IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error
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IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error

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IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error

IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error

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IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error IB Chemistry on Uncertainty, Error Analysis, Random and Systematic Error Presentation Transcript

  • Tutorial on Random and Systematic errors. Prepared by Lawrence Kok http://lawrencekok.blogspot.com
  • Measurement Every measurement – associated with an error No measurement is 100% precise or accurate. 3 Types of Measurement Not Precise + Not Accurate Precise + Accurate Accurate NOT accurate low systematic error NOT precise High systematic High random error Precise + Not Accurate Precise low random error Not accurate High systematic error
  • Measurement Every measurement – associated with an error No measurement is 100% precise or accurate. 3 Types of Measurement Not Precise + Not Accurate Precise + Accurate 2 Types of Errors Precise + Not Accurate Systematic Error Random Error Affects accuracy Affects precision high systematic error Accurate NOT accurate low systematic error NOT precise High systematic High random error Precise low random error Not accurate High systematic error high random error
  • Measurement Every measurement – associated with an error No measurement is 100% precise or accurate. 3 Types of Measurement Not Precise + Not Accurate 2 Types of Errors Precise + Accurate Precise + Not Accurate Systematic Error Random Error Affects accuracy Affects precision high systematic error Accurate NOT accurate low systematic error NOT precise High systematic High random error Precise low random error Not accurate High systematic error 2 Types of Errors Random Error • • • • Systematic Error Measurement random Instrument imprecise/uncertainty Fluctuation reading burette/pipette Small sample size/trials Statistical fluctuation of measurement/reading by someone/unpredictable VS Measurement too high/ low Instrument not calibrated Faulty apparatus (zero error) Incorrect measurement Imperfect instrument Procedure/method incorrect/predictable • • • • • Accurate + Precise Accuracy Measurement value close to correct value VS Precise Measurement value close to each other high random error
  • 2 Types of Errors Systematic Error Affects accuracy • • • • • • Random Error High systematic error Measurement too high/ low Instrument not calibrated Faulty apparatus (zero error) Incorrect measurement Imperfect instrument Procedure/method incorrect Predictable High random error • • • • • Affects precision Measurement random Instrument imprecise/uncertainty Fluctuation reading burette/pipette Small sample size/trials Statistical fluctuation of measurement/reading by someone Unpredictable
  • 2 Types of Errors Systematic Error Affects accuracy • • • • • • Random Error High systematic error Measurement too high/ low Instrument not calibrated Faulty apparatus (zero error) Incorrect measurement Imperfect instrument Procedure/method incorrect Predictable lower Correct value higher Direction error – always one side (higher/lower) High random error • • • • • Affects precision Measurement random Instrument imprecise/uncertainty Fluctuation reading burette/pipette Small sample size/trials Statistical fluctuation of measurement/reading by someone Unpredictable lower Correct value higher Direction error – always random
  • 2 Types of Errors Systematic Error Affects accuracy • • • • • • Random Error High random error High systematic error Measurement too high/ low Instrument not calibrated Faulty apparatus (zero error) Incorrect measurement Imperfect instrument Procedure/method incorrect Predictable lower Correct value • • • • • Direction error – always one side (higher/lower) Calibrating equipment for zero error Correct value higher Direction error – always random Can be reduced Can be identified/eliminated Improve measuring technique Measurement random Instrument imprecise/uncertainty Fluctuation reading burette/pipette Small sample size/trials Statistical fluctuation of measurement/reading by someone Unpredictable lower higher Affects precision Improve expt design Using precise instrument By repeating more trials/average ✗ Calorimetry expt Prevent heat loss using insulator Heating expt Cool down before weighing ✓
