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CHAPTER ONE: SCIENTIFIC NOTATION, SIGNIFICANT FIGURES, UNITS, DENSITY AND % COMPOSITION Note: These notes and examples are intended for self-study and will be examined. Some examples will be discussed in the lectures. REFERENCE Brown, LeMay, Burnsten, Murphy, Langford and Sagatys, 2nd Edition, Chapter 1.4 - 1.5 SCIENTIFIC NOTATION A number in scientific notation may be converted into any value of n by moving the decimal point (d.p.). Example 1 5.68762 × 102 56.8762 × 101 0.00568762 × 105 56876.2 × 10-2 Calculations with numbers expressed in scientific notation Multiplication and Division Multiplication: (N1 × 10n1) × (N2 × 10n2) = (N1 × N2) x 10n1+n2 Division: (N1 × 10n1) / (N2 × 10n2) = (N1 / N2) x 10n1-n2 Addition and Subtraction First convert all numbers to the same value of n (ie. the same power of 10): Addition: (N1 × 10n1) + (N2 × 10n1) + (N3 × 10n1) = (N1 + N2 + N3) × 10n1 Subtraction: (N1 × 10n1) - (N2 × 10n1) = (N1 - N2) × 10n1 Note: The value of n to which the numbers are converted does not affect the final answer: Example 2 1043.36 - 12.13 = 1.04336 × 103 - 1.213 × 101 Choose n = 1 Choose n = 3 1043.36 - 12.13 = 104.336 × 101 - 1.213 x 101 1043.36 - 12.13 = 1.04336 × 103 - 0.01213 × 103 = (104.336 - 1.213) × 101 = (1.04336 - 0.01213) × 103 = 103.123 × 101 = 1.03123 × 103 = 1031.23 = 1031.23 1 SIGNIFICANT FIGURES Determining the number of significant figures in a given measurement 1. Any figure that is not zero is significant: 845 mL Note: 2. ____ s.f. 1243.29 mg ____ s.f. changing to scientific notation: 1.24329 x 103 mg ____ s.f. changing the units: 1.24329 g ____ s.f. The number of s.f. is not affected by: Zero figures between non-zero figures are significant: 1906 mL 3. ____ s.f. ____ s.f. Zero figures to the left of the first non-zero figure are not significant: 0.008 kg Note: 40501.09 J ____ s.f. 0.003798 L ____ s.f. changing to scientific notation: 3.798 × 10-3 L ____ s.f. changing the units: 3.798 mL ____ s.f. The number of s.f. is not affected by: The zeroes are used only to show the position of the decimal point. 4. Exact numbers (from counting numbers of objects) have an number of s.f. by definition. 5. Physical constants/quantities defined to be exact numbers have their properties: 1 atm 101.325 kPa 760 mmHg 6. 0 oC 32 oF 273.15 K all _____ s.f. Zero figures to the right of the last non-zero figure are significant only if the number has a decimal point: 300.0 mm 7. ____ s.f. 0.0300 mm ____ s.f. In measurements without a decimal point, the number of s.f. is ambiguous: 1200 mg is unclear; could be 1200 ± 100 mg 2 s.f. 1200 ± 10 mg 3 s.f. 1200 ± 1 mg 4 s.f. This occurs when measurements are supplied from a source not using the s.f. system (eg. an older text); obtain a second opinion (demonstrator, lecturer) if an estimation cannot be made from the context of the reference. All questions supplied in tuts, tests & exams should (hopefully!) avoid this. Writing such ambiguous measurements may be avoided in two ways: 1. 2. using scientific notation 1200 ± 100 mg 2 s.f. 1.2 × 103 mg 1200 ± 10 mg 3 s.f. 1.20 × 103 mg 1200 ± 1 mg 4 s.f. 1.200 × 103 mg using a decimal point without any following figures: 1200 ± 1 mg 4 s.f. 1200. mg 2 Calculations using measurements with different numbers of significant figures Calculators do NOT count significant figures; this must be done by you. Multiplication/Division The number of s.f. in the final answer is equal to the least of the numbers of s.f. in each of the original measurements. Note: As the number of s.f. in a measurement remains unchanged by conversion to scientific notation or vice-versa, s.f. counting in multiplication and/or division calculations can be done directly irrespective of notation: Example 3 0.4323 × 232.36 multiply them on calculator: 100.44923 look at s.f. in each measurement 0.4323 _____ s.f. 232.36 lesser of these is ____ answer must have ____ s.f. _____ s.f. _______ Note: rounding off In scientific notation: (4.323 × 10-1) × (2.3236 × 102) on calculator: s.f. ____, ____ _____ 10.044923 × 101 10.044923 × 101 __________ __________ mixed: s.f. 0.4323 × (2.3236 × 102) on calculator: 100.44923 ____, ____ _____ 100.44923 __________ on calculator: 3(!!) Example 4 27.315/9.105 s.f. Note: ____, ____ _____ _________ 3 Figures are not always rounded off; in certain cases they must be added on. Example 5 1.279/250.0 -3 1.279/250 1.279/250. 