Information about IB Chemistry on Uncertainty, Significant figures and Scientific notation

IB Chemistry on Uncertainty, Significant figures and Scientific notation

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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