Information about IB Chemistry on Uncertainty Calculation and significant figures

IB Chemistry on Uncertainty Calculation and significant figures

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)

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

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)

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 4.2 2.32 + 0.6157 7.1357 least number decimal places 1.367 - 1.34 0.027 uncertain least number decimal places uncertain 4.7832 1.234 + 2.02 8.0372 12.587 4.25 + 0.12 16.957 uncertain 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 1247 134.5 450 + 78 1909.5 68.7 - 68.42 0.28 uncertain least number decimal places least number decimal places uncertain 1.0236 - 0.97268 0.05092 7.987 - 0.54 7.447 Convert to same exponent x 104 476.8 47.68 + 23.2 x 103 x 103 + 23.2 x 103 500.0 x 103 least number decimal places uncertain uncertain least number decimal places 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 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 x 104 476.8 47.68 + 23.2 x 103 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 - multiplication/division • Answer contains no more significant figures than the least accurately known number. 12.34 3.22 x 1.8 71.52264 16.235 0.217 x 5 17.614975 923 ÷ 20312 0.045441 least sf (2sf) least sf (1sf) least sf (3sf) 23.123123 x 1.3344 30.855495 4.52 ÷ 6.3578 7.1093775 1300 x 57240 74412000 least sf (5sf) least sf (3sf) 21.45 x 0.023 0.49335 0.00435 x 4.6 0.02001 least sf (2sf) Scientific notation 2.8723 x I.6 4.59568 least sf (2sf) least sf (2sf) 6305 ÷ 0.010 630500 least sf (2sf) least sf (2sf) I.3*103 x 5.724*104 7.4412 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 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) round down 0.0454 1300 x 57240 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 63000 6.3 x 105 I.3*103 x 5.724*104 7.4412 x 107 round down 74000000 7.4 x 107 Click here for practice notes on sig figures

Scientific notation How many significant figures Written as a=1-9 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=1-9 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 prcise Click here practice scientific notation Click here practice scientific notation ✔

Significant figures and Uncertainty 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 round down 41.6 4/3 – constant π – constant Their sf is not taken (not a measurement) least sf (3sf)

Significant figures and Uncertainty 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 Their sf is not taken (not a measurement) round down 41.6 Recording measurement using uncertainty of equipment Radius, r = (2.15 ±0.02) cm 4 Volume = p r 3 3 4 Volume = ´3.14 ´ 2.153 = 41.60 3 least sf (3sf)

Significant figures and Uncertainty 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 Their sf is not taken (not a measurement) least sf (3sf) round down 41.6 Recording measurement using uncertainty of equipment Radius, r = (2.15 ±0.02) cm Treatment of Uncertainty Multiplying or dividing measured quantities 4 Volume = p r 3 3 % uncertainty = sum of % uncertainty of individual quantities Radius, r = (2.15 ±0.02) %uncertainty radius (%Δr) = 0.02 x 100 = 0.93% 2.15 % uncertainty V = 3 x % uncertainty r % ΔV = 3 x % Δr * For measurement raised to power of n, multiply % uncertainty by n * Constant, pure/counting number has no uncertainty and sf not taken 4 Volume = p r 3 3 4 Volume = ´3.14 ´ 2.153 = 41.60 3 0.02 ´100% = 0.93% 2.15 Measurement raised to power of 3, multiply % uncertainty by 3 %DV = 3´ %Dr %DV = 3´ 0.93 = 2.79% Volume = (41.60 ± 2.79%) %Dr = AbsoluteDV = 2.79 ´ 41.60 =1.16 100 Volume = (41.60 ±1.16) Volume = (42 ±1)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Radius, r = 3.0 cm Circumference, C = 2πr C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495 round up 19 2 – constant π – constant Their sf is not taken (not a measurement) least sf (2sf)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Radius, r = 3.0 cm Circumference, C = 2πr C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495 2 – constant π – constant Their sf is not taken (not a measurement) least sf (2sf) round up 19 Recording measurement using uncertainty of equipment Radius, r = (3.0 ±0.2) cm Circumference = 2p r Circumference = 2´3.14´3.0 =18.8495

