Measurements and Their Uncertainty 3.1

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Measurements and Their Uncertainty 3.1

3.1

© Copyright Pearson Prentice Hall

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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.


Measurements and Their Uncertainty

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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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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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Why Use Scientific Notation?

602000000000000000000000

6.02x1023

0.000000000000000000000000000000911

9.11x10-31



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Scientific Notation

1x100 = 1 1x10-1 = 0.1

1x101 = 10 1x10-2 = 0.01

1x102 = 100 1x10-3 = 0.001

1x103 = 1,000 1x10-4 = 0.0001

1x104 = 10,000

1x105 = 100,000



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Scientific Notation

Numbers in Scientific notation must be between 1 and 10, with only one digit to the left of the decimal.

Ex. 2.5x107 not 25x106 or .25x108



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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. 35x107 3.5x108

.27x106 2.7x105



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Scientific Notation

Express the following in scientific notation.

a. 40 c. 0.4 e. 4004 g. 0.004

b. 400 d. 404 f. 4400 h. 0.0404



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Express the following as whole numbers or decimals.

a.6.1 x 102 e. 6.01 x 10-4

b. 6.01 x 103 f. 6.01 x 104

c.6.0 x 10-2 g. 2.4 x 103

d. 6.6 x 10 h. 5.43 x 10-5



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Multiplying with Scientific Notation

Multiplying with Scientific Notation

1.Multiply the numbers

2.Add the exponents

2.5x105 • 3.2x106 = 8.0x1011



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Dividing with Scientific Notation

Dividing with Scientific Notation

1.Divide the numbers

2.Subtract the exponents

8.0x109 / 4.0x103 = 2.0x106



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Multiplying and Dividing with Sci Not

Perform the following calculations, expressing your answers in scientific notation.

a.(6.0 x 104)(2.0 x 105) e. (8.0 x 103) / (2.0 x 106)

b.(6.0 x 10-3)(3.0 x 105) f. (3.0 x 104) / (6.0 x 10-2)

b.(4.0 x 104)(2.0 x 10-6) g. (2.0 x 10-3) / (4.0 x 10-8)

d. (6.0 x 106) / (2.0 x 104)



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Adding and Subtracting with Scientific Notation

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

2.4x105 + 4.6x106 .24x106 + 4.6x106 = 4.84x106



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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 108) + ( 2.6 x 109)= __________

(2.5 x 107) - (5.0 x 108) = __________

(7.4 x 105) - ( 2.7 x 106)= __________



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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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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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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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The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.



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Accuracy, Precision, and Error3.1



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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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Significant Figures in Measurements


Why must measurements be reported to the correct number of significant figures?

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Significant Figures in Measurements3.1



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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.

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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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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.



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(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.



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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.


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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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Practice• 3. Determine the number of significant figures

in each of the following measurements:

• a) 65.42 g b) 385 L c) 0.14 mL

• d) 709.2 m e) 5006.12 kg f) 400 dm

• g) 260. mm h) 0.47 cg i) 0.0068 k

• j) 7.0 cm3 k) 36.00 g l) 0.0070 kg

• m) 100.6040 Ln) 340.0 cm



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Significant Figures in Calculations

Significant Figures in Calculations

How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

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> 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.

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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.


SAMPLE PROBLEM

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SAMPLE PROBLEM

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Practice Problems


for Sample Problem 3.1



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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.


SAMPLE PROBLEM

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SAMPLE PROBLEM

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Practice Problems for Sample Problem 3.2




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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.


SAMPLE PROBLEM

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SAMPLE PROBLEM

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Practice Problems for Sample Problem 3.3




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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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Section Quiz

-or-Continue to: Launch:

Assess students’ understanding of the concepts in Section 3.1.

Section Assessment


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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. 842.6 104 = 8.426 106

d. 0.00452 106 = 4.52 109


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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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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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Measurements and Their Uncertainty 3.1

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