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# Section 3.1 Measurements and Their Uncertainty

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Section 3.1 Measurements and Their Uncertainty
OBJECTIVES: Distinguish among accuracy, precision, and error of a measurement.

Section 3.1 Measurements and Their Uncertainty
OBJECTIVES: Determine the number of significant figures in a measurement and in a calculated answer.

Section 3.1 Measurements and Their Uncertainty
OBJECTIVES: Convert measurements to scientific notation.

Measurements We make measurements every day: buying products, sports activities, and cooking Qualitative measurements are words, such as heavy or hot Quantitative measurements involve numbers (quantities), and depend on: The reliability of the measuring instrument the care with which it is read – this is determined by YOU!

Accuracy, Precision, and Error
It is necessary to make good, reliable measurements in the lab Accuracy – how close a measurement is to the true value Precision – how close the measurements are to each other (reproducibility)

Precision and Accuracy
Precise, but not accurate Neither accurate nor precise Precise AND accurate

Accuracy, Precision, and Error
Accepted value = the correct value based on reliable references Experimental value = the value measured in the lab

Accuracy, Precision, and Error
Error = accepted value – exp. value Can be positive or negative Percent error = the absolute value of the error divided by the accepted value, then multiplied by 100% | error | accepted value x 100% % error =

Why Is there Uncertainty?
Measurements are performed with instruments, and no instrument can read to an infinite number of decimal places Which of the balances below has the greatest uncertainty in measurement?

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.

Which measurement has the most significant figures?
What is the measured value? What is the measured value? What is the measured value?

Rules for Counting Significant Figures
Non-zeros always count as significant figures: 3456 has 4 significant figures

Rules for Counting Significant Figures
Zeros Leading zeroes do not count as significant figures: has 3 significant figures

Rules for Counting Significant Figures
Zeros Captive zeroes always count as significant figures: 16.07 has 4 significant figures

Rules for Counting Significant Figures
Zeros Trailing zeros are significant only if the number contains a written decimal point: 9.300 has 4 significant figures

Rules for Counting Significant Figures
Two special situations have an unlimited number of significant figures: Counted items 23 people, or 425 thumbtacks Exactly defined quantities 60 minutes = 1 hour

Sig Fig Practice #1 How many significant figures in the following?
5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs These all come from some measurements 3.29 x 103 s  3 sig figs cm  2 sig figs 3,200,000 mL  2 sig figs This is a counted value 5 dogs  unlimited

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.

Rounding Calculated Answers
Decide how many significant figures are needed (more on this very soon) Round to that many digits, counting from the left Is the next digit less than 5? Drop it. Next digit 5 or greater? Increase by 1

Sample Problem (Rounding)
Round off the numbers below to the number of significant figures in parenthesis. (Four) (Two) 8792 (Two)

Rounding Calculated Answers
Addition and Subtraction The answer should be rounded to the same number of decimal places as the least number of decimal places in the problem.

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)

Sample Problem (Addition)
Add the numbers below and record the answer with the correct number of significant figures. 12.52 m m m

Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m + 7.0 m
100.0 g g 76.27 g 76.3 g 0.02 cm cm 2.391 cm 2.39 cm 713.1 L L L 709.2 L lb lb lb lb 2.030 mL mL 0.16 mL 0.160 mL *Note the zero that has been added.

Rounding Calculated Answers
Multiplication and Division Round the answer to the same number of significant figures as the least number of significant figures in the problem.

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)

Sample Problem (Addition)
In the problems below, calculate the number of significant figures that the correct answer would contain. 7.55 m * 0.34 m 2.10 m * 0.70 m m / 9 m

Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m x 7.0 m
100.0 g ÷ 23.7 cm3 g/cm3 4.22 g/cm3 0.02 cm x cm cm2 0.05 cm2 710 m ÷ 3.0 s m/s 240 m/s lb x 3.23 ft lb·ft 5870 lb·ft 1.030 g x 2.87 mL g/mL 2.96 g/mL

Powers of 10 (multiples of ten)
Scientific Notation Powers 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.

Powers of 10 (multiples of ten)
Scientific Notation Powers 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

Powers of 10 (multiples of ten)
Scientific Notation Powers 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

Powers of 10 Practice #1 Power of 10 Answer Word Meaning 103 1,000
Thousand 102 100 Hundred 101 10 Ten 10-1 0.1 Tenth 10-2 0.01 Hundredth Thousandth 10-3 0.001

Powers of 10 Practice #2 Power of 10 Answer Word Meaning 109
1,000,000,000 Billion 106 1,000,000 Million 10-6 Millionth 10-9 Billionth 100 1 One (NOT ZERO)

Scientific Notation Mathematicians 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.

Scientific Notation Scientific 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 103 4.0 X 10-5

Scientific Notation Converting from longhand (conventional) notation to Scientific Notation Scientific 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).

Scientific Notation Converting from longhand (conventional) notation to Scientific Notation 1. 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 Figure and 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.

Scientific Notation Converting from longhand (conventional) notation to Scientific Notation After the decimal has been moved, drop any zeros that were not significant from the original number. 3.7654

Scientific Notation Converting from longhand (conventional) notation to Scientific Notation Using 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 108 If 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.

Scientific Notation Practice #1 (Longhand to Scientific Notation)
Longhand (Conventional) Scientific Notation 72300 7.23 X 104 7.23 X 10-4 7. 7. X 100

Scientific Notation Converting Scientific Notation to longhand (conventional) notation Converting 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 = 5620 4.0 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.

Scientific Notation Practice #2 (Scientific Notation to Longhand)
Longhand (Conventional) 5.1 X 106 5,100,000 5.1 X 10-6 5.0 X 100 5.0

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