Significant figures and Uncertainty in measurement Recording measurement using significant figures Radius, r = 3.0 cm Circumference, C = 2πr C = 2 x π x (3.0) = 2 x 3.14 x 3.0 = 18.8495 2 – constant π – constant Their sf is not taken (not a measurement) least sf (2sf) round up 19 Recording measurement using uncertainty of equipment Radius, r = (3.0 ±0.2) cm Treatment of Uncertainty Multiplying or dividing measured quantities Circumference = 2p r % uncertainty = sum of % uncertainty of individual quantities Radius, r = (3.0 ±0.2) %uncertainty radius (%Δr) = 0.2 x 100 = 6.67% 3.0 % uncertainty C = % uncertainty r % ΔC = % Δr * Constant, pure/counting number has no uncertainty and sf not taken Circumference = 2p r Circumference = 2´3.14´3.0 =18.8495 0.2 ´100% = 6.67% 3.0 %Dc = %Dr %Dc = 6.67% Circumference = (18.8495 ± 6.67%) %Dr = AbsoluteDC = 6.67 ´18.8495 =1.25 100 Circumference = (18.8495 ±1.25) Circumference = (19 ±1)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Time, t = 2.25 s Displacement, s = ½ gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625 g and ½ – constant Their sf is not taken (not a measurement) least sf (3sf) round down 24.8 1 Displacement, s = ´ 9.8x(2.25) 2

Significant figures and Uncertainty in measurement Recording measurement using significant figures Time, t = 2.25 s Displacement, s = ½ gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625 g and ½ – constant Their sf is not taken (not a measurement) least sf (3sf) round down 24.8 Recording measurement using uncertainty of equipment Time, t = (2.25 ±0.01) cm 1 Displacement, s = ´ 9.8x(2.25) 2 1 Displacement, s = gt 2 2 1 Displacement, s = ´ 9.8x2.25x2.25 = 24.80625 2

Significant figures and Uncertainty in measurement Recording measurement using significant figures Time, t = 2.25 s Displacement, s = ½ gt2 g and ½ – constant Their sf is not taken (not a measurement) s = 1/2 x 9.8 x (2.25)2 = 24.80625 least sf (3sf) round down 24.8 Recording measurement using uncertainty of equipment Time, t = (2.25 ±0.01) cm 1 Displacement, s = ´ 9.8x(2.25) 2 1 Displacement, s = gt 2 2 1 Displacement, s = ´ 9.8x2.25x2.25 = 24.80625 2 0.01 ´100% = 0.4% 2.25 Measurement raised to power of 2, multiply % uncertainty by 2 %Ds = 2 ´ %Dt %Ds = 2 ´ 0.4% = 0.8% Displacement = (24.80 ± 0.8%) %Dt = Treatment of Uncertainty 1 2 Multiplying or dividing measured quantities Displacement, s = gt 2 % uncertainty = sum of % uncertainty of individual quantities Time, t = (2.25 ±0.01) %uncertainty time (%Δt) = 0.01 x 100 = 0.4% 2.25 % uncertainty s = 2 x % uncertainty t % Δs = 2 x % Δt * For measurement raised to power of n, multiply % uncertainty by n AbsoluteDs = 0.4 ´ 24.80 = 0.198 100 Displacement = (24.80 ± 0.198) Displacement = (24.8 ± 0.2)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Length, I = 1.25 m L g T = 2p T = 2 x π x √(1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 round down 2.24 least sf (3sf) 2, π and g – constant Their sf is not taken (not a measurement)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Length, I = 1.25 m L g T = 2p 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 uncertainty of equipment T = 2p Length, I = (1.25 ±0.05) m T = 2p L g 1.25 = 2.24 9.8 2, π and g – constant Their sf is not taken (not a measurement)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Length, I = 1.25 m L g T = 2p least sf (3sf) T = 2 x π x √(1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 2, π and g – constant Their sf is not taken (not a measurement) round down 2.24 Recording measurement using uncertainty of equipment T = 2p Length, I = (1.25 ±0.05) m T = 2p L g 1.25 = 2.24 9.8 0.05 ´100% = 4% 1.25 Measurement raised to power of 1/2, 1 %DT = ´ %Dl multiply % uncertainty by 1/2 2 %DT = 2% T = (2.24 ± 2%) %Dl = Treatment of Uncertainty Multiplying or dividing measured quantities T = 2p L g % uncertainty = sum of % uncertainty of individual quantities Length, I = (1.25 ±0.05) %uncertainty length (%ΔI) = 0.05 x 100 = 4% 1.25 % uncertainty T = ½ x % uncertainty I % ΔT = ½ x % ΔI * For measurement raised to power of n, multiply % uncertainty by n AbsoluteDT = 2 ´ 2.24 = 0.044 100 T = (2.24 ± 0.044) T = (2.24 ± 0.04)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Area, A = I x h Length, I = 4.52 cm Height, h = 2.0 cm 4.52 2.0 9.04 x least sf (2sf) round down 9.0

