# IB Chemistry on Uncertainty, Significant figures and Scientific notation

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Information about IB Chemistry on Uncertainty, Significant figures and Scientific notation
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Published on March 4, 2014

Author: wkkok1957

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IB Chemistry on Uncertainty, Significant figures and Scientific notation

Tutorial on Uncertainty, Significant figures and Scientific notation . Prepared by Lawrence Kok http://lawrencekok.blogspot.com

Significant figures Used in measurements Degree of precision Show digits believed to be correct/certain + 1 estimated/uncertain All reads 80 80 80.0 80.00 80.000 least precise Certain 23.00 Uncertain 5 23.005g more precise Number sf necessary to express a measurement • Consistent with precision of measurement • Precise equipment = Measurement more sf • Last digit always an estimate/uncertain measurement 15.831g (15.831 ± 0.001)g (5 sig figures) Deals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, it’s accurate ( instrument may not be accurate)

Significant figures Used in measurements Degree of precision Show digits believed to be correct/certain + 1 estimated/uncertain All reads 80 80 80.0 80.00 80.000 least precise Certain 23.00 Uncertain 5 Zeros bet (significant) 4.109 = 4sf 902 = 3sf 5002.05 = 6sf measurement 15.831g 23.005g more precise (15.831 ± 0.001)g (5 sig figures) Rules for significant figures All non zero digit (significant) 31.24 = 4 sf 563 = 3 sf 23 = 2sf Number sf necessary to express a measurement • Consistent with precision of measurement • Precise equipment = Measurement more sf • Last digit always an estimate/uncertain Zeros after decimal point (significant) 4.580 = 4 sf 9.30 = 3sf 86.90000 = 7sf 3.040 = 4sf 67.030 = 5sf Zero right of decimal point and following a non zero digit (significant) 0.00500 = 3sf 0.02450 = 4sf 0.04050 = 4sf 0.50 = 2sf Deals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, it’s accurate ( instrument may not be accurate) Zeros to left of digit (NOT significant) 0.0023 = 2sf 0.000342 = 3sf 0.00003 = 1sf Zero without decimal (ambiguous) 80 = may have 1 or 2 sf 500 = may have 1 or 3 sf Click here and here for notes on sig figures

Significant figures 1 22 Smallest division = 0.1 22 Max = 21.63 2 Certain 21.6 Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01 3 Certain = 21.6 4 Uncertain = 21.62 ±0.01 5 (21.62 ±0.01) Measurement = Certain digits + 1 uncertain digit Min = 21.61 Answer = 21.62 (4 sf) 21.6 (certain) 2 (uncertain)

Significant figures 1 22 Smallest division = 0.1 22 Max = 21.63 2 Certain 21.6 Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01 3 Certain = 21.6 4 Uncertain = 21.62 ±0.01 5 (21.62 ±0.01) Measurement = Certain digits + 1 uncertain digit Min = 21.61 Answer = 21.62 (4 sf) 21.6 (certain) 1 Smallest division = 1 2 Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = ±0.1 3 Certain = 36 4 Uncertain = 36.5 ±0.1 5 Measurement = Certain digits + 1 uncertain digit 2 (uncertain) Certain 36 Max = 36.6 (36.5 ±0.1) Min = 36.4 Answer = 36.5 (3 sf) 36. 5 (certain) (uncertain)

Significant figures 1 Smallest division = 10 Max = 47 2 Certain 40 Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = ±1 3 Certain = 40 4 Uncertain = 46 ±1 5 (46 ±1) Measurement = Certain digits + 1 uncertain digit Min = 45 Answer = 46 (2 sf) 4 (certain) 6 (uncertain)

Significant figures 1 Smallest division = 10 Max = 47 2 Certain 40 Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = ±1 3 Certain = 40 4 Uncertain = 46 ±1 5 (46 ±1) Measurement = Certain digits + 1 uncertain digit Min = 45 Answer = 46 (2 sf) 4 (certain) 1 Certain 3.4 Smallest division = 0.1 2 Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01 3 Certain = 3.4 4 Uncertain = 3.41±0.01 5 Measurement = Certain digits + 1 uncertain digit 6 (uncertain) Max = 3.42 (3.41 ±0.01) Min = 3.40 Answer = 3.41 (3sf) 3.4 (certain) 1 (uncertain)

