1.2 Measurement and Scientific Notation What Is a Measurement? quantitative observation comparison to an agreed- upon standard every measurement has a number and a unit Examples: 4.5 g 5.082 kg 25.0 ºC 0.0004 lb 4.3210 meters number + unit it is a statement of magnitude: (very small, small, large, very large) Here we use scientific notation it is a statement of accuracy: (very accurate = very close to the real value) Here we use significant figures AVC-CHEM-CH

Scientific notation helps to write very large or very small numbers. A number in scientific notation contains a coefficient and a power of 10. coefficient power of ten 1.5 x 102 7.35 x 10-4 Writing Numbers in Scientific Notation To write a number in scientific notation, the decimal point is placed after the first digit. The spaces moved are shown as a power of ten. 52 000. = 5.2 x 104 0.00378 = 3.78 x 10-3 4 spaces left 3 spaces right S-10 AVC-CHEM-CH

Figure 01-T02 Title: Some Powers of 10 Caption: 01_T02.JPG

1. Write the following measurements in scientific notation: SAMPLE PROBLEM 1.2 Scientific Notation 1. Write the following measurements in scientific notation: 0.000 16 L b. 5 220 000 m 2. Write the following as standard numbers: a. 7.2 x 10–3 m b. 2.4 x 105 g

1.3 Measured Numbers and Significant Figures A measuring tool Is used to determine a quantity such as height or the mass of an object. Provides numbers for a measurement called measured numbers. Every measured number has a degree of uncertainty. The more uncertain, the less accurate. S-10 AVC-CHEM-CH

Reading a Meter Stick Determine the length of the wood. S-10 AVC-CHEM-CH

Reading a Meter Stick 4.7 cm Determine the length of the wood. S-10 AVC-CHEM-CH

Reading a Meter Stick Determine the length of the piece of wood. S-10 AVC-CHEM-CH

Reading a Meter Stick The markings on the meter stick are read as: The first digit 4 plus the second digit 4.5 The last digit is obtained by estimating. 4.56 The end of the wood might be estimated between 4.55 –4.57 for instruments marked with a scale, you get the last digit by estimating between the marks, if possible, mentally divide the space into 10 equal spaces, then estimate how many spaces are used. S-10 AVC-CHEM-CH

Zero as a Measured Number . l3. . . . l . . . . l4. . . . l . . . . l5. . cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm. The 4 and 5 are for sure, we must estimate the last digit. S-10 AVC-CHEM-CH

Practice Figure 01-14-06UN Title: Mesurement Problem Caption: Measure the length of each of the objects in figure (a), (b), and (c) using the metric rule in the figure. Indicate the number of significant figures for each and the estimated digit for each. 01_14-06UN.JPG

Known + Estimated numbers = Significant figures or Significant numbers In the length reported as 2.73 cm, The digits 2 and 7 are certain (known). The final digit 3 was estimated (uncertain). All three digits (2.73) are significant including the estimated digit. S-10 AVC-CHEM-CH

General, Organic, and Biological Chemistry

Practice A. Which answer(s) contains 3 significant figures? 1) 0.4760 2) 0.00476 3) 4.76 x 103 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. The number of significant figures in 5.80 x 102 is 1) one 2) two 3) three S-10 AVC-CHEM-CH

An exact number is obtained When objects are counted Counting objects Exact Numbers Not every number is a measured number, non-measured numbers are said to be exact. An exact number is obtained When objects are counted Counting objects 2 soccer balls 4 pizzas From numbers in a defined relationship. Defined relationships 1 foot = 12 inches 1 meter = 100 cm From integer values in equations. In the equation for the radius of a circle, the 2 is exact radius of a circle = diameter of a circle 2 S-10 AVC-CHEM-CH

SAMPLE PROBLEM 1.3 Significant Figures Identify each of the following numbers as measured or exact, and give the number of significant figures in each measured number: 42.2 g three eggs 5.0 x 10–4 cm 450 000 km 3.500 x 105 s

1.4 Significant Figures in Calculations When we carry out a mathematical operation such as: 4.25 3 sf X 0.10 2 sf  least 0.425 3 sf One can not increase significant figures (reduce the uncertainty) by means of a mathematical operation. This can only be done by means of the measuring instrument. To remove non-significant numbers one must round-off the number. S-10 AVC-CHEM-CH

