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Slide 1 of 48 Measurements and Their Uncertainty 3.1 3.1

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© Copyright Pearson Prentice Hall Slide 2 of 48 3.1 Measurements and Their Uncertainty On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Each day of its mission, Spirit recorded measurements for analysis. In the chemistry laboratory, you must strive for accuracy and precision in your measurements.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 3 of 48 3.1 Using and Expressing Measurements A measurement is a quantity that has both a number and a unit. Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 4 of 48 3.1 Using and Expressing Measurements In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 5 of 48 Why Use Scientific Notation? 602000000000000000000000 6.02x10 23 0.000000000000000000000000000000911 9.11x10 -31

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 6 of 48 Scientific Notation 1x10 0 =11x10 -1 = 0.1 1x10 1 =101x10 -2 = 0.01 1x10 2 =1001x10 -3 = 0.001 1x10 3 =1,0001x10 -4 = 0.0001 1x10 4 =10,000 1x10 5 =100,000

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 7 of 48 Scientific Notation Numbers in Scientific notation must be between 1 and 10, with only one digit to the left of the decimal. Ex. 2.5x10 7 not 25x10 6 or.25x10 8

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 8 of 48 Scientific Notation If the decimal must be moved, remember that making the number smaller makes the exponent bigger and making the number larger makes the exponent smaller. Ex. 35x10 7 3.5x10 8.27x10 6 2.7x10 5

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 9 of 48 Scientific Notation Express the following in scientific notation. a.40c. 0.4e. 4004g. 0.004 b. 400d. 404f. 4400 h. 0.0404

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 10 of 48 Express the following as whole numbers or decimals. a.6.1 x 10 2 e. 6.01 x 10 -4 b. 6.01 x 10 3 f. 6.01 x 10 4 c.6.0 x 10 -2 g. 2.4 x 10 3 d. 6.6 x 10h. 5.43 x 10 -5

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 11 of 48 Multiplying with Scientific Notation 1.Multiply the numbers 2.Add the exponents 2.5x10 5 3.2x10 6 = 8.0x10 11

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 12 of 48 Dividing with Scientific Notation 1.Divide the numbers 2.Subtract the exponents 8.0x10 9 / 4.0x10 3 = 2.0x10 6

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 13 of 48 Multiplying and Dividing with Sci Not Perform the following calculations, expressing your answers in scientific notation. a.(6.0 x 10 4 )(2.0 x 10 5 )e. (8.0 x 10 3 ) / (2.0 x 10 6 ) b.(6.0 x 10 -3 )(3.0 x 10 5 )f. (3.0 x 10 4 ) / (6.0 x 10 -2 ) c.(4.0 x 10 4 )(2.0 x 10 -6 )g. (2.0 x 10 -3 ) / (4.0 x 10 -8 ) d. (6.0 x 10 6 ) / (2.0 x 10 4 )

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 14 of 48 Adding and Subtracting with Scientific Notation 1.Move the decimal on one of the numbers so that both are the same exponent. 2.Add or subtract the numbers and keep the same exponent 2.4x10 5 + 4.6x10 6.24x10 6 + 4.6x10 6 = 4.84x10 6

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 15 of 48 Adding and Subtracting Sci Not Perform the following calculations, expressing your answers in scientific notation. (3.5 x 10 -4 ) + (5.0 x 10 -3 ) = __________ (6.5 x 10 8 ) + ( 2.6 x 10 9 )= __________ (2.5 x 10 7 ) - (5.0 x 10 8 ) = __________ (7.4 x 10 5 ) - ( 2.7 x 10 6 )= __________

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 16 of 48 3.1 Accuracy, Precision, and Error Accuracy and Precision Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 17 of 48 3.1 Accuracy, Precision, and Error To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 18 of 48 3.1 Accuracy, Precision, and Error

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 19 of 48 3.1 Accuracy, Precision, and Error Determining Error The accepted value is the correct value based on reliable references. The experimental value is the value measured in the lab. The difference between the experimental value and the accepted value is called the error.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 20 of 48 3.1 Accuracy, Precision, and Error The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 21 of 48 Accuracy, Precision, and Error 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 22 of 48 3.1 Accuracy, Precision, and Error Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 23 of 48 Significant Figures in Measurements Why must measurements be reported to the correct number of significant figures? 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 24 of 48 Significant Figures in Measurements 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 25 of 48 Significant Figures in Measurements Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated. 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 26 of 48 Significant Figures 1.All nonzero digits are significant. 2.Zeroes between two significant figures are themselves significant. 3.Zeroes at the beginning of a number are never significant. 4.Zeroes at the end of a number are significant if a decimal point is written in the number.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 27 of 48 Significant Figures in Measurements 3.1 The Atlantic Pacific Rule (1) Pacific – "P" is for decimal point is present. If a decimal point is present, count significant digits starting with the first non-zero digit on the left (Pacific side of the US). Examples: (a) 0.004713 has 4 significant digits. (b) 18.00 also has 4 significant digits. Pacific Atlantic