Significant figures 1 Smallest division = 0.05 Max = 0.48 0.1 2 0.2 0.3 0.4 0.5 Certain 0.45 Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = ±0.005 (±0.01) 3 Certain = 0.45 4 Uncertain = 0.47 ± 0.01 5 (0.47 ±0.01) Measurement = Certain digits + 1 uncertain digit Min = 0.46 Answer = 0.47 (2 sf) 0.4 (certain) 7 (uncertain)

Significant figures 1 Smallest division = 0.05 Max = 0.48 0.1 2 0.2 0.3 Certain 0.45 Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = ±0.005 (±0.01) 3 Certain = 0.45 4 Uncertain = 0.47 ± 0.01 5 0.4 (0.47 ±0.01) Measurement = Certain digits + 1 uncertain digit Min = 0.46 0.5 Answer = 0.47 (2 sf) 0.4 (certain) 7 (uncertain) Measurement 1 Smallest division = 0.1 2 Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = ±0.01 3 Certain = 5.7 4 Uncertain = 5.72 ± 0.01 (5.72 ±0.01) Answer = 5.72 (3sf) 5.7 (certain) 2 (uncertain) 1 Smallest division = 1 2 Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = ±0.1 3 Certain = 3 4 Uncertain = 3.0 ± 0.1 (3.0 ±0.1) Answer =3.0 (2 sf) 3 0 (certain) (uncertain)

Scientific notation How many significant figures Written as a = 1 to 9 Scientific - notation = a ´10b b = integer Number too big/small 3 sf 6,720,000,000 Size sand = 6.72 ´109 4 sf 0.0000000001254 =1.254 ´10-10 3 sf Speed of light 300000000 = 3.00 ´108

Scientific notation How many significant figures Written as a = 1 to 9 Number too big/small b = integer 3 sf Scientific - notation = a ´10b 6,720,000,000 Size sand = 6.72 ´109 4 sf 0.0000000001254 =1.254 ´10-10 3 sf Speed of light 300000000 How many significant figures 4.66 x 10 6 4.660 x 10 6 4 sf 4.6600 x 10 6 4660000 3 sf 5 sf Click here practice scientific notation Click here practice scientific notation = 3.00 ´108

Scientific notation How many significant figures Written as a = 1 to 9 Number too big/small b = integer 3 sf Scientific - notation = a ´10b 6,720,000,000 Size sand = 6.72 ´109 4 sf 0.0000000001254 =1.254 ´10-10 3 sf Speed of light = 3.00 ´108 300000000 Scientific notation 80 3 ways to write 80 How many significant figures 4.66 x 4660000 10 6 3 sf 4.660 x 10 6 5 sf 80 – 8 x 101 – (1sf) Digit 8 uncertain It can be 70 to 90 80. 80. – 8.0 x 101 – (2sf) Digit 8 is certain It can be 79 to 81 80.0 80.0 – 8.00 x 101 – (3sf) Digit 80 is certain It can be 79.9 or 80.1 4 sf 4.6600 x 10 6 80 90 or 9 x 101 80 or 8 x 101 70 or 7 x 101 81 or 8.1 x 101 80 or 8.0 x 101 79 or 7.9 x 101 80.1 or 8.01 x 101 80.0 or 8.00 x 101 79.9 or 7.99 x 101 More precise Click here practice scientific notation Click here practice scientific notation ✔

