1.Tutorial on Uncertainty, Error analysis and significant figures .Prepared by Lawrence Kok http://lawrencekok.blogspot.com

2. Significant figuresUsed in measurementsDegree of precisionShow digits believed to be correct/certain + 1 estimated/uncertainAll reads 8080 80.0 80.00 80.000least preciseCertain 23.00Uncertain 523.005g more preciseNumber sf necessary to express a measurement Consistent with precision of measurement Precise equipment = Measurement more sf Last digit always an estimate/uncertainmeasurement 15.831g(15.831 0.001)g (5 sig figures) 3. Significant figuresUsed in measurementsDegree of precisionShow digits believed to be correct/certain + 1 estimated/uncertainAll reads 8080 80.0 80.00 80.000least preciseCertain 23.00Uncertain 5Zeros bet (significant) 4.109 = 4sf 902 = 3sf 5002.05 = 6sfmeasurement 15.831g23.005g more precise(15.831 0.001)g (5 sig figures)Rules for significant figuresAll non zero digit (significant) 31.24 = 4 sf 563 = 3 sf 23 = 2sfNumber sf necessary to express a measurement Consistent with precision of measurement Precise equipment = Measurement more sf Last digit always an estimate/uncertainZeros after decimal point (significant) 4.580 = 4 sf 9.30 = 3sf 86.90000 = 7sf 3.040 = 4sf 67.030 = 5sfZero right of decimal point and following a non zero digit (significant) 0.00500 = 3sf 0.02450 = 4sf 0.04050 = 4sf 0.50 = 2sfDeals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, its accurate ( instrument may not be accurate)Zeros to left of digit (NOT significant) 0.0023 = 2sf 0.000342 = 3sf 0.00003 = 1sfZero without decimal (ambiguous) 80 = may have 1 or 2 sf 500 = may have 1 or 3 sf 4. Significant figuresUsed in measurementsDegree of precisionShow digits believed to be correct/certain + 1 estimated/uncertainAll reads 8080 80.0 80.00 80.000least preciseCertain 23.00Uncertain 5Zeros bet (significant) 4.109 = 4sf 902 = 3sf 5002.05 = 6sfmeasurement 15.831g23.005g more precise(15.831 0.001)g (5 sig figures)Rules for significant figuresAll non zero digit (significant) 31.24 = 4 sf 563 = 3 sf 23 = 2sfNumber sf necessary to express a measurement Consistent with precision of measurement Precise equipment = Measurement more sf Last digit always an estimate/uncertainZeros after decimal point (significant) 4.580 = 4 sf 9.30 = 3sf 86.90000 = 7sf 3.040 = 4sf 67.030 = 5sfZero right of decimal point and following a non zero digit (significant) 0.00500 = 3sf 0.02450 = 4sf 0.04050 = 4sf 0.50 = 2sfDeals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, its accurate ( instrument may not be accurate)Zeros to left of digit (NOT significant) 0.0023 = 2sf 0.000342 = 3sf 0.00003 = 1sfZero without decimal (ambiguous) 80 = may have 1 or 2 sf 500 = may have 1 or 3 sfClick here and here for notes on sig figures 5. Significant figures 122Smallest division = 0.122Max = 21.63 2Certain 21.6Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 21.64Uncertain = 21.62 0.015(21.62 0.01)Measurement = Certain digits + 1 uncertain digitMin = 21.61 Answer = 21.62 (4 sf) 21.6 (certain)2 (uncertain) 6. Significant figures 122Smallest division = 0.122Max = 21.63 2Certain 21.6Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 21.64Uncertain = 21.62 0.015(21.62 0.01)Measurement = Certain digits + 1 uncertain digitMin = 21.61 Answer = 21.62 (4 sf) 21.6 (certain)1Smallest division = 12Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = 0.13Certain = 364Uncertain = 36.5 0.15Measurement = Certain digits + 1 uncertain digit2 (uncertain)Certain 36Max = 36.6(36.5 0.1) Min = 36.4 Answer = 36.5 (3 sf) 36. 