Section 2.3 Uncertainty in Data Pages 47-49

Types of Observations and Measurements We make QUALITATIVE observations of reactions — changes in color and physical state. We also make QUANTITATIVE MEASUREMENTS, which involve numbers. 2

Nature of Measurement Measurement – quantitative observation consisting of two parts: Number Scale (unit) Examples: 20 grams 6.63 × 10-34 joule·seconds 3

Accuracy vs. Precision ACCURATE = CORRECT PRECISE = CONSISTENT Accuracy - how close a measurement is to the accepted value Precision - how close a series of measurements are to each other ACCURATE = CORRECT PRECISE = CONSISTENT

Accuracy vs. Precision

Precision and Accuracy in Measurements In the real world, we never know whether the measurement we make is accurate We make repeated measurements, and strive for precision We hope (not always correctly) that good precision implies good accuracy 6

Percent Error your value given value Indicates accuracy of a measurement your value given value

Percent Error A student determines the density of a substance to be 1.40 g/mL. Find the % error if the accepted value of the density is 1.36 g/mL. (correct sig figs)

Section 2.3 Significant Figures or Digits Pages 50-54

Uncertainty in Measurement A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

Why Is there Uncertainty? Measurements are performed with instruments No instrument can read to an infinite number of decimal places

Significant Figures 2.31 cm Indicate precision of a measurement. Recording Sig Figs Sig figs in a measurement include the known digits plus a final estimated digit 2.31 cm

Significant Figures What is the length of the cylinder? 13

Significant figures 14 The cylinder is 6.3 cm…plus a little more The next digit is uncertain; 6.36? 6.37? We use three significant figures to express the length of the cylinder. 14

When you are given a measurement to work with in a chemistry problem you may not know the type of instrument that was used to make the measurement so you must apply a set of rules in order to determine the number of significant digits that are in the measurement.

Rules for Counting Significant Figures Nonzero integers always count as significant figures. 3456 has 4 significant figures

Rules for Counting Significant Figures Zeros - Leading zeros do not count as significant figures. 0.0486 has 3 significant figures

Rules for Counting Significant Figures Zeros - Captive zeros 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 decimal point. 9.300 has 4 significant figures 9,300 has 2 significant figures

Rules for Counting Significant Figures Exact Numbers do not limit the # of sig figs in the answer. They have an infinite number of sig figs. Counting numbers: 12 students Exact conversions: 1 m = 100 cm “1” in any conversion: 1 in = 2.54 cm

Sig Fig Practice #1 1.0070 m  5 sig figs 17.10 kg  4 sig figs How many significant figures in each of the following? 1.0070 m  5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs 3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

Significant Numbers in Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) multiplying or dividing 2) adding or subtracting

Rules for Significant Figures in Mathematical Operations Multiplication and Division Use the same number of significant figures in the result as the data with the fewest significant figures. 1.827 m x 0.762 m = 1.392174 m2 (calculator) = 1.39 m2 (three sig. fig.) 453.6 g / 21 people = 21.6 g/person (calculator) = 21.60 g/person (four sig. fig.) (Question: why didn’t we round to 22 g/person?)

Rounding Numbers in Chemistry If the digit to the right of the last sig fig is less than 5, do not change the last sig fig. 2.532  2.53 If the digit to the right of the last sig fig is greater than 5, round up the last sig fig. 2.536  2.54 If the digit to the right of the last sig fig is a 5 followed by a nonzero digit, round up the last sig fig. 2.5351  2.54 If the digit to the right of the last sig fig is a 5 followed by zero or no other number, look at the last sig fig. If it is odd round it up; if it is even do not round up. 2.5350  2.54 2.5250  2.52

Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m x 7.0 m 100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3 0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft 1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL

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. Use the same number of decimal places in the result as the data with the fewest decimal places. 49.146 m + 72.13 m – 9.1434 m = ? = 112.1326 m (calculator) = 112.13 m (2 decimal places)

Adding and Subtracting with Trailing Zeros The answer has the same number of trailing zeros as the measurement with the greatest number of trailing zeros. 110 one trailing zero 2500 two trailing zeros + 230.3 2840.3 answer 2800 two trailing zeros

Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m + 7.0 m 100.0 g - 23.73 g 76.27 g 76.3 g 0.02 cm + 2.371 cm 2.391 cm 2.39 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1818.2 g + 3.37 g 1821.57 g 1821.6 g 2.030 mL - 1.870 mL 0.16 mL 0.160 mL