Chapter 2: Measurements & Calculations

Chapter 2: Measurements & Calculations LA-PRIVATE:sg:sg.02_Measurements_and_Calculations.docx (9/1/14)

Chemistry Measurements & Calculations p.1 TABLE OF CONTENTS I. SCIENTIFIC METHOD... 2 II. METRIC UNITS (LE SYSTÈME INTERNATIONAL D UNITÉS)... 3 III. SCIENTIFIC NOTATION... 5 IV. SIGNIFICANT FIGURES... 6 V. DIMENSIONAL ANALYSIS... 8 VI. EVALUATION OF SCIENTIFIC DATA... 9 A. ACCURACY AND PRECISION... 9 B. PERCENT ERROR... 10 C. PROPORTIONALITY: DIRECT or INVERSE... 11

Chemistry Measurements & Calculations p.2 I. SCIENTIFIC METHOD The scientific method the logical approach to solving problems. Generally, all variables except one are held constant. The experimenter changes that one variable, the independent variable 1, which affects the dependent variable. The control is what would happen without changing the independent variable. The system is the specific portion of matter, space and time that is studied. For example, if I want to see the effects of salt water on plant growth, the control may be a set amount of distilled water shown to produce growth in a plant. The water, plant, soil, sunlight, temperature, etc comprises the system. The independent variable is the amount or concentration of salt in the water and the dependent variable is the amount that the plant grows. The scientific method is comprised of the following parts: 1. Making an observation. 2. Formulating a hypothesis. This step is not merely a guess but applying basic scientific principles to explain the situation and make a prediction about the outcome of this study. 3. Designing and carrying out the experiment = testing the hypothesis. During this part, observations are recorded, analyzed, and interpreted. Observations are data either quantitative (i.e., contain a number) or qualitative (i.e., do not contain a number). Results are the values resulting from calculations done on data. For example, the temperatures of a solution before and after a chemical reaction are data 2. The difference in temperature due to the chemical reaction is a result. 4. Applying the basic scientific principles to the results of the experiment, used to construct a model and/or explain the results of the study is theorizing. 5. The study is not complete until the investigation is communicated to other people, such as writing a report. OBSERVING o collecting data o measuring o experimenting o communicating FORMULATING HYPOTHESES o decipher/identify basic scientific principles o classifying o inferring o predicting TESTING THEORIZING PUBLISHING RESULTS o design experiment o construct o communicate o perform model experiment o predict o collect data (observations: quantitative & qualitative) o analyze Data do not support hypothesis: revise or reject hypothesis. Results confirmed or proved incorrect by other scientists. 1 One way to differentiate between the independent and dependent variables is that the independent variable is the variable that I change. The dependent variable depends on what I do. For example, I give the plant a certain amount of water. The height the plant grows depends on how much water I give it. 2 Data are plural; datum is singular. For example, the temperature of the room is a datum; the temperature and humidity of the room are data.

Chemistry Measurements & Calculations p.3 II. METRIC UNITS (Le Système International d Unités) Observations are critical in scientific discovery. Quantitative observations involve numbers; qualitative observations do not. For example, the temperature of the water is 97.4 o C is quantitative. The water is hot is a qualitative statement. Quantitative measurements must have both a numerical value and a unit e.g., 2.54 cm. In 1960, scientists from all over the world agreed on SI (a.k.a., metric) units as the world s standard. The metric system uses a standard unit with a prefixes in multiples of ten. Base Units: Table 1. SI Base Units Quantity Base Unit COMMENTS Time second (s) 60 s = 1 min; 60 min = 1 hr; 24 hr = 1 day Length meter (m) 2.54 cm = 1 inch Mass kilogram (kg) the average human mass is ~ 150 lb or 70 kg Temperature Kelvin (K) Celsius ( o C) also is commonly used Amount of Substance mole (mol) 1 mol = 6.022 x 10 23 things Electric Current ampere (A) Luminous Intensity candela (cd) Table 2. Prefixes Used With SI Units Prefix Symbol FACTOR Scientific Notation giga- G 1 000 000 000 10 9 20 GB (gigabyte) computer hard drive mega- M 1 000 000 10 6 4 MB (megabyte) camera memory kilo- k 1 000 10 3 10 km (kilometer) track race hecto- h 100 10 2 hg (hectagram) deka- da 10 10 1 dag (dekagram) base unit deci- d 0.1 = 1/10 10 1 10 dl (deciliter) blood sample centi- c 0.01 = 1/100 10 2 32 cm (centimeters) milli- mm 0.001 = 1/1 000 10 3 29 mm (millimeters) micro- µ (or u) 0.000 001 = 1/1 000 000 10 6 ~5 µl is the volume of blood that is taken by a mosquito nano- n 0.000 000 001 1/1 000 000 000 10 9 Example Mnemonic: (1.) Many // kids have dropped over dead converting metric // units! mega kilo hecto deka base deci centi milli micro (2) King Hector doesn t usually drink cold milk kilo hecta deka UNIT deci centi milli

