Presentation on theme: "Section 3.1 Measurements and Their Uncertainty"— Presentation transcript:
1 Section 3.1 Measurements and Their Uncertainty OBJECTIVES:Distinguish among accuracy, precision, and error of a measurement.
2 Section 3.1 Measurements and Their Uncertainty OBJECTIVES:Determine the number of significant figures in a measurement and in a calculated answer.
3 Section 3.1 Measurements and Their Uncertainty OBJECTIVES:Convert measurements to scientific notation.
4 MeasurementsWe make measurements every day: buying products, sports activities, and cookingQualitative measurements are words, such as heavy or hotQuantitative measurements involve numbers (quantities), and depend on:The reliability of the measuring instrumentthe care with which it is read – this is determined by YOU!
5 Accuracy, Precision, and Error It is necessary to make good, reliable measurements in the labAccuracy – how close a measurement is to the true valuePrecision – how close the measurements are to each other (reproducibility)
6 Precision and Accuracy Precise, but not accurateNeither accurate nor precisePrecise AND accurate
7 Accuracy, Precision, and Error Accepted value = the correct value based on reliable referencesExperimental value = the value measured in the lab
8 Accuracy, Precision, and Error Error = accepted value – exp. valueCan be positive or negativePercent error = the absolute value of the error divided by the accepted value, then multiplied by 100%| error |accepted valuex 100%% error =
9 Why Is there Uncertainty? Measurements are performed with instruments, and no instrument can read to an infinite number of decimal placesWhich of the balances below has the greatest uncertainty in measurement?
10 Significant Figures in Measurements Significant figures in a measurement include all of the digits that are known, plus one more digit that is estimated.Measurements must be reported to the correct number of significant figures.
11 Which measurement has the most significant figures? What is the measured value?What is the measured value?What is the measured value?
12 Rules for Counting Significant Figures Non-zeros always count as significant figures:3456 has4 significant figures
13 Rules for Counting Significant Figures ZerosLeading zeroes do not count assignificant figures:has3 significant figures
15 Rules for Counting Significant Figures ZerosTrailing zeros are significant only if the number contains a written decimal point:9.300 has4 significant figures
16 Rules for Counting Significant Figures Two special situations have an unlimited number of significant figures:Counted items23 people, or 425 thumbtacksExactly defined quantities60 minutes = 1 hour
17 Sig Fig Practice #1 How many significant figures in the following? 5 sig figs17.10 kg 4 sig figs100,890 L 5 sig figsThese all come from some measurements3.29 x 103 s 3 sig figscm 2 sig figs3,200,000 mL 2 sig figsThis is a counted value5 dogs unlimited
18 Significant Figures in Calculations In general a calculated answer cannot be more precise than the least precise measurement from which it was calculated.Ever heard that a chain is only as strong as the weakest link?Sometimes, calculated values need to be rounded off.
19 Rounding Calculated Answers Decide how many significant figures are needed (more on this very soon)Round to that many digits, counting from the leftIs the next digit less than 5? Drop it.Next digit 5 or greater? Increase by 1
20 Sample Problem (Rounding) Round off the numbers below to the number of significant figures in parenthesis.(Four)(Two)8792 (Two)
21 Rounding Calculated Answers Addition and SubtractionThe answer should be rounded to the same number of decimal places as the least number of decimal places in the problem.
22 Rules for Significant Figures in Mathematical Operations Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement.= (1 dec. place)
23 Sample Problem (Addition) Add the numbers below and record the answer with the correct number of significant figures.12.52 m m m
24 Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m + 7.0 m 100.0 g g76.27 g76.3 g0.02 cm cm2.391 cm2.39 cm713.1 L LL709.2 Llb lblblb2.030 mL mL0.16 mL0.160 mL*Note the zero that has been added.
25 Rounding Calculated Answers Multiplication and DivisionRound the answer to the same number of significant figures as the least number of significant figures in the problem.
26 Rules for Significant Figures in Mathematical Operations Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation.6.38 x 2.0 =12.76 13 (2 sig figs)
27 Sample Problem (Addition) In the problems below, calculate the number of significant figures that the correct answer would contain.7.55 m * 0.34 m2.10 m * 0.70 mm / 9 m
28 Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m x 7.0 m 100.0 g ÷ 23.7 cm3g/cm34.22 g/cm30.02 cm x cmcm20.05 cm2710 m ÷ 3.0 sm/s240 m/slb x 3.23 ftlb·ft5870 lb·ft1.030 g x 2.87 mLg/mL2.96 g/mL
29 Powers of 10 (multiples of ten) Scientific NotationPowers of 10 (multiples of ten)It is important in both science and mathematics to know the powers of ten.Many systems, including the metric system, are based on multiples of ten.By being able to recognize these multipliers, you will have an easier time converting units in the metric and SI systems and using scientific notation.