To round 45.832 to 3 significant figures 45.8 32 Rules for Rounding Off If the first digit to be dropped is 4 or less, it and all following digits are simply dropped from the number. To round 45.832 to 3 significant figures 45.8 32 drop the digits 32 = 45.8 If the first digit to be dropped is 5 or greater, the last retained digit is increased by 1. To round 2.4884 to 2 significant figures 2.4 884 drop the digits 884 and increase the 4 by 1 = 2.5 S-10 AVC-CHEM-CH

SAMPLE PROBLEM 1.4 Rounding Off Round off each of the following numbers to three significant figures: a. 35.7823 m b. 0.002 627 L c. 3.8268 x 103 g d. 1.2836 kg

Adding Significant Zeros Sometimes a calculated answer requires more significant digits. Here one or more zeros are added. 4.0 x 1.0 = 4 needs to reported as 4.0 Calculated Zeros added to answer give 3 significant figures 4 4.00 1.5 1.50 0.2 0.200 Round 1001 to 1 sig. fig 1000 Round 1001 to 2 sig. figs 1.0 x 103 S-10 AVC-CHEM-CH

Calculations with Measured Numbers In calculations with measured numbers, significant figures or decimal places are counted to determine the number of figures in the final answer. Multiplication and Division When multiplying or dividing use The same number of significant figures as the measurement with the fewest significant figures. Rounding to obtain the correct number of significant figures. Example: 110.5 x 0.048 = 5.304 = 5.3 (rounded) 4 SF 2 SF calculator 2 SF S-10 AVC-CHEM-CH

SAMPLE PROBLEM 1.5 Significant Figures in Multiplication and Division Perform the following calculations of measured numbers. Give the answers with the correct number of significant figures: a. b. c. d.

Addition and Subtraction When adding or subtracting use The same number of decimal places as the measurement with the fewest decimal places. Rounding rules to adjust the number of digits in the answer. 25.2 one decimal place + 1.34 two decimal places 26.54 calculated answer 26.5 answer with one decimal place S-10 AVC-CHEM-CH

SAMPLE PROBLEM 1.6 Significant Figures in Addition and Subtraction Perform the following calculations, and give the answers with the correct number of decimal places: a. 27.8 cm + 0.235 cm b. 104.45 mL – 0.838 mL + 46 mL c. 153.247 g – 14.82 g

1.1 Units of Measurement In every measurement, a number is followed by a unit. Number and Unit 35 m 0.25 lb The units used in most of the world, and everywhere by scientists, are those found in the metric system. (~ 1790) In an effort to improve the uniformity of units used in the sciences, the metric system was modified and called the International System of Units (Système International) or SI. (~ 1960) Measurement Metric SI Length meter (m) meter (m) Volume liter (L) cubic meter (m3) Mass gram (g) kilogram (kg) Time second (s) second (s) Temperature Celsius (C) Kelvin (K) Fundamental units or Base units All other units are called derived. S-10 AVC-CHEM-CH

Unit x 10  increases its value 1 x 10 =10 = 1x101 The metric system or SI (international system) is A decimal system based on 10. Regardless of the unit, it can be increased or decrease by a factor of 10 Unit x 10  increases its value 1 x 10 =10 = 1x101 1 x 10 x 10 =100 = 1x102 1 x 10 x 10 x 10 =1000 = 1x103 Unit ÷ 10  decreases its value 1/10 =0.10 = 1x10-1 1/10x10 =0.010 = 1x10-2 1/10x10x10 =0.001 = 1x10-3 S-10 AVC-CHEM-CH

1.5 Prefixes and Equalities A prefix Makes a unit larger or smaller than the initial unit by one or more factors of 10. 1m x 10 x 10 x 10 = 1x103 m =1000 m An increment of 1000 or 1x103 is referred to as Kilo 1 kilometer is 1000 m 1 km = 1000 m In front of a unit increases or decreases the size of that unit. Indicates a numerical value. prefix value 1 kilometer = 1000 meters 1 kilogram = 1000 grams S-10 AVC-CHEM-CH