  • Random and Systematic Error Measuring circumference using a ruler 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 = 6.6% 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% = 6.6% 3.0 %Dc = %Dr %Dc = 6.6% Circumference = (18.8495 ± 6.7%) %Dr = AbsoluteDC = 6.6 ´18.8495 =1.25 100 Circumference = (18.8495 ±1.25) Circumference = (19 ±1)
  • Random and Systematic Error Measuring circumference using a ruler 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 = 6.6% 3.0 % uncertainty C = % uncertainty r % ΔC = % Δr * Constant, pure/counting number has no uncertainty and sf not taken Random and Systematic Error Correct value = 20.4 Expt value = 19 ±6.7% Circumference = 2p r Circumference = 2´3.14´3.0 =18.8495 0.2 ´100% = 6.6% 3.0 %Dc = %Dr %Dc = 6.6% Circumference = (18.8495 ± 6.7%) %Dr = AbsoluteDC = 6.6 ´18.8495 =1.25 100 Circumference = (18.8495 ±1.25) Circumference = (19 ±1) %Percentage Error = 6.7% %Error = ( expt - correct )´100% correct 19 - 20.4 %Error = ( ) ´100% = 6.7% 20.4 Circumference = (18.8495± 6.7%) % Random Error %Random Error 6.6% %Systematic Error 0.1%
  • Random and Systematic Error Measuring circumference using a ruler 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 = 6.6% 3.0 % uncertainty C = % uncertainty r % ΔC = % Δr * Constant, pure/counting number has no uncertainty and sf not taken Random and Systematic Error Correct value = 20.4 Expt value = 19 ±6.7% Circumference = 2p r Circumference = 2´3.14´3.0 =18.8495 0.2 ´100% = 6.6% 3.0 %Dc = %Dr %Dc = 6.6% Circumference = (18.8495 ± 6.7%) %Dr = AbsoluteDC = 6.6 ´18.8495 =1.25 100 Circumference = (18.8495 ±1.25) Circumference = (19 ±1) %Percentage Error = 6.7% %Error = ( expt - correct )´100% correct 19 - 20.4 %Error = ( ) ´100% = 6.7% 20.4 Circumference = (18.8495± 6.7%) % Random Error %Random Error 6.6% High random error Way reduce random error %Systematic Error 0.1% Small systematic error Step/procedure correct
  • Random and Systematic Error Measuring displacement using a stopwatch Recording measurement using uncertainty of equipment Time, t = (2.25 ±0.01) cm Treatment of Uncertainty 1 2 Multiplying or dividing measured quantities Displacement, s = 2 gt % 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 1 Displacement, s = gt 2 2 1 Displacement, s = ´ 9.8x2.25x2.25 = 24.80 2 0.01 ´100% = 0.4% 2.25 Measurement raised to power of 2, %Ds = 2 ´ %Dt multiply % uncertainty by 2 %Ds = 2 ´ 0.4% = 0.8% Displacement = (24.80 ± 0.8%) %Dt = AbsoluteDs = 0.4 ´ 24.80 = 0.198 100 Displacement = (24.80 ± 0.198) Displacement = (24.8 ± 0.2)
  • Random and Systematic Error Measuring displacement using a stopwatch Recording measurement using uncertainty of equipment Time, t = (2.25 ±0.01) cm Treatment of Uncertainty 1 2 Multiplying or dividing measured quantities Displacement, s = 2 gt % 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 1 Displacement, s = gt 2 2 1 Displacement, s = ´ 9.8x2.25x2.25 = 24.80 2 0.01 ´100% = 0.4% 2.25 Measurement raised to power of 2, %Ds = 2 ´ %Dt multiply % uncertainty by 2 %Ds = 2 ´ 0.4% = 0.8% Displacement = (24.80 ± 0.8%) %Dt = AbsoluteDs = 0.4 ´ 24.80 = 0.198 100 Random and Systematic Error Correct value = 23.2 Expt value = 24.8 ±0.8% expt - correct %Error = ( )´100% correct %Error = ( 24.8 - 23.2 ) ´100% = 0.7% 23.2 Displacement = (24.80 ± 0.8%) % Random Error Displacement = (24.80 ± 0.198) Displacement = (24.8 ± 0.2) %Percentage Error = 0.7% %Random Error 0.8%
  • Random and Systematic Error Measuring displacement using a stopwatch Recording measurement using uncertainty of equipment Time, t = (2.25 ±0.01) cm Treatment of Uncertainty 1 2 Multiplying or dividing measured quantities Displacement, s = 2 gt % 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 1 Displacement, s = gt 2 2 1 Displacement, s = ´ 9.8x2.25x2.25 = 24.80 2 0.01 ´100% = 0.4% 2.25 Measurement raised to power of 2, %Ds = 2 ´ %Dt multiply % uncertainty by 2 %Ds = 2 ´ 0.4% = 0.8% Displacement = (24.80 ± 0.8%) %Dt = AbsoluteDs = 0.4 ´ 24.80 = 0.198 100 Random and Systematic Error Correct value = 23.2 Expt value = 24.8 ±0.8% expt - correct %Error = ( )´100% correct %Error = ( 24.8 - 23.2 ) ´100% = 0.7% 23.2 Displacement = (24.80 ± 0.8%) % Random Error Displacement = (24.80 ± 0.198) Displacement = (24.8 ± 0.2) %Percentage Error = 0.7% %Random Error 0.8% % error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable but need to reduce random error
  • Random and Systematic Error Measuring period using a ruler Recording measurement using uncertainty of equipment Length, I = (1.25 ±0.05) m 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 l % ΔT = ½ x % ΔI * For measurement raised to power of n, multiply % uncertainty by n L g 1.25 T = 2p = 2.24 9.8 T = 2p 0.05 ´100% = 4% 1.25 1 power %DT = ´ %Dl Measurement raised to by 1/2 of 1/2, multiply % uncertainty 2 %DT = 2% T = (2.24 ± 2%) %Dl = AbsoluteDT = 2 ´ 2.24 = 0.044 100 T = (2.24 ± 0.044) T = (2.24 ± 0.04)