30 Powers of 10 (multiples of ten) Scientific NotationPowers of 10 (multiples of ten)For positive exponents, any power of ten can be thought of mathematically as ten multiplied together the number times equivalent to the exponent.For example:The exponent in this example is 3. The numerical value of 103 is equal to ten multiplied together 3 times (10 * 10 * 10).103 = 10 * 10 * 10 = 1,000
31 Powers of 10 (multiples of ten) Scientific NotationPowers of 10 (multiples of ten)For negative exponents, any power of ten can be thought of as placing one over the equivalent positive power of ten (THINK RECIPROCAL!!!)Negative exponents indicate the reciprocal of the positive exponents.For example:The exponent in this example is -3. The numerical value of 10-3 is equal to the reciprocal of ten multiplied by itself 3 times (1/(10*10*10)).10-3 = 1 / 103 = 1 / (10 * 10 * 10) = 1 / 1,000 = .001
32 Powers of 10 Practice #1 Power of 10 Answer Word Meaning 103 1,000 Thousand102100Hundred10110Ten10-10.1Tenth10-20.01HundredthThousandth10-30.001
33 Powers of 10 Practice #2 Power of 10 Answer Word Meaning 109 1,000,000,000Billion1061,000,000Million10-6Millionth10-9Billionth1001One(NOT ZERO)
34 Scientific NotationMathematicians and Scientists often deal with quantities that are extremely large or extremely small.It can be a cumbersome and difficult task to write out and express these quantities in their original longhand notation.For example,The speed of light, c, is estimated to be approximately 300,000,000 m/s.In chemistry, Avogadro’s constant represents the number of particles in one mole of a substance. The number is ~60,220,000,000,000,000,000,000 particles per mole.The number of grams of a given substance in a mixture might be g. In the above examples, it is easy to see that these numbers can be long and confusing.
35 Scientific NotationScientific Notation is a shorthand method of expressing numbers based on the powers of 10 (Once again, it is very important to know the powers of ten.)Students in chemistry must know how to convert from longhand notation to scientific notation and vice versa.Examples:5.62 X 1034.0 X 10-5
36 Scientific NotationConverting from longhand (conventional) notation to Scientific NotationScientific Notation is expressed by taking the first significant digit of a number and placing a decimal after it.For example:In the number 376,540,000, three (3) is the first significant digit. Therefore, to express the number in scientific notation, you must first move the decimal to the right of the first significant digit (3).
37 Scientific NotationConverting from longhand (conventional) notation to Scientific Notation1. Locate the original decimal. If the decimal is not written it is understood to be at the end (far right) of the number.2. Locate the first Significant Figureand place a decimal to the right of it. In this case, the first Significant Figure is the 3.3. Count how many places (powers of ten) you moved the decimal.Next, it is important to note how many places the decimal was moved over. In the above example, the decimal was moved 8 places. Each place represents a power of ten (10n). Therefore, you can note that the change between the modified number and the original number was 108 or 100,000,000.The decimal was moved 8 places.
38 Scientific NotationConverting from longhand (conventional) notation to Scientific NotationAfter the decimal has been moved, drop any zeros that were not significant from the original number.3.7654
39 Scientific NotationConverting from longhand (conventional) notation to Scientific NotationUsing the format _._______ X 10n, convert the number to scientific notation. (The exponent, n, will correspond to the number of places the original decimal was moved…in this case 8.)X 108If the original number was a large number (> 1), then the exponent is positive.If the original number was small (between zero and 1), the exponent is negative.If the original number was zero (=0) or you did not move the decimal, then the exponent is zero.
40 Scientific Notation Practice #1 (Longhand to Scientific Notation) Longhand (Conventional)Scientific Notation723007.23 X 1047.23 X 10-47.7. X 100
41 Scientific NotationConverting Scientific Notation to longhand (conventional) notationConverting from Scientific Notation to longhand (conventional) notation is simply a matter of taking a number and multiplying it by a power of ten.Examples:5.62 X 103 = 56204.0 X 10-5 =Multiplying by powers of 10 is simply a matter of moving the decimal.For positive exponents, move the decimal to the right.For negative exponents, move the decimal to the left.
42 Scientific Notation Practice #2 (Scientific Notation to Longhand) Longhand (Conventional)5.1 X 1065,100,0005.1 X 10-65.0 X 1005.0