5 (certain) (uncertain) 7. Significant figures 1Smallest division = 10Max = 47 2Certain 40Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = 13Certain = 404Uncertain = 46 15(46 1)Measurement = Certain digits + 1 uncertain digitMin = 45 Answer = 46 (2 sf) 4 (certain)6 (uncertain) 8. Significant figures 1Smallest division = 10Max = 47 2Certain 40Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = 13Certain = 404Uncertain = 46 15(46 1)Measurement = Certain digits + 1 uncertain digitMin = 45 Answer = 46 (2 sf) 4 (certain)1Certain 3.4Smallest division = 0.12Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 3.44Uncertain = 3.410.015Measurement = Certain digits + 1 uncertain digit6 (uncertain)Max = 3.42(3.41 0.01) Min = 3.40 Answer = 3.41 (3sf) 3.4 (certain)1 (uncertain) 9. Significant figures 1Smallest division = 0.05Max = 0.480.1 20.2 0.3 0.4 0.5Certain 0.45Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = 0.005 (0.01)3Certain = 0.454Uncertain = 0.47 0.015(0.47 0.01)Measurement = Certain digits + 1 uncertain digitMin = 0.46 Answer = 0.47 (2 sf) 0.4 (certain)7 (uncertain) 10. Significant figures 1Smallest division = 0.05Max = 0.480.1 20.2 0.3Certain 0.45Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = 0.005 (0.01)3Certain = 0.454Uncertain = 0.47 0.0150.4(0.47 0.01)Measurement = Certain digits + 1 uncertain digitMin = 0.460.5Answer = 0.47 (2 sf) 0.4 (certain)7 (uncertain)Measurement 1Smallest division = 0.12Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 5.74Uncertain = 5.72 0.01(5.72 0.01) Answer = 5.72 (3sf) 5.7 (certain)2 (uncertain)1Smallest division = 12Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = 0.13Certain = 34Uncertain = 3.0 0.1(3.0 0.1) Answer =3.0 (2 sf) 3 0 (certain) (uncertain) 11. 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/subtracted23.112233 1.3324 + 0.25 24.694633uncertain least number decimal places4.2 2.32 + 0.6157 7.1357least number decimal places1.367 - 1.34 0.027uncertain least number decimal placesuncertain4.7832 1.234 + 2.02 8.037212.587 4.25 + 0.12 16.957uncertain least number decimal placesuncertain least number decimal places2.300 x 103 + 4.59 x 103 6.890 x 103least number decimal places1247 134.5 450 + 78 1909.568.7 - 68.42 0.28uncertainleast number decimal placesleast number decimal places uncertain1.0236 - 0.97268 0.050927.987 - 0.54 7.447Convert to same exponent x 104 476.847.68 + 23.2 x 103x 103 + 23.2 x 103 500.0 x 103least number decimal placesuncertainuncertain least number decimal placesleast number decimal places 12. 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/subtracted23.112233 1.3324 + 0.25 24.694633uncertain least number decimal placesround down4.7832 1.234 + 2.02 8.0372uncertain least number decimal placesround down1247 134.5 450 + 78 1909.5uncertainleast number decimal places1.0236 - 0.97268 0.050924.2 2.32 + 0.6157 7.13578.04 least number decimal places uncertainround downround up0.03uncertain least number decimal places68.7 - 68.42 0.280.0509 least number decimal places uncertain7.987 - 0.54 7.447uncertain least number decimal placesround upround downround up0.316.967.1 1.367 - 1.34 0.027191012.587 4.25 + 0.12 16.957uncertainround downround up24.69least number decimal placesuncertain least number decimal places2.300 x 103 + 4.59 x 103 6.890 x 103least number decimal places7.45 Convert to same exponent x 104 476.847.68 + 23.2 x 103x 103 + 23.2 x 103 500.0 x 103round up6.89 x 103500.0 x 103 5.000 x 105least number decimal places 13. 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 71.5226416.235 0.217 x 5 17.614975923 20312 0.045441least sf (2sf)least sf (1sf)least sf (3sf)23.123123 x 1.3344 30.8554954.52 6.3578 7.10937751300 x 57240 74412000least sf (5sf)least sf (3sf)21.45 x 0.023 0.493350.00435 x 4.6 0.02001least sf (2sf) Scientific notation2.8723 x I.6 4.59568least sf (2sf)least sf (2sf)6305 0.010 630500least sf (2sf)least sf (2sf)I.3*103 x 5.724*104 7.4412 x 107Click here for practice notes on sig figures 14. 