Rules for sig figures addition /subtraction: • Last digit retained is set by the first doubtful digit. • Number decimal places be the same as least number of decimal places in any numbers being added/subtracted 23.112233 1.3324 + 0.25 24.694633 uncertain least number decimal places round down 4.7832 1.234 + 2.02 8.0372 uncertain least number decimal places round down 1247 134.5 450 + 78 1909.5 uncertain least number decimal places 1.0236 - 0.97268 0.05092 4.2 2.32 + 0.6157 7.1357 8.04 least number decimal places uncertain round down round up 0.03 uncertain least number decimal places 68.7 - 68.42 0.28 0.0509 least number decimal places uncertain 7.987 - 0.54 7.447 uncertain least number decimal places round up round down round up 0.3 16.96 7.1 1.367 - 1.34 0.027 1910 12.587 4.25 + 0.12 16.957 uncertain round down round up 24.69 least number decimal places uncertain least number decimal places 2.300 x 103 + 4.59 x 103 6.890 x 103 least number decimal places 7.45 Convert to same exponent 47.68 x 104 + 23.2 x 103 476.8 x 103 + 23.2 x 103 500.0 x 103 round up 6.89 x 103 500.0 x 103 5.000 x 105 least number decimal places

Rules for sig figures addition /subtraction: • Last digit retained is set by the first doubtful digit. • Number decimal places be the same as least number of decimal places in any numbers being added/subtracted 23.112233 1.3324 + 0.25 24.694633 uncertain least number decimal places round down 4.7832 1.234 + 2.02 8.0372 uncertain least number decimal places round down 1247 134.5 450 + 78 1909.5 uncertain least number decimal places 1.0236 - 0.97268 0.05092 8.04 4.2 2.32 + 0.6157 7.1357 least number decimal places uncertain round down 0.03 3 x 10-2 (1sf) 1.910 x 103 (4sf) uncertain least number decimal places 5.09 x 10-2 (3sf) least number decimal places 68.7 - 68.42 0.28 uncertain 7.987 - 0.54 7.447 uncertain least number decimal places round up round down 0.3 16.96 7.1 x 100 (2sf) round up 12.587 4.25 + 0.12 16.957 0.0509 round up 7.1 1.367 - 1.34 0.027 1910 8.04 x 100 (3sf) 2.469 x 101 (4sf) uncertain round down round up 24.69 least number decimal places 1.696 x 101 (4sf) uncertain least number decimal places 2.300 x 103 + 4.59 x 103 6.890 x 103 7.45 3 x 10-1 (1sf) least number decimal places 7.45 x 100 (3sf) Convert to same exponent 47.68 x 104 + 23.2 x 103 476.8 x 103 + 23.2 x 103 500.0 x 103 round up 6.89 x 103 6.89 x 103 (3sf) 500.0 x 103 5.000 x 105 least number decimal places

Rules for sig figures - multiplication/division • Answer contains no more significant figures than the least accurately known number. 12.34 3.22 x 1.8 71.52264 least sf (2sf) round up 23.123123 x 1.3344 30.855495 least sf (5sf) 21.45 x 0.023 0.49335 round down round down 30.855 72 16.235 0.217 x 5 17.614975 least sf (1sf) round up 4.52 ÷ 6.3578 7.1093775 least sf (3sf) 923 ÷ 20312 0.045441 least sf (3sf) 1300 x 57240 round down 74412000 4.6 0.00435 x 4.6 0.02001 least sf (2sf) round down 7.11 0.020 least sf (2sf) Scientific notation least sf (2sf) round up 0.49 round up 20 2.8723 x I.6 4.59568 least sf (2sf) 6305 ÷ 0.010 630500 least sf (2sf) round down 630000 I.3*103 x 5.724*104 7.4412 x 107 round down 0.0454 74000000 7.4 x 107 Click here for practice notes on sig figures

Rules for sig figures - multiplication/division • Answer contains no more significant figures than the least accurately known number. 12.34 3.22 x 1.8 71.52264 23.123123 x 1.3344 30.855495 least sf (2sf) round up 21.45 x 0.023 0.49335 least sf (5sf) round down round down 30.855 72 7.2 x 101 (2sf) 3.0855 x 101 (5sf) 16.235 0.217 x 5 17.614975 least sf (1sf) round up 4.52 ÷ 6.3578 7.1093775 0.00435 x 4.6 0.02001 923 ÷ 20312 0.045441 1300 x 57240 round down 74412000 least sf (2sf) 0.020 7.11 x 100 (3sf) least sf (3sf) 4.6 x 100 (2sf) round down 7.11 2 x 101 (1sf) 4.6 4.9 x 10-1 (2sf) least sf (3sf) 2.0 x 10-2 (2sf) least sf (2sf) Scientific notation least sf (2sf) round up 0.49 round up 20 2.8723 x I.6 4.59568 least sf (2sf) 6305 ÷ 0.010 630500 least sf (2sf) round down 630000 6.3 x 105 (2sf) I.3*103 x 5.724*104 7.4412 x 107 round down 0.0454 74000000 7.4 x 107 4.54 x 10-2 (3sf) Click here for practice notes on sig figures