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 28 of 48 Significant Figures in Measurements 3.1 The Atlantic Pacific Rule (2) Atlantic – "A" is for decimal point is absent. If there is no decimal point, start counting significant digits with the first non-zero digit on the right (Atlantic side of the US). Examples: (a) 140,000 has 2 significant digits. (b) 20060 has 4 significant digits. Pacific Atlantic

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 29 of 48 Significant Figures in Measurements 3.1 The Atlantic Pacific Rule Imagine a map of the U.S.; (1)If the decimal is absent count from the Atlantic side. (2)If the decimal point is present, count from the Pacific side. (3)In both cases, start counting with the first non- zero digit. Pacific Atlantic

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© Copyright Pearson Prentice Hall Slide 30 of 48

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© Copyright Pearson Prentice Hall Slide 31 of 48 Practice Problems Problem Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial. for Conceptual Problem 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 32 of 48 Practice 3. Determine the number of significant figures in each of the following measurements: a) 65.42 gb) 385 L c) 0.14 mL d) 709.2 me) 5006.12 kgf) 400 dm g) 260. mmh) 0.47 cgi) 0.0068 k j) 7.0 cm3k) 36.00 gl) 0.0070 kg m) 100.6040 Ln) 340.0 cm

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 33 of 48

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 34 of 48

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 35 of 48

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 36 of 48 Significant Figures in Calculations How does the precision of a calculated answer compare to the precision of the measurements used to obtain it? 3.1

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Slide 37 of 48 © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Significant Figures in Calculations In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated. 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 38 of 48 3.1 Significant Figures in Calculations Rounding To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 39 of 48 3.1

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 40 of 48 3.1

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 41 of 48 3.1

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 42 of 48 3.1

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© Copyright Pearson Prentice Hall Slide 43 of 48 Practice Problems Problem Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial. for Sample Problem 3.1

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 44 of 48 3.1 Significant Figures in Calculations Addition and Subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 45 of 48 3.2

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 46 of 48 3.2

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 47 of 48 3.2

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 48 of 48 3.2

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© Copyright Pearson Prentice Hall Slide 49 of 48 Practice Problems for Sample Problem 3.2 Problem Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 50 of 48 3.1 Significant Figures in Calculations Multiplication and Division In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 51 of 48 3.3

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 52 of 48 3.3

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 53 of 48 3.3

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© Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 54 of 48 3.3

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© Copyright Pearson Prentice Hall Slide 55 of 48 Practice Problems for Sample Problem 3.3 Problem Solving 3.8 Solve Problem 8 with the help of an interactive guided tutorial.

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© Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 56 of 48 Sig Fig Practice 1) 8.57 cm x 2.7cm = _____2) 2.8385 m x 3.152m = _____ 3) 72 g / 1.52 ml = _____4) 232.5g + 68.963g + 3.75g = _____ 5) 8.525g x 0.00086g = _____6) 45.5m + 6.86m +157m = _____ 7) 10.564 s - 6.3 s = _____8) 98.6 kg - 5.63 kg = _____

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© Copyright Pearson Prentice Hall Slide 57 of 48 Section Quiz -or- Continue to: Launch: Assess students’ understanding of the concepts in Section 3.1. Section Assessment

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© Copyright Pearson Prentice Hall Slide 58 of 48 3.1 Section Quiz 1. In which of the following expressions is the number on the left NOT equal to the number on the right? a.0.00456 10 –8 = 4.56 10 –11 b.454 10 –8 = 4.54 10 –6 c.842.6 10 4 = 8.426 10 6 d.0.00452 10 6 = 4.52 10 9

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© Copyright Pearson Prentice Hall Slide 59 of 48 3.1 Section Quiz 2. Which set of measurements of a 2.00-g standard is the most precise? a.2.00 g, 2.01 g, 1.98 g b.2.10 g, 2.00 g, 2.20 g c.2.02 g, 2.03 g, 2.04 g d.1.50 g, 2.00 g, 2.50 g

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© Copyright Pearson Prentice Hall Slide 60 of 48 3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement? a.2 b.3 c.4 d.5 3.1 Section Quiz

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