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 71.52264least sf (2sf)round up23.123123 x 1.3344 30.855495least sf (5sf)21.45 x 0.023 0.49335round downround down30.85572 16.235 0.217 x 5 17.614975least sf (1sf)round up4.52 6.3578 7.1093775least sf (3sf)923 20312 0.045441least sf (3sf)round down0.04541300 x 57240 744120004.60.00435 x 4.6 0.02001least sf (2sf)round down7.110.020 least sf (2sf) Scientific notationleast sf (2sf)round up0.49round up202.8723 x I.6 4.59568least sf (2sf)6305 0.010 630500least sf (2sf)round down630006.3 x 105I.3*103 x 5.724*104 7.4412 x 107round down740000007.4 x 107Click here for practice notes on sig figures 15. Scientific notation How many significant figuresWritten as a=1-9Number too big/smallb = integer3 sfScientific - notation = a 10b6,720,000,000Size sand= 6.72 109 4 sf0.0000000001254=1.254 10-10 3 sfSpeed of light300000000How many significant figures4.66 x 10 6 4.660 x 10 64 sf4.6600 x 10 646600003 sf5 sfClick here practice scientific notationClick here practice scientific notation= 3.00 108 16. Scientific notation How many significant figuresWritten as a=1-9Number too big/smallb = integer3 sfScientific - notation = a 10b6,720,000,000Size sand= 6.72 109 4 sf0.0000000001254=1.254 10-10 3 sfSpeed of light= 3.00 108300000000Scientific notation80 3 ways to write 80How many significant figures4.66 x 466000010 63 sf4.660 x 10 65 sf80 8 x 101 (1sf) Digit 8 uncertain It can be 70 to 9080. 80. 8.0 x 101 (2sf) Digit 8 is certain It can be 79 to 8180.0 80.0 8.00 x 101 (3sf) Digit 80 is certain It can be 79.9 or 80.14 sf4.6600 x 10 68090 or 9 x 101 80 or 8 x 101 70 or 7 x 10181 or 8.1 x 101 80 or 8.0 x 101 79 or 7.9 x 10180.1 or 8.01 x 101 80.0 or 8.00 x 101 79.9 or 7.99 x 101 More prciseClick here practice scientific notationClick here practice scientific notation 17. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 2.15 cmVolume, V = 4/3r3 V = 4/3 x x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.60 round down41.64/3 constant constant Their sf is not taken (not a measurement) least sf (3sf) 18. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 2.15 cmVolume, V = 4/3r3 V = 4/3 x x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.604/3 constant constant Their sf is not taken (not a measurement)round down41.6 Recording measurement using uncertainty of equipmentRadius, r = (2.15 0.02) cm4 Volume = p r 3 3 4 Volume = 3.14 2.153 = 41.60 3least sf (3sf) 19. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 2.15 cmVolume, V = 4/3r3 V = 4/3 x x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.604/3 constant constant Their sf is not taken (not a measurement) least sf (3sf)round down41.6 Recording measurement using uncertainty of equipmentRadius, r = (2.15 0.02) cmTreatment of Uncertainty Multiplying or dividing measured quantities4 Volume = p r 3 3% uncertainty = sum of % uncertainty of individual quantities Radius, r = (2.15 0.02) %uncertainty radius (%r) = 0.02 x 100 = 0.93% 2.15 % uncertainty V = 3 x % uncertainty r % V = 3 x % r * For measurement raised to power of n, multiply % uncertainty by n * Constant, pure/counting number has no uncertainty and sf not taken4 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 100Volume = (41.60 1.16) Volume = (42 1) 20. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 3.0 cmCircumference, C = 2r C = 2 x x (3.0) = 2 x 3.14 x 3.0 = 18.8495 round up192 constant constant Their sf is not taken (not a measurement) least sf (2sf) 21. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 3.0 cmCircumference, C = 2r C = 2 x x (3.0) = 2 x 3.14 x 3.0 = 18.84952 constant constant Their sf is not taken (not a measurement) least sf (2sf)round up19 Recording measurement using uncertainty of equipmentRadius, r = (3.0 0.2) cmCircumference = 2p r Circumference = 23.143.0 =18.8495 22. Significant figures and Uncertainty in measurement Recording measurement using significant figuresRadius, r = 3.0 cmCircumference, C = 2r C = 2 x x (3.0) = 2 x 3.14 x 3.0 = 18.84952 constant constant Their sf is not taken (not a measurement) least sf (2sf)round up19 Recording measurement using uncertainty of equipmentRadius, r = (3.0 0.2) cmTreatment of Uncertainty Multiplying or dividing measured quantitiesCircumference = 2p r% uncertainty = sum of % uncertainty of individual quantities Radius, r = (3.0 0.2) %uncertainty radius (%r) = 0.2 x 100 = 6.67% 3.0 % uncertainty C = % uncertainty r % C = % r * Constant, pure/counting number has no uncertainty and sf not takenCircumference = 2p r Circumference = 23.143.0 =18.84950.2 100% = 6.67% 3.0 %Dc = %Dr %Dc = 6.67% Circumference = (18.8495 6.67%) %Dr =AbsoluteDC =6.67 18.8495 =1.25 100Circumference = (18.8495 1.25) Circumference = (19 1) 23. Significant figures and Uncertainty in measurement Recording measurement using significant figuresTime, t = 2.25 sDisplacement, s = gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625g and constant Their sf is not taken (not a measurement)least sf (3sf)round down24.81 Displacement, s = 9.8x(2.25) 2 24. Significant figures and Uncertainty in measurement Recording measurement using significant figuresTime, t = 2.25 sDisplacement, s = gt2 s = 1/2 x 9.8 x (2.25)2 = 24.80625g and constant Their sf is not taken (not a measurement)least sf (3sf)round down24.8 Recording measurement using uncertainty of equipmentTime, t = (2.25 0.01) cm1 Displacement, s = 9.8x(2.25) 21 Displacement, s = gt 2 2 1 Displacement, s = 9.8x2.25x2.25 = 24.80625 2 25. Significant figures and Uncertainty in measurement Recording measurement using significant figuresTime, t = 2.25 sDisplacement, s = gt2g and constant Their sf is not taken (not a measurement)s = 1/2 x 9.8 x (2.25)2 = 24.80625least sf (3sf)round down24.8 Recording measurement using uncertainty of equipmentTime, t = (2.25 0.01) cm1 Displacement, s = 9.8x(2.25) 21 Displacement, s = gt 2 2 1 Displacement, s = 9.8x2.25x2.25 = 24.80625 20.01 100% = 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 = gt2% uncertainty = sum of % uncertainty of individual quantities Time, t = (2.25 0.01) %uncertainty time (%t) = 0.01 x 100 = 0.4% 2.25 % uncertainty s = 2 x % uncertainty t % s = 2 x % t * For measurement raised to power of n, multiply % uncertainty by nAbsoluteDs =0.4 24.80 = 0.198 100Displacement = (24.80 0.198) Displacement = (24.8 0.2) 26. Significant figures and Uncertainty in measurement Recording measurement using significant figuresLength, I = 1.25 mL gT = 2pT = 2 x x (1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 round down2.24least sf (3sf) 2, and g constant Their sf is not taken (not a measurement) 27. Significant figures and Uncertainty in measurement Recording measurement using significant figuresLength, I = 1.25 mL gT = 2pleast sf (3sf)T = 2 x x (1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 round down2.24 Recording measurement using uncertainty of equipmentT = 2pLength, I = (1.25 0.05) mT = 2pL g 1.25 = 2.24 9.82, and g constant Their sf is not taken (not a measurement) 28. Significant figures and Uncertainty in measurement Recording measurement using significant figuresLength, I = 1.25 mL gT = 2pleast sf (3sf)T = 2 x x (1.25/9.8) = 2 x 3.14 x 0.35714 = 2.243992, and g constant Their sf is not taken (not a measurement)round down2.24 Recording measurement using uncertainty of equipmentT = 2pLength, I = (1.25 0.05) mT = 2pL g 1.25 = 2.24 9.80.05 100% = 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 quantitiesT = 2pL g% uncertainty = sum of % uncertainty of