Rules for sig figures – Multiplication/Division/Addition/Subtraction Answer contains no more significant figures than the least accurately known number. Avoid rounding off error. • • 0.0000673 x 291 ÷ 0.125 = ? least sf (3sf) 0.0000673 x 291 0.0195843 0.0000673 x 291 0.0195843 Intermediate step – leave more sf DO NOT ROUND UP/DOWN intermediate steps Leave extra sf to avoid rounding off error 0.0195843 ÷ 0.125 0.15632 ROUND UP/DOWN intermediate step 0.0196 ÷ 0.125 0.1568 round up round down 0.156 1.56 x 10-1 ✔ (3sf) Rounding off error 0.157 ✗ 10-1 (3sf) 1.57 x

Rules for sig figures – Multiplication/Division/Addition/Subtraction Answer contains no more significant figures than the least accurately known number. Avoid rounding off error. • • 0.0000673 x 291 ÷ 0.125 = ? least sf (3sf) 0.0000673 x 291 0.0195843 0.0000673 x 291 0.0195843 Intermediate step – leave more sf DO NOT ROUND UP/DOWN intermediate steps Leave extra sf to avoid rounding off error 0.0195843 ÷ 0.125 0.15632 ROUND UP/DOWN intermediate step 0.0196 ÷ 0.125 0.1568 round up round down 1.56 x 10-1 Rounding off error ✔ 0.156 (3sf) 0.157 ✗ 10-1 (3sf) 1.57 x (21.5 + 21.53 + 22.548 ) x 8.45 = ? least number decimal places 21.5 21.53 + 22.548 65.578 Intermediate step – leave more sf 65.578 x 8.45 554.1341 round down 554 least number decimal places ✔ 5.54 x 102 (3sf) DO NOT ROUND UP/DOWN Leave extra sf to avoid rounding off error least sf (3sf) ROUND UP/DOWN intermediate step least sf (3sf) 21.5 21.53 22.548 65.578 65.5 x 8.45 553.47 round down 553 ✗ 5.53 x 102 (3sf)

Significant figures in measurement Recording measurement using significant figures Volume, V = 4/3πr3 Radius, r = 2.15 cm Recording measurement using significant figures Circumference, C = 2πr Radius, r = 3.0 cm Recording measurement using significant figures Displacement, s = ½ gt2 Time, t = 2.25 s Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106) Speed, s = (G + H) Z

Significant figures in measurement Recording measurement using significant figures Radius, r = 2.15 cm Volume, V = 4/3πr3 V = 4/3 x π x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.60 Recording measurement using significant figures C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495 Recording measurement using significant figures Time, t = 2.25 s Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106) least sf (2sf) g and ½ – constant sf is not taken (not a measurement) Displacement, s = ½ gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625 least sf (3sf) Speed, s = (G + H) Z 20 + 16 = 36 ÷ 106 0.339 least sf (3sf) 2 and π – constant sf is not taken (not a measurement) Circumference, C = 2πr Radius, r = 3.0 cm 4/3 – constant π – constant sf is not taken (not a measurement) least sf (2sf)

Significant figures in measurement Recording measurement using significant figures Radius, r = 2.15 cm Volume, V = 4/3πr3 V = 4/3 x π x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.60 4/3 – constant π – constant sf is not taken (not a measurement) least sf (3sf) round down 41.6 Recording measurement using significant figures Radius, r = 3.0 cm 2 and π – constant sf is not taken (not a measurement) Circumference, C = 2πr C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495 least sf (2sf) round up 19 Recording measurement using significant figures Time, t = 2.25 s g and ½ – constant sf is not taken (not a measurement) Displacement, s = ½ gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625 least sf (3sf) round down 24.8 Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106) Speed, s = (G + H) Z 20 + 16 = 36 ÷ 106 0.339 least sf (2sf) round down 0.34