Chemistry Measurements & Calculations p.4 Table 3. Common Metric Units Mass (gram) Length (meter) Volume (liter) 1,000 g = 1 kg 1,000 m = 1 km 1,000 L = 1 kl 1 g = 100 cg 1 m = 100 cm 1 L = 100 cl 1 g = 1000 mg 1 m = 1000 mm 1 L = 1000 ml 1 g = 1,000,000 µg 1 m = 1,000,000 µm 1 L = 1,000,000 µl Derived Units: Volume: We commonly use milliliters (1/1000 L) and liters in chemistry. You are probably already familiar with liters because bottled water is frequently sold in 0.5 L sizes and large soda bottles are 2-L common. Density: Density is an intensive property and, as such, frequently used to identify unknown substances. Density is determined from the following formula: mass density = volume The units for density are frequently in g/ml for solids and liquids and g/cm 3 for gases. Example: A piece of metal, with a mass of 147 g, is placed in a 50-mL graduated cylinder. The water rises from 20 ml to 41 ml. What is the density of the metal? 147g 147g density = = = 7.0g / ml 41mL 20mL 21mL Table 4. Common and Approximate Conversion Factors. 1 inch = 2.54 cm 12 inch = 1 foot 5,280 feet = 1 mile 1 kg 2.2 pounds (lb) 1 liter 1 quart 1 ml = 1 cm 3 ( 1 g of water) K = o C + 273 1 mole = 6.022 x 10 23 things (1 cm 3 is sometimes abbreviated 1 cc e.g., in medicine)

Chemistry Measurements & Calculations p.5 III. SCIENTIFIC NOTATION Scientific notation expresses numbers as a multiple of two parts: an integer between 1 and 10 (the significand) and ten raised to a power (exponent) in the form of a x 10 b. When the number is less one, the power of ten is negative. Using scientific notation makes working with values much, much easier. For example, the mass of a proton is 0.000 000 000 000 000 000 000 000 001 672 62 kg or, in scientific notation, 1.672 62 x 10 27 kg. Example: (A) The diameter of the Sun is 1 392 000 km. Express this value in scientific notation. Move the decimal place to produce a factor between one and ten. 1392000 The decimal place is moved 6 times to the left = 1.392 x 10 6 kg (B) The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 910 939 kg. Express this value in scientific notation. Because this value is less than one, we move the decimal place to the right, and the exponent will be a negative integer. 0.000 000 000 000 000 000 000 000 000 000 910 939 9.109 39 x 10 31 kg 1 2 3 4 5 6 7 8 9 10 = 3 x 10 30 + 1 Addition & Subtraction with Scientific Notation To add or subtract values expressed in scientific notation, the exponents must be the same. For example, 765 plus 9 is 774. To perform is operation using scientific notation, 765 = 7.65 x 10 2 and 9 = 9 x 10 0. One or the other number must be changed so that each value is set to the same exponent. Pick one. If you picked (765 x 10 0 ) + (9 x 10 0 ), you d get 774 x 10 0, which you could then restate in correct scientific notation to 7.74 x 10 2, which is 774 and the correct answer. A similar conversion to the same exponent is required for subtraction using scientific notation. Multiplication and Division with Scientific Notation This is much easier. For multiplication, multiply the significands and add the exponents. For example, to multiply 30 000 by 700 000, express the values in scientific notation (3 x 10 4 ) x (7 x 10 5 ). Multiply the significands (3 x 7 = 21) and add the exponents (4 + 5 = 9) 21 x 10 9. Expressed in correct scientific notation, the answer is 2.1 x 10 10. Check: 30,000 x 700,000 = 21,000,000,000 = 2.1 x 10 10. Similarly, for division, divide the significands and subtract the exponents. For example: (6 x 10 4 g) / (2 x 10 3 cm 3 ): The significand is 6 / 2 = 3, and the exponent is 4 3 = 1. So, the answer is 3 x 10 g/cm 3. (The 1 is omitted when the exponent is one and left simply as 10.)