individual quantities Length, I = (1.25 0.05) %uncertainty length (%I) = 0.05 x 100 = 4% 1.25 % uncertainty T = x % uncertainty I % T = x % I * For measurement raised to power of n, multiply % uncertainty by nAbsoluteDT =2 2.24 = 0.044 100T = (2.24 0.044) T = (2.24 0.04) 29. Significant figures and Uncertainty in measurement Recording measurement using significant figuresArea, A = I x hLength, I = 4.52 cm Height, h = 2.0 cm4.52 2.0 9.04xleast sf (2sf)round down9.0 30. Significant figures and Uncertainty in measurement Recording measurement using significant figuresArea, A = I x hLength, I = 4.52 cm Height, h = 2.0 cm4.52 2.0 9.04xleast sf (2sf)round down9.0 Recording measurement using uncertainty of equipmentLength, I = (4.52 0.02) cm Height, h = (2.0 0.2)cm3Area, A = Length,l height, hArea = 4.52 2.0 = 9.04 31. Significant figures and Uncertainty in measurement Recording measurement using significant figuresArea, A = I x hLength, I = 4.52 cm Height, h = 2.0 cm4.52 2.0 9.04xleast sf (2sf)round down9.0 Recording measurement using uncertainty of equipmentLength, I = (4.52 0.02) cm Height, h = (2.0 0.2)cm3Area, A = Length,l height, hArea = 4.52 2.0 = 9.040.02 100% = 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 quantitiesArea, A = Length,l height,h% uncertainty = sum of % uncertainty of individual quantities Length, l = (4.52 0.02) %uncertainty length (%l) = 0.02 x 100 = 0.442% 4.52 Height, h = (2.0 0.2) %uncertainty height (%h) = 0.2 x 100 = 10% 2.0 % uncertainty A = % uncertainty length + % uncertainty height % A = % I + %hAbsoluteDA =Area = (9.0 0.9)10.442 9.04 = 0.9 100 32. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMoles, n = Conc x VolConc, c = 2.00 M Volume, v = 2.0 dm32.00 2.0 4.00xleast sf (2sf)round down4.0 33. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMoles, n = Conc x VolConc, c = 2.00 M Volume, v = 2.0 dm32.00 2.0 4.00xleast sf (2sf)round down4.0 Recording measurement using uncertainty of equipmentConc, c = (2.00 0.02) cm Volume, v = (2.0 0.1)dm3Mole, n = Conc, c Volume, vMole = 2.00 2.0 = 4.00 34. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMoles, n = Conc x VolConc, c = 2.00 M Volume, v = 2.0 dm32.00 2.0 4.00xleast sf (2sf)round down4.0 Recording measurement using uncertainty of equipmentConc, c = (2.00 0.02) cm Volume, v = (2.0 0.1)dm3Mole, n = Conc, c Volume, vMole = 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 quantitiesMole, n = Conc, c Vol, v% uncertainty = sum of % uncertainty of individual quantities Conc, c = (2.00 0.02) %uncertainty conc (%c) = 0.02 x 100 = 1% 2.00 Volume, v = (2.0 0.1) %uncertainty volume (%v) = 0.1 x 100 = 5% 2.0 % uncertainty n = % uncertainty conc + % uncertainty volume % n = % c + %v%Dn = 1% + 5% = 6% Mole = (4.00 6%) AbsoluteDn =6 4.00 = 0.24 100Mole = (4.00 0.24) Mole = (4.0 0.2) 35. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass, m = 482.63g Volume, v = 258 cm3Density = Mass Volume 482.63 258 1.870658 round down1.87least sf (3sf) 36. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass, m = 482.63g Volume, v = 258 cm3Density = Mass Volume 482.63 258 1.870658least sf (3sf)round down1.87 Recording measurement using uncertainty of equipmentMass, m = (482.63 1)g Volume, v = (258 5)cm3Density, D = Density, D =Mass Volume482.63 =1.870658 258 37. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass, m = 482.63g Volume, v = 258 cm3Density = Mass Volume 482.63 258 1.870658least sf (3sf)round down1.87 Recording measurement using uncertainty of equipmentMass, m = (482.63 1)g Volume, v = (258 5)cm3Treatment of Uncertainty Multiplying or dividing measured quantitiesDensity, D =Mass Volume% uncertainty = sum of % uncertainty of individual quantities Mass, m = (482.63 1) %uncertainty mass (%m) = 1 x 100 = 0.21% 482.63 Volume, V = (258 5) %uncertainty vol (%V) = 5 x 100 = 1.93% 258 % uncertainty density = % uncertainty mass + % uncertainty volume % D = % m + %VDensity, D = Density, D =Mass Volume482.63 =1.870658 2581 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 100Density = (1.87 0.04) 38. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass water = 2.00 g Temp = 2.0 CEnthalpy, H = mcTx2.00 4.18 2.0 16.72c constant sf is not taken (not a measurement)least sf (2sf)round up17 39. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass water = 2.00 g Temp = 2.0 CEnthalpy, H = mcTx2.00 4.18 2.0 16.72c constant sf is not taken (not a measurement)least sf (2sf)round up17Recording measurement using uncertainty of equipmentEnthalpy, H = m c DTMass water = (2.00 0.02)g Temp = (2.0 0.4) CEnthalpy, H = 2.00 4.18 2.0 =16.72 40. Significant figures and Uncertainty in measurement Recording measurement using significant figuresMass water = 2.00 g Temp = 2.0 CEnthalpy, H = mcTx2.00 4.18 2.0 16.72c constant sf is not taken (not a measurement)least sf (2sf)round up17Recording measurement using uncertainty of equipmentMass water = (2.00 0.02)g Temp = (2.0 0.4) CTreatment of Uncertainty Multiplying or dividing measured quantitiesEnthalpy, H = m c DT Enthalpy, H = 2.00 4.18 2.0 =16.72Enthalpy, H = m c DT% uncertainty = sum of % uncertainty of individual quantities Mass, m = (2.00 0.02) %uncertainty mass (%m) = 0.02 x 100 = 1% 2.00 Temp = (2.0 0.4) %uncertainty temp (%T) = 0.4 x 100 = 20% 2.0 % uncertainty H = % uncertainty mass + % uncertainty temp % H = % m + %T0.02 100% = 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 100Enthalpy = (16.72 3.51) Enthalpy = (17 4) 41. Treatment of uncertainty in measurementAdding or subtracting Max absolute uncertainty is the SUM of individual uncertaintiesInitial mass beaker, M1 = (20.00 0.01) g Final mass beaker + water, M2 = (22.00 0.01)gInitial Temp, T1 = (21.2 0.2)C Final Temp, T2 = (23.2 0.2)CMass water, m = (M2 M1) Absolute uncertainty, m = (0.01 + 0.01) = 0.02Diff Temp T = (T2 T1) Absolute uncertainty, T = (0.2 + 0.2) = 0.4Mass water, m = (22.00 20.00) = 2.00 Absolute uncertainty, m = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 0.02)gDiff Temp T = (23.2 21.2) = 2.0 Absolute uncertainty, T = (0.2 + 0.2) = 0.4 Diff Temp, T = (2.0 0.4)Mass water, m = (2.00 0.02)gTemp = (2.0 0.4) CAddition/Subtraction/Multiply/DivideMultiplying or dividing Max %uncertainty is the SUM of individual %uncertainties 42. Treatment of uncertainty in measurementAdding or subtracting Max absolute uncertainty is the SUM of individual uncertaintiesInitial mass beaker, M1 = (20.00 0.01) g Final mass beaker + water, M2 = (22.00 0.01)gInitial Temp, T1 = (21.2 0.2)C Final Temp, T2 = (23.2 0.2)CAddition/Subtraction/Multiply/DivideMultiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertaintyEnthalpy, H = (M2-M1) x c x (T2-T1) Mass water, m = (M2 M1) Absolute uncertainty, m = (0.01 + 0.01) = 0.02Diff Temp T = (T2 T1) Absolute uncertainty, T = (0.2 + 0.2) = 0.4 Multiplication Add % uncertaintyMass water, m = (22.00 20.00) = 2.00 Absolute uncertainty, m = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 0.02)gMass water, m = (2.00 0.02)gDiff Temp T = (23.2 21.2) = 2.0 Absolute uncertainty, T = (0.2 + 0.2) = 0.4 Diff Temp, T = (2.0 0.4) Temp = (2.0 0.4) CEnthalpy, H = m c DT Enthalpy, H = 2.00 4.18 2.0 =16.72 43. Treatment of uncertainty in measurementAdding or subtracting Max absolute uncertainty is the SUM of individual uncertaintiesInitial mass beaker, M1 = (20.00 0.01) g Final mass beaker + water, M2 = (22.00 0.01)gAddition/Subtraction/Multiply/DivideMultiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertaintyInitial Temp, T1 = (21.2 0.2)C Final Temp, T2 = (23.2 