IB Chemistry on Uncertainty calculation and significant figures

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IB Chemistry on Uncertainty calculation and significant figures

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IB Chemistry on Uncertainty calculation and significant figures

  1. 1. 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 Certain 23.00 least precise 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
  2. 2. 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 2 (uncertain) Certain 36 3 Certain = 36 4 Measurement = Certain digits + 1 uncertain digit (36.5 ±0.1) Uncertain = 36.5 ±0.1 5 Max = 36.6 Min = 36.4 Answer = 36.5 (3 sf) 36. 5 (certain) (uncertain)
  3. 3. 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)
  4. 4. 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) Certain = 0.45 Uncertain = 0.47 ± 0.01 5 0.5 3 4 0.4 (0.47 ±0.01) Measurement = Certain digits + 1 uncertain digit Min = 0.46 Answer = 0.47 (2 sf) 0.4 (certain) 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) 7 (uncertain)
  5. 5. 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
  6. 6. 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
  7. 7. 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  6.72109 Size sand 4 sf 0.0000000001254  1.2541010 3 sf Speed of light  3.00108 300000000 Scientific notation 80 80 How many significant figures 4.66 x 4660000 10 6 3 sf 4.660 x 10 6 5 sf 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 – 8 x 101 – (1sf) Digit 8 uncertain It can be 70 to 90 3 ways to write 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 ✔
  8. 8. 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 Volume, V = 4/3πr3 % 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 Volume, V = 4/3πr3 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 %V  3  %r %V  3  0.93  2.79% Volume  (41.60  2.79%) %r  AbsoluteV  2.79  41.60  1.16 100 Volume  (41.60  1.16) Volume  (42  1)
  9. 9. 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, C = 2πr % 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, C = 2πr Circumference  2  3.14 3.0  18.8495 0.2 100%  6.67% 3.0 %c  %r %c  6.67% Circumference  (18.8495  6.67%) %r  AbsoluteC  6.67 18.8495  1.25 100 Circumference  (18.8495  1.25) Circumference  (19  1)
  10. 10. 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 = 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 %s  2  %t %s  2  0.4%  0.8% Displacement  (24.80  0.8%) %t  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 Absolutes  0.4  24.80  0.198 100 Displacement  (24.80  0.198) Displacement  (24.8  0.2)
  11. 11. Significant figures and Uncertainty in measurement Recording measurement using significant figures Length, I = 1.25 m L g T  2 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  2 Length, I = (1.25 ±0.05) m T  2 L g 1.25  2.24 9. 8 0.05 100%  4% 1.25 1 Measurement raised to power of 1/2, %T   %l multiply % uncertainty by 1/2 2 %T  2% %l  Treatment of Uncertainty Multiplying or dividing measured quantities T  2 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 T  (2.24  2%) AbsoluteT  2  2.24  0.044 100 T  (2.24  0.044) T  (2.24  0.04)
  12. 12. 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 = I x h Area  4.52 2.0  9.04 0.02 100%  0.442% 4.52 0.2 %h  100%  10% 2.0 %A  %l  %h %l  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 %A  0.442%  10%  10.442% Area  (9.04  10.442%) AbsoluteA  Area  (9.0  0.9) 10.442  9.04  0.9 100
  13. 13. 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) M 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 %v  100%  5% 2.0 %n  %c  %v %c  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 %n  1%  5%  6% Mole  (4.00  6%) Absoluten  6  4.00  0.24 100 Mole  (4.00  0.24) Mole  (4.0  0.2)
  14. 14. 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 %V  100%  1.93% 258 %D  %m  %V %m  %D  0.21%  1.93%  2.14% Density  (1.87  2.14%) AbsoluteD  2.14 1.87  0.04 100 Density  (1.87  0.04)
  15. 15. 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  T Enthalpy, H  2.00 4.18 2.0  16.72 Enthalpy, H  m  c  T % 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 %T  100%  20% 2.0 %H  %m  %T %m  %H  1%  20%  21% Enthalpy  (16.72  21%) AbsoluteH  21 16.72  3.51 100 Enthalpy  (16.72  3.51) Enthalpy  (17  4)
  16. 16. 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  T % 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  T Enthalpy, H  2.00 4.18 2.0  16.72 0.02 100%  1% 2.00 0.4 %T  100%  20% 2.0 %H  %m  %T %m  %H  1%  20%  21% Enthalpy  (16.72  21%) AbsoluteH  21 16.72  3.51 100 Enthalpy  (16.72  3.51) Enthalpy  (17  4)
  17. 17. Significant figures and Uncertainty in measurement tI2 Energy  1/ 2 v 4.52 x 3.0 x 3.0 = 40.68 ÷ 1.414 28.769 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 tI2 Energy, E  1/ 2 v % 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 % t   100%  0.442% 4.52 0 .6 % I   100%  20% 3 .0 0.2 % v   100%  10% 2.0 1 %E  %t  2  % I   %v 2 %E   0.02 0.6 1 0.2 100%    2  100%     100% 4.52 3.0 2 2.0 %E  0.442%  40%  5%  45.442%  45% Energy, E  (28.638  45%) AbsoluteE  Energy, E  (29  13) 45  28.638  13 100 
  18. 18. 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 %(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 %Z  1.0 100%  2.77% 36 1.0 100%  0.94% 106 %S  %(G  H )  %Z %S  2.77%  0.94%  3.71% Speed, s  (0.339  3.71%) AbsoluteS  3.71  0.339  0.012 100 Speed, s  (0.339  0.012) Speed, s  (0.34  0.01)

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