01_T06.JPG Figure 01-T06 Title: Metric and SI Prefixes Caption:

General, Organic, and Biological Chemistry An equality States the same measurement in two different units. General, Organic, and Biological Chemistry

Exact and Measured Numbers in Equalities Equalities between units of The same system are definitions and use exact numbers. 1 m = 1000 mm both 1 and 1000 are exact and not used to determine significant figures. Different systems (metric and U.S.) use measured numbers and count as significant figures. 1 lb = 454 g Here, 454 has 3 sig. figs. and the 1 is considered exact. An exception: The unit of an inch Is equal to exactly 2.54 centimeters in the metric (SI) system. 1 in. = 2.54 cm S-10 AVC-CHEM-CH

Is a ratio obtained from an equality. Equality: 1 in. = 2.54 cm Unit equalities provide two conversion factors. What is a Conversion Factor ? A conversion factor Is a ratio obtained from an equality. Equality: 1 in. = 2.54 cm Can be inverted to give two conversion factors for every equality. 1 in. and 2.54 cm 2.54 cm 1 in. May be obtained from a table of equalities (see p. 22 table 1.9) May be obtained from information in a word problem. Example 1: The price of one pound (1 lb) of red peppers is $2.39. 1 lb red peppers and $2.39 $2.39 1 lb red peppers Example 2: The cost of one gallon (1 gal) of gas is $2.94. 1 gallon of gas and $2.94 $2.94 1 gallon of gas S-10 AVC-CHEM-CH

Any ratio can be used as a conversion factor. Percent as a Conversion Factor Percent is a ratio 1% = 1 100 2% = 2 100% = 100 Example: A food contains 30% (by mass) fat. 30 g fat and 100 g food 100 g food 30 g fat Density as a Conversion Factor Density is a ratio D = mass = g volume mL For a substance with a density of 3.8 g/mL, the equality is: 3.8 g = 1 mL and gives: 3.8 g and 1 mL 1 mL 3.8 g S-10 AVC-CHEM-CH

Practice Write the equality and conversion factors for each of the following: A. meters and centimeters B. jewelry that contains 18% gold one gallon of gas is $ 2.95 hours and minutes Density of water is 1.00 S-10 AVC-CHEM-CH

1.7 Problem Solving Strategic problem solving starts with: 1. Identifying the initial and final unit. Initial unit “given” Final unit “want” 2. Select equalities to be used as conversion factors. 3. Apply the sequence: Given x cf(s) = want Problem: A person has a height of 180 cm. What is the height in inches? conversion factor 2.54 cm = 1 in Use conversion factor to cancel the initial unit and provide the final unit. Given x cf(s) = want 180 cm x 1 in = 71 in 2.54 cm AVC-CHEM-CH

Practice A rattlesnake is 2.44 m long. How many centimeters long is the snake? 1) 2440 cm 2) 244 cm 3) 24.4 cm S-10 AVC-CHEM-CH

Practice What is 165 lb in kg? AVC-CHEM-CH S-10

Using Two or More Factors Often, two or more conversion factors are required to obtain the unit needed for the answer. Example: How many minutes are in 1.4 days? 1.4 days x 24 hr x 60 min = 2.0 x 103 min 1 day 1 hr Be sure to check your unit cancellation in the setup. The units in the conversion factors must cancel to give the correct unit for the answer. AVC-CHEM-CH S-10

Practice A bucket contains 4.65 L of water. How many gallons of water is that? S-10 AVC-CHEM-CH

Practice If a ski pole is 3.0 feet in length, how long is the ski pole in mm? S-10 AVC-CHEM-CH

Practice If olive oil has a density of 0.92 g/mL, how many liters of olive oil are in 285 g of olive oil? 1) 0.26 L 2) 0.31 L 3) 310 L AVC-CHEM-CH

Practice How many lb of sugar are in 120 g of candy if the candy is 25% (by mass) sugar? S-10 AVC-CHEM-CH

SAMPLE PROBLEM 1.13 Clinical Factors from a Word Problem STUDY CHECK An antibiotic dosage of 500 mg is ordered. If the antibiotic is supplied in liquid form as 250 mg in 5.0 mL, how many mL would be given?