Significant figures and Uncertainty in measurement Recording measurement using significant figures Area, A = I x h Length, I = 4.52 cm Height, h = 2.0 cm 4.52 2.0 9.04 x least sf (2sf) round down 9.0 Recording measurement using uncertainty of equipment Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3 Area, A = Length,l ´ height, h Area = 4.52 ´ 2.0 = 9.04

Significant figures and Uncertainty in measurement Recording measurement using significant figures Area, A = I x h Length, I = 4.52 cm Height, h = 2.0 cm 4.52 2.0 9.04 x least sf (2sf) round down 9.0 Recording measurement using uncertainty of equipment Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3 Area, A = Length,l ´ height, h Area = 4.52 ´ 2.0 = 9.04 0.02 ´100% = 0.442% 4.52 0.2 %Dh = ´100% = 10% 2.0 %DA = %Dl + %Dh %DA = 0.442% +10% = 10.442% Area = (9.04 ±10.442%) %Dl = Treatment of Uncertainty Multiplying or dividing measured quantities Area, A = Length,l ´height,h % uncertainty = sum of % uncertainty of individual quantities Length, l = (4.52 ±0.02) %uncertainty length (%Δl) = 0.02 x 100 = 0.442% 4.52 Height, h = (2.0 ±0.2) %uncertainty height (%Δh) = 0.2 x 100 = 10% 2.0 % uncertainty A = % uncertainty length + % uncertainty height % ΔA = % ΔI + %Δh AbsoluteDA = Area = (9.0 ± 0.9) 10.442 ´ 9.04 = 0.9 100

Significant figures and Uncertainty in measurement Recording measurement using significant figures Moles, n = Conc x Vol Conc, c = 2.00 M Volume, v = 2.0 dm3 2.00 2.0 4.00 x least sf (2sf) round down 4.0

Significant figures and Uncertainty in measurement Recording measurement using significant figures Moles, n = Conc x Vol Conc, c = 2.00 M Volume, v = 2.0 dm3 2.00 2.0 4.00 x least sf (2sf) round down 4.0 Recording measurement using uncertainty of equipment Conc, c = (2.00 ±0.02) cm Volume, v = (2.0 ±0.1)dm3 Mole, n = Conc, c ´Volume, v Mole = 2.00 ´ 2.0 = 4.00