Chemistry Measurements & Calculations p.6 IV. SIGNIFICANT FIGURES << How old is a dinosaur if I learned five years ago it was seven million years old? >> What is the length of the nail using the left ruler? What is the length of the nail using the right ruler? Values in chemistry class represent real measurements. They are only as accurate as the measuring device. For example, let s say you have some gold but don t know how much. You find someone in town who can determine the mass to the nearest ounce. S/he determines that you have 2 ounces. It is worth $890 at $445/ounce. However, another person, down the street, has a more expensive balance that can determine the mass within ± 0.01 ounce. S/he determines that you actually have 2.45 ounces 0.45 ounces that slipped under the wire of the first person s balance. Now, you have $200 that you wouldn t have had with the less accurate balance. Significant figures include all known digits plus one estimated digit. For example, the length of the measured nail in the left figure is somewhere between 4 cm and 5 cm. Using the ruler at the right, the nail is somewhere between 4.1 and 4.2 which could be read as 4.15 cm. The known digits are 4.1; the unknown is 0.05 cm. Table 5. Rules For Recognizing Significant Figures Rule Example (Number of Significant Figures) 1. Non-zero numbers are always significant. 73.2 g (3 s.f.) 2. Zeros between non-zero numbers are always significant. 60.5 g (3 s.f.) 3. All final zeros to the right of the decimal place are 6.20 g (3 s.f.) significant. 4. Zeros that act as placeholders are not significant. (Convert to scientific notation to remove placeholders.) a. Decimal place zero(s) - non-zero digits a. 0.0253 g (3 s.f.) b. Zeros at right of number, without ending decimal. b. 4320 g (3 s.f.) c. Zeros at right of number but ending with decimal place. c. 4320. g (4 s.f.) 5. Counting numbers and defined constants have an infinite number of significant figures. 6 molecules ( s.f.) 60 s = 1 min ( s.f.)

Chemistry Measurements & Calculations p.7 The Atlantic-Pacific rule is one general way to recognize which digits are significant. If a decimal point is Present, ignore zeros on the Pacific (left) side. If the decimal point is Absent, ignore zeros on the Atlantic (right) side. Everything else is significant. 3 Pacific side = decimal Present Atlantic side = decimal Absent Number ( sf ) Number ( sf ) 0.0019 ( 2 sf ) 802 000 ( 3 sf ) 65.01 ( 4 sf ) 400 000 ( 1 sf ) 0.0300 (3 sf ) 32 105 ( 5 sf ) 700. ( 3 sf ) 700 ( 1 sf ) Problems 1. Underline and determine the number of significant figures in each of the following values: A. 0.3080 g B. 5600 m C. 0.00007 L D. 900. 000 001 cm 3 E. 428 o C F. 160. s G. 2005 H. 9.0000 Arithmetic Calculations with Significant Figures. Adding or Subtracting: When adding or subtracting, the answer contains the same number of decimal places as the least certain value. EXAMPLE: 39.1 cm + 103.492 cm = 142.6 cm significant figures: (1/10 ths ) (1/1000 ths ) (1/10 ths ) EXAMPLE: 391 L + 100 L = 500 L significant figures: (1 s) (100 s) (100 s) Multiplying and Dividing: When multiplying and dividing, the answer contains the same number of significant figures as the least certain value. EXAMPLE: 39.1 cm x 103.492 cm = 4050 cm 2 significant figures: (3) (6) (3 the last zero isn t significant) N.B. Units multiply just as numbers do: e.g., cm x cm = cm 2 EXAMPLE: 391 L x 100 L = 400 L significant figures: (3) (1) (1) 3 H. M. Stone, "Atlantic-Pacific sig figs (INS)", J. Chem. Educ., 66, 829 (1989).

