convert between the units boxed in Chapter 2 21 Metric Equivalents We can write unit equations for the conversion between different metric units. The prefix kilo- means 1000 basic units, so 1 kilometer is 1000 meters. What is the unit equation? 1 km 1000 m What are the two conversion factors for this unit equation? 1 km or 1000 m 1000 m 1 km Chapter

12 Metric-Metric Conversion Problem What is the mass in grams of a 325 mg aspirin tablet? Step 1: Determine what is given: 325 mg. Step 2: Determine want you want:??? grams. Step 3: Write out your plan to convert from mg to grams. Step 4: Select conversion factor(s): 1 mg 1 x 10-3 g 1 x mg -3 g g 1 mg Chapter 2 23 Two Metric-Metric Conversions A hospital has 125 deciliter bags of blood plasma. What is the volume in milliliters? Step 1: Determine what you have: 125 dl Step 2: Determine what you want:??? ml. Step 3: Write out your plan to convert from dl to ml. Step 4: Select conversion factor(s): 1 dl 0.1 L AND 1 ml 1 x 10-3 L 125 dl 0.1 L 1 ml 12,500 ml 1 dl 1 x 10-3 L Chapter

13 Another Example The mass of the Earth is kg. What is the mass expressed in megagrams, Mg? Step 1: Determine what is given: kg Step 2: Determine what you want:??? Mg Step 3: Write out your plan to convert from kg to Mg. Step 4: Select conversion factor(s): 1 kg 1000 g AND 1 Mg 1,000,000 g 1000 g 1 Mg kg Mg 1 kg g Chapter 2 25 Measured numbers are the numbers you obtain when you measure a quantity such as your height or weight. Measured Numbers A measurement is a number with a unit attached. It is not possible to make exact measurements, and all measurements have uncertainty. Chapter

14 Uncertainty: Accuracy and Precision Uncertainty is expressed in two ways: Accuracy and Precision Accuracy is how close to the true value a given measurement is. Precision is how well a number of independent measurements agree with one another. Chapter 2 27 Significant Figures One way to indicate uncertainty in a measurement is to use significant figures (or significant digits). The number of significant figures in a measurement is the number of digits that are known accurately plus one uncertain digit. The uncertain digit is always the last digit that is written! Counting Significant Figures: Rule 1: Count the number of digits in a measurement from the left to right: Start with the first non-zero digit (ignore the decimal place!) Do not count place-holder zeros Rule 2: The rules for significant digits apply only to measurements and not to exact numbers. Chapter

15 Place-holder Zeros Zeros found at the beginning of a number ARE never significant. Therefore, 0.5 cm, 0.05 cm, and cm all have one significant digit. Zeros found at the end of a number with no decimal point ARE NOT significant. Therefore, 50 cm, 500 cm, and 5000 cm all have one significant digit. All other zeros are significant Therefore, 50.0 cm, cm, and 501cm all have three significant digits. Chapter 2 29 Exact Numbers When we count (not measure!) something, it is an exact number. Exact numbers are also obtained when you compare two units in a measurement system Example: 1 minute 60 seconds Significant digit rules do not apply to exact numbers. Other Examples?? Chapter

16 Rounding Numbers All numbers from a measurement are significant. However, we often generate nonsignificant digits when performing calculations, which are removed by rounding off numbers. Rules for Rounding Numbers: Rule 1: If the first nonsignificant digit is less than 5, drop all nonsignificant digits. Rule 2: If the first nonsignificant digit is greater than or equal to 5, increase the last significant digit by 1 and drop all nonsignificant digits. Chapter 2 31 Adding & Subtracting Measurements When adding or subtracting measurements, always look at the number of decimal places in each number the number with the least number of decimal places has the most uncertainty and determines the number of significant digits! Lets add three mass measurements. The measurement 5 g has the least number of decimal places and thus the greatest uncertainty (± 1 g) The answer has to have the same number of decimal places as the number with the least: 15 g Chapter 2 32 g g g g 16

17 Multiplying & Dividing Measurements When multiplying or dividing measurements, you look at the total number of significant digits (don t look at decimal places here!) Here, the number of sig figs in the answer is limited by the value with the fewest total significant figures. Lets multiply two length measurements. The measurement 2.3 cm has the fewest significant digits, two. The correct answer is 12 cm cm X 2.3 cm cm 2 Chapter 2 33 Metric and English Units The English system is still very common in the United States and we often have to convert between these units. Mass Volume 1 lb 454 g 1 qt 4 cups (cu) 1 oz 28.3 g 1 qt 2 pts 1 kg 2.20 lb 1 qt 32 fl. oz. Length 1 gal 4 qts 1 ft 12 in 1 ml 1 cm 3 1 yd 3 ft 1 L 1.06 qt 1 mile 5280 ft 1 gal L 1 in 2.54 cm Other 1 m 39.4 in 1 calorie (cal) Joules (J) 1 ft 30.5 cm 1 atm 760 mm Hg or torr 1 km mi 1 atm kpa You will need to bring these equalities to class for example problems! You don t need to memorize these equalities. Chapter 2 They will be given to you on the quizzes and exams