Significant figures and Uncertainty in measurement Recording measurement using significant figures Moles, n = Conc x Vol Conc, c = 2.00 M Volume, v = 2.0 dm3 2.00 2.0 4.00 x least sf (2sf) round down 4.0 Recording measurement using uncertainty of equipment Conc, c = (2.00 ±0.02) cm Volume, v = (2.0 ±0.1)dm3 Mole, n = Conc, c ´Volume, v Mole = 2.00 ´ 2.0 = 4.00 0.02 ´100% = 1% 2.00 0.1 %Dv = ´100% = 5% 2.0 %Dn = %Dc + %Dv %Dc = Treatment of Uncertainty Multiplying or dividing measured quantities Mole, n = Conc, c ´Vol, v % uncertainty = sum of % uncertainty of individual quantities Conc, c = (2.00 ±0.02) %uncertainty conc (%Δc) = 0.02 x 100 = 1% 2.00 Volume, v = (2.0 ±0.1) %uncertainty volume (%Δv) = 0.1 x 100 = 5% 2.0 % uncertainty n = % uncertainty conc + % uncertainty volume % Δn = % Δc + %Δv %Dn = 1% + 5% = 6% Mole = (4.00 ± 6%) AbsoluteDn = 6 ´ 4.00 = 0.24 100 Mole = (4.00 ± 0.24) Mole = (4.0 ± 0.2)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass, m = 482.63g Volume, v = 258 cm3 Density = Mass Volume 482.63 ÷ 258 1.870658 round down 1.87 least sf (3sf)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass, m = 482.63g Volume, v = 258 cm3 Density = Mass Volume 482.63 ÷ 258 1.870658 least sf (3sf) round down 1.87 Recording measurement using uncertainty of equipment Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3 Density, D = Density, D = Mass Volume 482.63 =1.870658 258

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass, m = 482.63g Volume, v = 258 cm3 Density = Mass Volume 482.63 ÷ 258 1.870658 least sf (3sf) round down 1.87 Recording measurement using uncertainty of equipment Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3 Treatment of Uncertainty Multiplying or dividing measured quantities Density, D = Mass Volume % uncertainty = sum of % uncertainty of individual quantities Mass, m = (482.63 ±1) %uncertainty mass (%Δm) = 1 x 100 = 0.21% 482.63 Volume, V = (258 ±5) %uncertainty vol (%ΔV) = 5 x 100 = 1.93% 258 % uncertainty density = % uncertainty mass + % uncertainty volume % ΔD = % Δm + %ΔV Density, D = Density, D = Mass Volume 482.63 =1.870658 258 1 ´100% = 0.21% 482.63 5 %DV = ´100% = 1.93% 258 %DD = %Dm + %DV %DD = 0.21% +1.93% = 2.14% Density = (1.87 ± 2.14%) %Dm = AbsoluteDD = 2.14 ´1.87 = 0.04 100 Density = (1.87 ± 0.04)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass water = 2.00 g ΔTemp = 2.0 C Enthalpy, H = mcΔT x 2.00 4.18 2.0 16.72 c – constant sf is not taken (not a measurement) least sf (2sf) round up 17

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass water = 2.00 g ΔTemp = 2.0 C Enthalpy, H = mcΔT x 2.00 4.18 2.0 16.72 c – constant sf is not taken (not a measurement) least sf (2sf) round up 17 Recording measurement using uncertainty of equipment Enthalpy, H = m ´ c ´ DT Mass water = (2.00 ±0.02)g ΔTemp = (2.0 ±0.4) C Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72

Significant figures and Uncertainty in measurement Recording measurement using significant figures Mass water = 2.00 g ΔTemp = 2.0 C Enthalpy, H = mcΔT x 2.00 4.18 2.0 16.72 c – constant sf is not taken (not a measurement) least sf (2sf) round up 17 Recording measurement using uncertainty of equipment Mass water = (2.00 ±0.02)g ΔTemp = (2.0 ±0.4) C Treatment of Uncertainty Multiplying or dividing measured quantities Enthalpy, H = m ´ c ´ DT Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72 Enthalpy, H = m ´ c ´ DT % uncertainty = sum of % uncertainty of individual quantities Mass, m = (2.00 ±0.02) %uncertainty mass (%Δm) = 0.02 x 100 = 1% 2.00 ΔTemp = (2.0 ±0.4) %uncertainty temp (%ΔT) = 0.4 x 100 = 20% 2.0 % uncertainty H = % uncertainty mass + % uncertainty temp % ΔH = % Δm + %ΔT 0.02 ´100% = 1% 2.00 0.4 %DT = ´100% = 20% 2.0 %DH = %Dm + %DT %Dm = %DH = 1% + 20% = 21% Enthalpy = (16.72 ± 21%) AbsoluteDH = 21 ´16.72 = 3.51 100 Enthalpy = (16.72 ± 3.51) Enthalpy = (17 ± 4)