Chemistry Measurements & Calculations p.8 Rounding: If the number ends in a number less than five, the number rounds down. If the number ends in a number greater than five, the number rounds up. If the number ends in a five, the number rounds to the even number. EXAMPLES: 39.1 cm rounds to 39 cm 26.45 µg rounds to 26.4 µg 46.89 L rounds to 46.9 L 26.55 µg rounds to 26.6 µg Problems 2. Write the following values in either conventional format or scientific notation (as directed): A.. 200 203.2 B. 1.04 x 10-3 C. 0.000 303 101 D. 3.0 x 10 8 m / s E. 6.626068 10-34 m 2 kg / s G. 60 220 000 000 000 000 000 000 atoms 3. Perform the following calculations WITHOUT using a calculator. (Answer in scientific notation). A. (2.0 x 10 3 ) * (3.0 x 10 2 ) = B. (4.0 x 10 9 ) * (1.0 x 10 12 ) = C. (3.0 x 10-4 ) * (3.0 x 10 4 ) = D. (9.0 x 10 23 ) * (10) = E. (5.0 x 10 3 ) / (2.0 x 10 2 ) = F. (5.0 x 10 3 ) / (2.0 x 10-2 ) = G. (6.0 x 10 6 ) / (3.0 x 10 3 ) = H.(1.0 x 10 3 ) + (3.0 x 10 2 ) = I. (7.0 x 10-1 ) + (1.0 x 10-1 ) = J. (4.0 x 10 4 ) - (2.0 x 10 4 ) = V. DIMENSIONAL ANALYSIS Dimensional analysis (also called factor-label method) is the mathematical format that chemists use most frequently to perform calculations. It is a very simple, although not always easy, technique that requires very little memorization but is very flexible and useful. It uses two fundamental arithmetic properties: (1) anything divided by itself equals one, and (2) anything multiplied by one equals itself. Two equivalent values can written as a fraction and is called a conversion factor. Using dimensional analysis, multiply the given value with the conversion factor(s) to obtain the desired unit. EXAMPLE: 100 pennies = 1 dollar Written as a conversion factor: and conversely: 1dollar 100 pennies 100 pennies 1dollar Now, consider that we want to know how many pennies are in 3.46 dollars. 3.46 dollars 1 100 pennies x 1dollar = 346 pennies

Chemistry Measurements & Calculations p.9 EXAMPLE: 5.712 mg =? kg To reduce the amount that you have to memorize, it is not necessary to remember the mg to kg conversion. Instead, memorize the conversions to the basic unit in this case, the grams. 1000 mg = 1 g 1000 g = 1 kg 5 6.712 mg 1g 1kg - x x = 5.712 x 10 1 1000 mg 1000 g kg N.B. For bookkeeping purposes, and to reduce mistakes, cross out units as they are eliminated. EXAMPLE: 3.0 x 10 8 m/s =? km/yr 1000 m = 1 km 60 s = 1 min 60 min = 1 hr 24 hr = 1 day 365 day = 1 yr 8 3 12.0x10 1s m 1km 60 s 60 min 24 hr 365 day x x x x x = 9.5 x 10 1000 m 1min 1hr 1day 1yr kmk/yr VI. EVALUATION OF SCIENTIFIC DATA A. ACCURACY AND PRECISION Accuracy the closeness of measurements to the accepted (actual, correct) value. Precision the closeness of measurements to each other. EXAMPLE: Two groups determine the mass of a single object. Each group makes three measurements of the same object. Their data are given below. DATA SET 1 DATA SET 2 12.90 g 11.50 g 12.93 g 12.33 g 12.92 g 11.99 g AVERAGE: 12.92 11.94 g On first inspection, data set 1 seems to have the best numbers. It is the more precise. However, is it the more accurate? If the actual mass of the object is 12.00 g, then data set 2 would be the more accurate although it is less precise.

Chemistry Measurements & Calculations p.10 The average is defined mathematically by the following equation: X _ Σ(X) = N The standard deviation, s or sd, a measure of the precision of a set of data, is determined by the _ 2 Σ(X- X) following equation: s =. N -1 Other common terms used in the statistical evaluation of data are range (the difference between the lowest and highest values) and median (the middle value). For example, the median of 1, 2, 3, 4, 5, 10, 100, 1000, and 2000 is 5. Fortunately, many statistical evaluations are easily determined using standard scientific and graphing calculators and computer spreadsheets. B. PERCENT ERROR (Experimental Value) - (Accepted Value) Percent Error = * 100 (Accepted Value) If the Percent Error is Negative = Experimental Value < Accepted Value Positive = Experimental Value > Accepted Value

Chemistry Measurements & Calculations p.11 C. PROPORTIONALITY: DIRECT or INVERSE An important qualitative evaluation of two related quantities is whether they are directly related or inversely related. If one increases as the other increases, they are directly related. If one increases as the other decreases (or vice versa), they are inversely related. Examples of a direct relationship is the amount of sunshine and plant growth, and the amount of time studying and your grade in chemistry. An inverse relationship would be pressure of air and volume as the volume of a balloon decreases, the pressure inside the balloon increases. Mass and volume are directly related. mass density = volume Mass vs. Volume of Aluminum Pressure and volume of a gas are inversely related. pressure x volume = constant Pressure vs. Volume for (Constant Temperature) Volume (ml) 60 50 40 30 20 10 0 0 50 100 150 Mass (g) Volume (ml) 600 500 400 300 200 100 0 0 200 400 600 Pressure (kpa) Some mathematical expressions for commonly encountered direct and inversely relationships are as follows for quantities x and y, and constant k: Direct Relationship: x = y * k x = k y Inverse Relationship: x * y = k k x = y