18 English-Metric Conversion A half gallon carton contains 64.0 fl oz of milk. How many milliliters of milk are in a carton? (1 qt 32 fl oz and 1 qt 946 ml ) Step 1: Determine what is given: 64.0 fl oz. Step 2: Determine what you want:??? ml Step 3: Write out your plan to convert fl oz. to ml Step 4: Select conversion factor(s): 1 qt 32 fl oz; 1 L 1.06 qt; 1 ml L Step 5: Round to three significant figures (WHY??) 1 qt 1 ml 64.0 fl oz 1 L 1,890 ml 32 fl oz 1.06 qt L Chapter 2 35 Compound Units Some measurements have a ratio of units. Can you think of any? For example, the speed limit on many highways is 55 miles per hour. How would you convert to this to meters per second? Convert one unit at a time using the correct conversion factors. Chapter

19 Compound Unit Problem A Corvette is traveling at 95 km/hour. What is the speed in meters per second? Step 1: Determine what is given: 95 km/h Step 2: Determine what you want:??? m/s. Step 3: Write out your plan to convert km to m AND h to s Step 4: Select conversion factor(s): 1 km 1000 m and 1 h 3600 s. Step 5: Round to two significant figures. (WHY??) 95 km hr 1000 m 1 hr 1 km 3600 s 26 m/s Chapter 2 37 The Percent Concept A percent, %, expresses the amount of a single portion compared to an entire sample. portion of interest ("Part") % 100% total sample ("Whole") The percent composition of a compound lists the mass percent of each element. % Mass of element Mass of compound X 100% Chapter

20 Calculating Percentages Sterling silver contains silver and copper. If a sterling silver chain contains 18.5 g of silver and 1.5 g of copper, what is the percent silver in sterling silver? Step 1: Identify the Part. Here it is the silver (18.5 g). Step 2: Identify the Whole. Here, the total sample (the sterling silver) is the mass of the silver (18.5 g) plus the mass of the copper (1.5 g). Step 3: Plug these values into the formula for percent composition and multiply by 100% g silver ( ) g 100% 92.5% silver Chapter 2 39 The Density Concept The density of an object is its mass per unit of volume Density (D) is defined as the mass (m) of an object divided by the volume (V) of the object. mass volume density (D) m V Density is a physical property of a chemical substance Density can vary with the physical state of a substance! What do you think is the most dense? A solid, liquid or gas? Chapter

21 Estimating Density We can estimate the density of a substance by comparing it to another object. A solid object will float on top a liquid with a higher density. What do you think happens if the solid has a higher density than the liquid? Chapter 2 41 Calculating Density What is the density of a platinum nugget that has a mass of g and a volume of 10.0 cm 3? Step 1: We have a mass ( g) and a volume (10.0 cm 3 ) Step 2: We want the density (g/cm 3 ) Step 3: Use the formula for density with your given values to solve for density Step 4: Round to three significant digits g 10.0 cm g/cm 3 D V m Chapter

22 Volume by Displacement If a solid has an irregular shape, its volume cannot be determined by measuring its dimensions. You can determine its volume indirectly by measuring the amount of water it displaces. This technique is called volume by displacement. The Volume of the Object Final Volume Initial Volume V Object V F -V I Chapter 2 43 Calculating Density A blob of aluminum has a mass of g. When the aluminum is placed in a graduated cylinder with a water level of ml the water level rises to ml. Find the density of the aluminum blob. Step 1: We have a mass ( g) and two volume measurements (before Al added (V I ): ml AND after Al added (V F ): ml) Step 2: We want the density (g/ml) Step 3: You must calculate the volume of the aluminum blob (V F -V I ) then use the formula for density with you given values to solve for density Step 4: Round to four significant digits m V Al V F V I D V ml ml g ml D g/ml ml Chapter

23 How to Solve a Quantitative Chemistry Problem Chapter 2 45 Another Density Problem An automobile battery contains 1275 ml of acid. If the density of battery acid is 1.84 g/ml, how many grams of acid are in an automobile battery? Step 1: We have volume (1275 ml) Step 2: We want mass (grams) Step 3: These quantities are related to one another by the formula for density so rearrange that formula and use the given value for density and volume to solve for our unknown mass Step 4: Round to three significant digits D V m Rearrange m D V 1.84 g 1275 ml 2350 g ml Chapter

24 More Density Problems There are additional practice problems available on eres and my website! D V m Chapter