5.116 × 10 5.116 × 10-3 ___, ___ ___ ___, ___ ___ ___, ___ ___ ____________ ____________ ____________ calculator: 5.116 × 10 (0.005116) s.f. answer -3 3 Example 6 What is the total mass of 7 identical test tubes each of mass 15.912g? calculator: 111.384 g s.f. ___, ___ ____ 111.384 g _________ Example 7 [0.0021281 × 104 × 63.2596(0.00045/100.062)] / [(2.345 × 10-6 (21.561/17.2)] calculator: 2059.5793 s.f. ____, ____, ____, ____, ____, ____, ____ ____ 2059.5793 ___________ (0.000450, 0.0004500) Note: Intermediate rounding off in complex calculations is unnecessary provided that all calculations are x and /; do complete calculation and round off only at end. Addition/Subtraction The number of decimal places in the final answer is equal to the least of the numbers of decimal places (d.p.) in each of the original measurements. Note: As conversion of a measurement to scientific notation or vice-versa changes its number of d.p., addition and/or subtraction of quantities in different notations will give incorrect results. Quantities must be converted to the same notation before d.p. can be counted: Example 8 (1.123 × 103) + 25.72 + 0.0651 normal: on calculator: 1148.7851 1148.7851 __________ 1123 + 25.72 + 0.0651 d.p. scientific: ___, ___, ___ ____ (1.123 × 103) + (2.572 × 101) + (6.51 x 10-2) choose n = 1 (112.3 + 2.572 + 0.00651) × 101 d.p. ___, ___, ___ ___ on calculator: 114.87851 × 101 114.8765 × 101 _________ _________ Example 9 d.p. scientific: d.p. 0.00177 - 0.0013 on calculator:* 4.7 × 10-4 ___, ___ ____ 0.00047 (1.77 - 1.3) × 10-3 on calculator:** 4.7 × 10-4 ___, ___ ____ 0.47 × 10-3 4 _________ _________ _________ Example 10 1130. + 4.9 1130.0 + 4.9 1130 + 4.9 calculator: 1134.9 1134.9 1134.9 d.p. ___, ___ ____ ___, ___ ____ see below answer ___________ ___________ ___________ In real terms 1130 (3 s.f.) means a number that is accurate only to the nearest 10; 1130 + 4.9 = 1134.9 which remains 1130 (3 s.f.) when rounded off to the nearest 10. However, 1130 (4 s.f.) is accurate to the nearest 1; 1130. + 4.9 = 1134.9 which becomes 1135(4 s.f.) when rounded off to the nearest 1. To summarise: Multiplication/Division: Count significant figures Addition/Subtraction: Count decimal places Addition/subtraction in scientific notation: (i) Convert all quantities to same notation &/or value of n before counting d.p. (ii) Convert answer to that same value of n before rounding off d.p. Combined calculations and when to round off S.f. & d.p. counting and rounding off must be done every time the operation changes from addition/subtraction to multiplication/division and vice versa. UNITS Determining Conversion Factors For Prefixes Imagine a ruler, with the various powers as the graduations, count the number of “jumps” that have to be made to go from the prefix given the prefix desired. The example uses length (m), but this applies to all units: Power 10 3 10 2 Unit km hm 101 100 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 dam m dm cm mm μm nm Converting km m: looking at the powers of both, km is 103 while m is 100, a difference (jump) of 3 ie. 1000, .: multiply km × 1000 = m, while from m km: divide m by 1000 = km. Converting km nm: looking at the powers of both, km is 103 while m is 10-9, involves a difference (jump) of 12 ie. 1000 000 000 000 (1012), .: multiply km × 1012 = nm, while from nm km: divide m by 1012 = km. 5 Question 1: How many significant figures are in each of the following measurements? (i) 5269 pm, (ii) 4.07 m, (iii) 42.300 cm, (iv) 0.0065 m, (v) 320 nm and (iv) 4.8 × 10-9 m Question 2: Convert all of the numbers in question 1 into micrometres (μm). Determining Conversion Factors For Temperatures and Pressures Temperature The main conversion will be converting °C to K, and vice versa. For: °C K: °C + 273.15 K K °C: K – 273.15 K = °C Pressure 1 atm = 760 mmHg = 101325 Pa Therefore the conversion factors are: 760 mmHg / 1 atm = 101325 Pa / 1 atm = 101325 Pa / 760 mmHg Select the relevant one, e.g., convert 1.045 atm into mmHg, and use dimensional analysis: x mmHg = 1.045 atm × (760 mmHg / 1 atm) = 794.2 mmHg Now convert this answer in mmHg into Pa: x Pa = 794.2 mmHg × (101325 Pa / 760 mmHg) = 1.059 × 105 Pa Finally, convert this into kPa. Question 3: Express the following temperatures: (i) 300.89 K in °C, (ii) 25.06 °C in K and (iii) 200.25 K in °C Question 4: Express: (i) 710.9 mmHg in Pa, (ii) 104328 Pa in atm and (iii) 102.5 kPa in mmHg DENSITY & PERCENT COMPOSITION Question 5: The density of diamond is 3.51 g cm-3. The international (but non-SI) unit for reporting the masses of diamonds is the “carat”, with 1 carat = 200.0 mg. What is the volume of a diamond of mass 0.300 carat? Question 6: A supersonic transport (SST) airplane consumes about 18000 L of kerosene per hour of flight. Kerosene has a density of 0.965 kg L-1. What mass of kerosene is consumed on a flight lasting 3.0 hours? Question 7: Silver iodide (AgI) is relatively insoluble in water. At 25 °C, only 214 μg (micrograms) will dissolve in 1.00 litre of water. How many litres of water are needed to dissolve 5.00 g of silver iodide? 6