0.2)CEnthalpy, H = (M2-M1) x c x (T2-T1) Mass water, m = (M2 M1) Absolute uncertainty, m = (0.01 + 0.01) = 0.02Diff Temp T = (T2 T1) Absolute uncertainty, T = (0.2 + 0.2) = 0.4 Multiplication Add % uncertaintyMass water, m = (22.00 20.00) = 2.00 Absolute uncertainty, m = (0.01 + 0.01) = 0.02 Mass water, m = (2.00 0.02)gMass water, m = (2.00 0.02)gTreatment of Uncertainty Multiplying or dividing measured quantitiesDiff Temp T = (23.2 21.2) = 2.0 Absolute uncertainty, T = (0.2 + 0.2) = 0.4 Diff Temp, T = (2.0 0.4) Temp = (2.0 0.4) CEnthalpy, H = m c DT% uncertainty = sum of % uncertainty of individual quantities Mass, m = (2.00 0.02) %uncertainty mass (%m) = 0.02 x 100 = 1% 2.00 Temp = (2.0 0.4) %uncertainty temp (%T) = 0.4 x 100 = 20% 2.0 % uncertainty H = % uncertainty mass + % uncertainty temp % H = % m + %TEnthalpy, H = m c DT Enthalpy, H = 2.00 4.18 2.0 =16.720.02 100% = 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 100Enthalpy = (16.72 3.51) Enthalpy = (17 4) 44. Significant figures and Uncertainty in measurement Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52st I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 1.414 28.769Energy =round up29least sf (2sf) 45. Significant figures and Uncertainty in measurement Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52st I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 1.414 28.769Energy =least sf (2sf)round up29 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 46. Significant figures and Uncertainty in measurementt I2 v1/2 4.52 x 3.0 x 3.0 = 40.68 1.414 28.769Energy =Recording measurement using significant figures Volt, v = 2.0 V Current, I = 3.0A Time, t = 4.52sleast sf (2sf)round up29 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 quantitiesEnergy, E =t I2 v1/2% uncertainty = sum of % uncertainty of individual quantities Time, t = (4.52 0.02) %uncertainty time (%t) = 0.02 x 100 = 0.442% 4.52 Current, I = (3.0 0.6) %uncertainty current (%I) = 0.6 x 100 = 20% 3.0 Volt, v = (2.00.2) %uncertainty volt (%v) = 0.2 x 100 = 10% 2.0 % E = % t + 2 %I + %V * For measurement raised to power of n, multiply % uncertainty by nt 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% 2.01 %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) 47. Significant figures and Uncertainty in measurement Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106)(G + H ) Z 20 + 16 = 36 106 0.339Speed, s =round down0.34least sf (2sf) 48. Significant figures and Uncertainty in measurement Recording measurement using significant figures(G + H ) Z 20 + 16 = 36 106 0.339Speed, s =G = (20 ) H = (16 ) Z = (106)least sf (2sf)round down0.34 Recording measurement using uncertainty of equipment G = (20 0.5) H = (16 0.5) Z = (106 1.0)Addition add absolute uncertaintyG+H = (36 1) Z = (106 1.0)Speed, s =(G + H ) ZSpeed, s =(20 +16) = 0.339 106 49. Significant figures and Uncertainty in measurement Recording measurement using significant figures(G + H ) Z 20 + 16 = 36 106 0.339Speed, s =G = (20 ) H = (16 ) Z = (106)least sf (2sf)round down0.34Speed, s =Recording measurement using uncertainty of equipment G = (20 0.5) H = (16 0.5) Z = (106 1.0)Addition add absolute uncertaintyG+H = (36 1) Z = (106 1.0)(G + H ) ZSpeed, 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) (%G+H) = 1 x 100 = 2.77% 36 Z = (106 1.0) %uncertainty Z (%z) = 1.0 x 100 = 0.94% 106 %uncertainty s = %uncertainty(G+H) + %uncertainty(Z) % s = % (G+H) + %z *Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties%DZ =1.0 100% = 2.77% 361.0 100% = 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 100Speed, s = (0.339 0.012) ScientificNotation = a 10 50. 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-CircleThanks 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