Treatment of uncertainty in measurement • Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + water, M2 = (22.00 ±0.01)g Initial Temp, T1 = (21.2 ±0.2)C Final Temp, T2 = (23.2 ±0.2)C Mass water, m = (M2 –M1) Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Diff Temp ΔT = (T2 –T1) Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Mass water, m = (22.00 –20.00) = 2.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 ±0.02)g Diff Temp ΔT = (23.2 –21.2) = 2.0 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (2.0 ±0.4) Mass water, m = (2.00 ±0.02)g ΔTemp = (2.0 ±0.4) C Addition/Subtraction/Multiply/Divide • Multiplying or dividing Max %uncertainty is the SUM of individual %uncertainties

Treatment of uncertainty in measurement • Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + water, M2 = (22.00 ±0.01)g Initial Temp, T1 = (21.2 ±0.2)C Final Temp, T2 = (23.2 ±0.2)C Addition/Subtraction/Multiply/Divide • Multiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertainty Enthalpy, H = (M2-M1) x c x (T2-T1) Mass water, m = (M2 –M1) Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Diff Temp ΔT = (T2 –T1) Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Multiplication Add % uncertainty Mass water, m = (22.00 –20.00) = 2.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 ±0.02)g Mass water, m = (2.00 ±0.02)g Diff Temp ΔT = (23.2 –21.2) = 2.0 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (2.0 ±0.4) ΔTemp = (2.0 ±0.4) C Enthalpy, H = m ´ c ´ DT Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72

Treatment of uncertainty in measurement • Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + water, M2 = (22.00 ±0.01)g Addition/Subtraction/Multiply/Divide • Multiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertainty Initial Temp, T1 = (21.2 ±0.2)C Final Temp, T2 = (23.2 ±0.2)C Enthalpy, H = (M2-M1) x c x (T2-T1) Mass water, m = (M2 –M1) Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Diff Temp ΔT = (T2 –T1) Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Multiplication Add % uncertainty Mass water, m = (22.00 –20.00) = 2.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 ±0.02)g Mass water, m = (2.00 ±0.02)g Treatment of Uncertainty Multiplying or dividing measured quantities Diff Temp ΔT = (23.2 –21.2) = 2.0 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (2.0 ±0.4) ΔTemp = (2.0 ±0.4) C Enthalpy, H = m ´ c ´ DT % uncertainty = sum of % uncertainty of individual quantities Mass, m = (2.00 ±0.02) %uncertainty mass (%Δm) = 0.02 x 100 = 1% 2.00 ΔTemp = (2.0 ±0.4) %uncertainty temp (%ΔT) = 0.4 x 100 = 20% 2.0 % uncertainty H = % uncertainty mass + % uncertainty temp % ΔH = % Δm + %ΔT Enthalpy, H = m ´ c ´ DT Enthalpy, H = 2.00 ´ 4.18´ 2.0 =16.72 0.02 ´100% = 1% 2.00 0.4 %DT = ´100% = 20% 2.0 %DH = %Dm + %DT %Dm = %DH = 1% + 20% = 21% Enthalpy = (16.72 ± 21%) AbsoluteDH = 21 ´16.72 = 3.51 100 Enthalpy = (16.72 ± 3.51) Enthalpy = (17 ± 4)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s t ´ I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 ÷ 1.414 28.769 Energy = round up 29 least sf (2sf)