Significant figures in measurement Recording measurement using significant figures Period, T = 2π √L √g Length, I = 1.25 m Recording measurement using significant figures Area, A = I x h Length, I = 4.52 cm Height, h = 2.0 cm Recording measurement using significant figures Conc, c = (2.00) M Volume, v = (2.0 )dm3 Moles, n = Conc x Vol

Significant figures in measurement Recording measurement using significant figures Length, I = 1.25 m Recording measurement using significant figures Length, I = 4.52 cm Height, h = 2.0 cm Recording measurement using significant figures Conc, c = (2.00) M Volume, v = (2.0 )dm3 Period, T = 2π √L √g least sf (3sf) T = 2 x π x √(1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 Area, A = I x h 4.52 x 2.0 9.04 least sf (2sf) Moles, n = Conc x Vol 2.00 x 2.0 4.00 least sf (2sf) 2, π and g – constant sf is not taken (not a measurement)

Significant figures in measurement Recording measurement using significant figures Length, I = 1.25 m Period, T = 2π √L √g least sf (3sf) T = 2 x π x √(1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 round down 2.24 Recording measurement using significant figures Length, I = 4.52 cm Height, h = 2.0 cm Area, A = I x h 4.52 x 2.0 9.04 least sf (2sf) round down 9.0 Recording measurement using significant figures Conc, c = (2.00) M Volume, v = (2.0 )dm3 Moles, n = Conc x Vol 2.00 x 2.0 4.00 least sf (2sf) round down 4.0 2, π and g – constant sf is not taken (not a measurement)

Significant figures in measurement Recording measurement using significant figures Density = Mass Volume Mass, m = 482.63g Volume, v = 258 cm3 Recording measurement using significant figures Enthalpy, H = m x c x ΔT Mass water, m = 2.00 g ΔTemp, ΔT = 2.0 C Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s t ´ I2 Energy = 1/2 v

Significant figures in measurement Recording measurement using significant figures Density = Mass Volume Mass, m = 482.63g Volume, v = 258 cm3 Recording measurement using significant figures 482.63 ÷ 258 1.870658 Enthalpy, H = m x c x ΔT Mass water, m = 2.00 g ΔTemp, ΔT = 2.0 C x Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s least sf (3sf) 2.00 4.18 2.0 16.72 t ´ I2 Energy = 1/2 v 4.52 3.0 x 3.0 40.68 ÷ 1.414 28.769 c (4.18) - constant sf is not taken (not a measurement) least sf (2sf) least sf (2sf)

Significant figures in measurement Recording measurement using significant figures Density = Mass Volume Mass, m = 482.63g Volume, v = 258 cm3 482.63 ÷ 258 1.870658 least sf (3sf) round down 1.87 Recording measurement using significant figures Enthalpy, H = m x c x ΔT Mass water, m = 2.00 g ΔTemp, ΔT = 2.0 C x 2.00 4.18 2.0 16.72 c (4.18) - constant sf is not taken (not a measurement) least sf (2sf) round up Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s 17 t ´ I2 Energy = 1/2 v 4.52 3.0 x 3.0 40.68 ÷ 1.414 28.769 round up 29 least sf (2sf)

Acknowledgements Thanks to source of pictures and video used in this presentation http://crescentok.com/staff/jaskew/isr/tigerchem/econfig/electron4.htm http://pureinfotech.com/wp-content/uploads/2012/09/periodicTable_20120926101018.png http://www.wikihow.com/Find-the-Circumference-and-Area-of-a-Circle Thanks to Creative Commons for excellent contribution on licenses http://creativecommons.org/licenses/ Prepared by Lawrence Kok Check out more video tutorials from my site and hope you enjoy this tutorial http://lawrencekok.blogspot.com

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