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Chem_AppC_math_handbook.fm Page 59 Wednesday, June 16, 2004 10:17 AM The significant figures of a measurement are those digits known with certainty plus the rightmost digit that is estimated. Every measurement has a certain number of significant figures. For instance, if you measure the air temperature to be 21C, this measurement has two significant figures. Counting Significant Figures Every nonzero digit in a measurement is significant. For example, the measurement 831 g has three significant figures. The rules for when to count zeros in measurements as significant are as follows: • Zeros in the middle of a number are always significant. A length that measures 507 m has three significant figures. • Zeros at the beginning of a number are not significant. The measurement 0.0056 m has two significant figures. • Zeros at the end of a number are only significant if they follow a decimal point. The measurement 35.00 g has four significant figures; the measurement 2400 g has two significant figures. Richard MEgna/ Fundamental Photographs Significant Figures The mass of this bowl of candy is 298.8 grams. The measurement has four significant figures. Counted values (17 beakers) and the numbers in defined relationships (100 cm 1 m) are exact numbers and are considered to have an unlimited number of significant figures. Exact numbers never affect the number of significant figures in the results of a calculation. Each of the measurements listed below has three significant figures. The significant figures are underlined. 0.305 g 456 mL 0.000457 g 70.4 mg 444,000 ng 5.64 103 km 0.00406 dm 1.30 102 m SAMPLE PROBLEM MH-2 Counting Significant Figures in Measurements How many significant figures are in each measurement? a. 3300C b. 110.5 kJ c. 0.000 0176 g d. 210,000 kcal e. 5 notebooks f. 0.90 lb Solution Follow the rules described above for counting significant figures. a. 2 b. 4 c. 3 d. 2 e. unlimited f. 2 Math Handbook R59 Chem_AppC_math_handbook.fm Page 60 Wednesday, June 16, 2004 10:17 AM Significant Figures in Calculations When measurements are used in a calculation, the answer you calculate must be rounded to the correct number of significant figures. How you round your answer depends on the mathematical operation used in the calculation. • In multiplication and division, the answer can have no more significant figures than the least number of significant figures in any measurement in the problem. • In addition and subtraction, the answer can have no more decimal places than the least number of decimal places in any measurement in the problem. When a calculated answer must be rounded to the appropriate number of significant figures, use the following rules: • If the first nonsignificant digit is less than 5, drop all nonsignificant digits. • If the first nonsignificant digit is 5, or greater than 5, increase the last significant digit by one and drop all nonsignificant digits. The Sample Problem below illustrates how to apply these rules when performing calculations involving measurements. SAMPLE PROBLEM MH-3 Rounding Calculated Answers Solve each problem, and round your answer to the correct number of significant figures. a. (5.3 m) (1.54 m) b. 23. 5 m 2.1 m 7.26 m c. 189.427 g 19.00 g 2 d. 0.497 m 1.50 m Solution Follow the rules described above for rounding calculated answers to the appropriate number of significant figures. a. (5.3 m) (1.54 m) 8.162 m2 8.2 m2 (5.3 m has two significant figures.) b. 23.5 m 2.1 m 7.26 m 32.86 m 32.9 m (2.1 has one decimal place.) c. 189.427 g 19.00 g 170.427 g 170.43 g (19.00 has two decimal places.) d. 0.497 m2 0.3313333 m 0.331 m 1.50 m (0.497 and 1.50 each have three significant figures.) R60 Appendix C Chem_AppC_math_handbook.fm Page 61 Wednesday, June 16, 2004 10:17 AM Practice the Math 1. How many significant figures are in each measurement? a. 0.723 m b. 14.0 g c. 123,000 m d. 6.00 102 g e. 0.005 12 kg f. 1050 cm 2. Round each of the measurements in Question 1 to two significant figures. 3. Multiply or divide the following measurements, and round your answer to the correct number of significant figures. a. 3.4 m 7.8 m b. 7.00 cm 9.8 cm c. 1.56 mm 0.864 mm 14.00 mm d. 6.88 m2 2.6 m e. 52.98 g 1.8 mL f. 0.14 kg 0.0131 L 4. Add or subtract the following measurements, and round your answer to the correct number of significant figures. a. 2.34 m 18.28 m b. 828.2 g 134 g c. 0.278 cm 0.0832 cm 0.15 cm d. 54.2 mg 12.66 mg e. 6.40 ng 0.450 ng 1.001 ng f. (5.2 102 dg) (1.82 103 dg) Applying Significant Figures to Chemistry 5. Determine the number of significant figures in each measurement. a. The density of mercury (Hg): 13.55 kg/L b. The number of milligrams (mg) in one gram (g): 1000 mg 1 g c. The number of protons in an atom of copper (Cu): 29 protons d. The mass of a silver (Ag) atom: 1.792 1022 g e. The melting point of gallium: 29.8C f. The concentration of a water solution of NaCl: 24% (m/v). 6. The data from Question 5 has been used to set up a solution to each of the following problems. Calculate each answer, and express each answer to the correct number of significant figures. Make sure to cancel units where appropriate. b. How many milligrams are in 6.321 grams? a. Calculate the mass of 0.45 L of mercury. 13.55 kg Hg ? 1 L Hg c. How many protons are in 7 copper atoms? 0.45 L Hg 7 Cu atoms 29 protons ? 1 Cu atom e. What is the melting point of gallium in kelvins (K)? K C 273 29.8 273 ? 1000 mg ? 1g d. What is the mass of 35 silver atoms? 6.321 g 1.792 10-22 g Ag ? 1 Ag atom f. How many grams of NaCl are in 25.8 mL of a 24% (m/v) NaCl solution? 35 Ag atoms 25.8 mL solution 24 g NaCl ? 100 mL solution 7. Calculate the perimeter ((2 length) (2 width)) and the area (length width) of a garden plot that measures 32.8 m by 15 m. Round each answer to the correct number of significant figures. Math Handbook R61