Significant figures and Uncertainty in measurement Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s t ´ I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 ÷ 1.414 28.769 Energy = least sf (2sf) round up 29 Recording measurement using uncertainty of equipment Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Time, t = (4.52 ± 0.02) t ´ I2 Energy, E = 1/2 v 4.52(3.0)2 Energy, E = = 28.638 2.01/2

Significant figures and Uncertainty in measurement t ´ I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 ÷ 1.414 28.769 Energy = Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s least sf (2sf) round up 29 Recording measurement using uncertainty of equipment Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Time, t = (4.52 ± 0.02) Treatment of Uncertainty Multiplying or dividing measured quantities Energy, E = t ´ I2 v1/2 % uncertainty = sum of % uncertainty of individual quantities Time, t = (4.52 ±0.02) %uncertainty time (%Δt) = 0.02 x 100 = 0.442% 4.52 Current, I = (3.0 ±0.6) %uncertainty current (%ΔI) = 0.6 x 100 = 20% 3.0 Volt, v = (2.0±0.2) %uncertainty volt (%Δv) = 0.2 x 100 = 10% 2.0 % ΔE = % Δt + 2 %ΔI + ½ %ΔV * For measurement raised to power of n, multiply % uncertainty by n t ´ I2 Energy, E = 1/2 v 4.52(3.0)2 Energy, E = = 28.638 2.01/2 0.02 %Dt = ´100% = 0.442% 4.52 0.6 %DI = ´100% = 20% 3.0 0.2 %Dv = ´100% = 10% 2.0 1 %DE = %Dt + 2 ´%I + ´%Dv 2 %DE = ( 0.02 0.6 1 0.2 ´100% ) + ( 2 ´ ´100% ) + ( ´ ´100% 4.52 3.0 2 2.0 %DE = 0.442%+ 40%+ 5% = 45.442% = 45% Energy, E = (28.638± 45%) AbsoluteDE = Energy, E = (29 ±13) 45 ´ 28.638 =13 100 )

Significant figures and Uncertainty in measurement Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106) (G + H ) Z 20 + 16 = 36 ÷ 106 0.339 Speed, s = round down 0.34 least sf (2sf)

Significant figures and Uncertainty in measurement Recording measurement using significant figures (G + H ) Z 20 + 16 = 36 ÷ 106 0.339 Speed, s = G = (20 ) H = (16 ) Z = (106) least sf (2sf) round down 0.34 Recording measurement using uncertainty of equipment G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0) ✔ Addition add absolute uncertainty G+H = (36 ± 1) Z = (106 ± 1.0) Speed, s = (G + H ) Z Speed, s = (20 +16) = 0.339 106

Significant figures and Uncertainty in measurement Recording measurement using significant figures (G + H ) Z 20 + 16 = 36 ÷ 106 0.339 Speed, s = G = (20 ) H = (16 ) Z = (106) least sf (2sf) round down 0.34 Speed, s = Recording measurement using uncertainty of equipment G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0) ✔ Addition add absolute uncertainty G+H = (36 ± 1) Z = (106 ± 1.0) (G + H ) Z Speed, s = (20 +16) = 0.339 106 %D(G + H ) = Treatment of Uncertainty Multiplying or dividing measured quantities (G + H ) Speed, s = Z % uncertainty = sum of % uncertainty of individual quantities (G + H) = (36 ±1) %uncertainty (G+H) (%ΔG+H) = 1 x 100 = 2.77% 36 Z = (106 ±1.0) %uncertainty Z (%Δz) = 1.0 x 100 = 0.94% 106 %uncertainty s = %uncertainty(G+H) + %uncertainty(Z) % Δs = % Δ(G+H) + %Δz *Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties %DZ = 1.0 ´100% = 2.77% 36 1.0 ´100% = 0.94% 106 %DS = %D(G + H)+%DZ %DS = 2.77%+ 0.94% = 3.71% Speed, s = (0.339 ± 3.71%) AbsoluteDS = 3.71 ´ 0.339 = 0.012 100 Speed, s = (0.339 ± 0.012) ScientificNotation = a ´10

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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