Recitation 1-Dimensional Analysis and Significan Figures - SBT

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Arizona State University *

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Feb 20, 2024

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CHM 113 – General Chemistry I Recitation: Dimensional Analysis and Significant Figures Names: Stephanie Becerra Date: 01/25/2024 lily Adamo Marcus Lee Yesenia Perez robles. Part 1: Scientific Notation Very large and very small measurements are often written in scientific notation rather than in decimal form. Here are some examples: Decimal Form Scientific Notation 52304.0 5.23040 x 10 4 0.004560 4.560 x 10 –3 1. Convert the following to decimal form. (a) 3.96 x 10 3 3960 (b) 7.2 x 10 –4 0.00072 2. Convert the following to scientific notation. (a) 0.0123 1.23 x 10 − 2 (b) 1010.5 1.0105 x 10 3 3. Perform the following operation. Express your answer in scientific notation. 2.2 × 10 3 9.1 × 10 − 5 = _______2.4 x 10 7 ______________ Part 2: Significant Figures What are significant figures? The number of significant figures in a measured value is the number of digits that are known with some degree of reliability. For example, consider how we would measure the height (in cm) of the larger peak shown in the figure. We see that the height is somewhere between 18 and 19 cm. We can estimate the height at 18.3 cm, but there is
some uncertainty in the last digit (the tenths place). There are three significant figures, two of which are certain (1 and 8) and one which is somewhat uncertain (3). Determining the Number of Significant Figures Every number represents a specific quantity with a certain degree of precision that depends on the manner in which it was determined. When we work with numbers, we frequently must determine how many significant figures they contain. How do we do that? First, we must remember that the non-zero digits are always significant, no matter where they occur. The only problem in counting significant figures, then, is deciding whether a zero is significant, using the following rules: The following table summarizes the significant figures in the numbers we just considered. The digits that are significant are highlighted. A zero alone in front of a decimal point is not significant; it is used simply to make sure we do not overlook the decimal point ( 0 .2806, 0 .002806). A zero to the right of the decimal point but before the first non-zero digit is simply a place marker and is not significant (0. 00 2806). A zero between non-zero numbers is significant (28 0 6, 0.0028 0 6). A zero at the end of a number and to the right of the decimal point is significant (0.002806 0 , 2806. 0 ). A zero at the end of a number and to the left of the decimal point (2806 0 ) may or may not be significant – we cannot tell by looking at the number. It may be precisely known, and thus significant, or it may simply be a place holder. If we encounter such a number, we have to make our best guess of the meaning intended. Number Count of significant figures 0. 2806 4 0.00 2806 4 2806 4 0.00 28060 5 2806.0 5 2806 00 4 or 5 or 6 To avoid creating an ambiguous number with zeroes at the end of the number, we should write the number in scientific notation so that the troublesome zero occurs to the right of the decimal point . In this case, it is simple to show whether the zero is significant (2.8060 × 10 4 ) or not (2.806 × 10 4 ). The power of ten, 10 4 , is not included in the count of significant figures, since it simply tells us the position of the decimal point. 4. In the following examples determine the number of significant figures in each number: (a) 0.0049 ___2___ (b) 20.560 ____5____ (c) 0.100 ______3____ Significant Figures in Calculations What happens to the number of significant figures in numbers that we calculate? We must also be concerned about the proper expression of numbers calculated from measured numbers. Multiplication and Division In a multiplication or division problem, the product or quotient must have the same number of significant figures as the least precise number in the problem. Consider the following example: 2 Arizona State University | College of Integrative Sciences and Arts, https://cisa.asu.edu
2.4 × 1.12 = ? The first number has two significant figures, and the second has three. Since the answer must match the number with the fewest significant figures, it should have two : 2.4 × 1.12 = 2.7 (not 2.69) Try the following calculations: 5. Express the answers to the following operations with the proper number of significant figures: (a) 7.27 × 3.0 = _________21.8_________ (b) 1.5 × 318 = _________477_________ (c) 6.30 / 0.0025 = __________2520________ Addition and Subtraction A sum or difference can only be as precise as the least precise number used in the calculation. Thus, we round off the sum or difference to the first uncertain digit. If we add 10.1 to 1.91314, for instance, we get 12.0 rather than 12.01314, since there is uncertainty in the tenths position of 10.1, and this uncertainty carries over to the answer. In this case, it is not the number of significant figures that is important but rather the place of the last significant digit. Consider another example: 0.0005032 + 1.0102 = ? The first number has four significant figures and the second number has five significant figures. However, this information does not determine the number of significant figures in the answer. If we line these numbers up, we can see which digits can be added to give an answer that has significance: 0.0005032 + 1.0102 –––––––––––––––– 1.0107 032 The digits after the 5 in the first number have no corresponding digits in the second number, so it is not possible to add them and obtain digits that are significant in the sum. The answer is 1.0107, which has five significant figures, dictated by the number with the greatest uncertainty (1.0102). Try the following calculations: 6. Express the answers to the following operations with the proper number of significant figures: (a) 2.5 + 0.32 = ______2.82_________ (b) 10.0 – 0.350 = __9.65_____________ Part 3: Unit Conversions Each unit conversion calculation requires the use of a conversion factor that relates the units you are converting between. 3 Arizona State University | College of Integrative Sciences and Arts, https://cisa.asu.edu Metric Unit Prefixes Mass measurements Giga- G 10 9 1Gg=10 9 g Mega- M 10 6 1Mg=10 6 g Kilo- k 10 3 1kg=10 3 g Centi- c 10 –2 1cg=10 –2 g Milli- m 10 –3 1mg=10 –3 g Micro- μ 10 –6 1 μ g=10 –6 g Nano- n 10 –9 1ng=10 –9 g Pico- p 10 –12 1pg=10 –12 g English – Metric Unit Relationships Length Mass 1 km = 0.62 mile 1 kg = 2.205 lb 1 in = 2.54 cm 1 lb = 453.6 g
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For example, the relationship between grams (g) and pounds (lb) is: 1 lb = 453.6 g This relationship can also be written as either of the following rations: ( 453.6 g 1 lb ) or ( 1 lb 453.6 g ) Example Calculation : How many grams does a 98 lb dog weigh? Solution using Dimensional Analysis Method : ( 98 lb ) × 453.6 g 1 lb = 44452.8 g = 44,000 g (with correct # of significant figures (Math Toolbox 1.2) = 4.4 x 10 4 g (and in scientific notation) 7. Show your work for each of the following unit conversions. Circle your answer . (a) A rubber band was stretched to a length of 30.0 inches. What is this length in units of feet? (30.0 in) x 1 ft 12 ∈ ¿¿ = 2.5 ft (b) What is the length of the stretched rubber band in (a) in units of centimeters (cm)? (30.0 in) x 2.54cm/1in = 76.2cm (c) Convert 1.25 g to mg. 1.25 g x 1000 mg/1g = 1250 mg (d) Convert 42 mm to km. (2 steps) 42mm x 1m/1000m = 0.42m 0.42m x 1km/1000m = 0.000042km 4 Arizona State University | College of Integrative Sciences and Arts, https://cisa.asu.edu

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