  • Random and Systematic Error Measuring period using a ruler Recording measurement using uncertainty of equipment Length, I = (1.25 ±0.05) m 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 l % ΔT = ½ x % ΔI * For measurement raised to power of n, multiply % uncertainty by n Random and Systematic Error Correct value = 2.15 Expt value = 2.24 ±2% L g 1.25 T = 2p = 2.24 9.8 T = 2p 0.05 ´100% = 4% 1.25 1 power %DT = ´ %Dl Measurement raised to by 1/2 of 1/2, multiply % uncertainty 2 %DT = 2% T = (2.24 ± 2%) %Dl = AbsoluteDT = 2 ´ 2.24 = 0.044 100 T = (2.24 ± 0.044) T = (2.24 ± 0.04) %Percentage Error = 4.2% %Error = ( expt - correct )´100% correct %Error = ( 2.24 - 2.15 ) ´100% = 4.2% 2.15 T = (2.24 ± 2%) % Random Error %Random Error = 2% %Systematic Error = 2.2%
  • Random and Systematic Error Measuring period using a ruler Recording measurement using uncertainty of equipment Length, I = (1.25 ±0.05) m 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 l % ΔT = ½ x % ΔI * For measurement raised to power of n, multiply % uncertainty by n Random and Systematic Error Correct value = 2.15 Expt value = 2.24 ±2% L g 1.25 T = 2p = 2.24 9.8 T = 2p 0.05 ´100% = 4% 1.25 1 power %DT = ´ %Dl Measurement raised to by 1/2 of 1/2, multiply % uncertainty 2 %DT = 2% T = (2.24 ± 2%) %Dl = AbsoluteDT = 2 ´ 2.24 = 0.044 100 T = (2.24 ± 0.044) T = (2.24 ± 0.04) %Percentage Error = 4.2% %Error = ( expt - correct )´100% correct %Error = ( 2.24 - 2.15 ) ´100% = 4.2% 2.15 T = (2.24 ± 2%) % Random Error %Random Error = 2% %Systematic Error = 2.2% % error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – way to reduce systematic error
  • Random and Systematic Error Measuring Area using ruler Recording measurement using uncertainty of equipment Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3 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 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%) %Dl = AbsoluteDA = Area = (9.0 ± 0.9) 10 ´ 9.04 = 0.9 100
  • Random and Systematic Error Measuring Area using ruler Recording measurement using uncertainty of equipment Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3 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 Random and Systematic Error 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%) %Dl = AbsoluteDA = Area = (9.0 ± 0.9) %Percentage Error = 9% Correct value = 22.7 Expt value = 24.8 ±0.87% %Error = ( expt - correct )´100% correct %Error = ( 24.8 - 22.7 )´100% = 9% 22.7 Area = (9.04 ±10%) % Random Error %Random Error = 10% 10 ´ 9.04 = 0.9 100
  • Random and Systematic Error Measuring Area using ruler Recording measurement using uncertainty of equipment Length, I = (4.52 ±0.02) cm Height, h = (2.0 ±0.2)cm3 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 Random and Systematic Error 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%) %Dl = AbsoluteDA = 10 ´ 9.04 = 0.9 100 Area = (9.0 ± 0.9) %Percentage Error = 9% Correct value = 22.7 Expt value = 24.8 ±0.87% %Error = ( Area, A = Length,l ´ height, h expt - correct )´100% correct 24.8 - 22.7 %Error = ( )´100% = 9% 22.7 Area = (9.04 ±10%) % Random Error %Random Error = 10% % error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable – need to reduce random error Reduce random error – HUGE (10%) – use precise instrument vernier calipers Vernier caliper
  • Random and Systematic Error Measuring moles using dropper and volumetric flask 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 %Dn = 1% + 5% = 6% Mole = (4.00 ± 6%) %Dc = Treatment of Uncertainty Mole, n = Conc ´Vol Multiplying or dividing measured quantities % uncertainty = sum of % uncertainty of individual quantity 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 AbsoluteDn = Mole = (4.00 ± 0.24) Mole = (4.0 ± 0.2) Dropper, volumetri c flask 6 ´ 4.00 = 0.24 100
  • Random and Systematic Error Measuring moles using dropper and volumetric flask 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 %Dn = 1% + 5% = 6% Mole = (4.00 ± 6%) %Dc = Treatment of Uncertainty Mole, n = Conc ´Vol Multiplying or dividing measured quantities % uncertainty = sum of % uncertainty of individual quantity 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 AbsoluteDn = Mole = (4.00 ± 0.24) Mole = (4.0 ± 0.2) Dropper, volumetri c flask 6 ´ 4.00 = 0.24 100 %Percentage Error = 10% Random and Systematic Error Correct value = 3.63 Expt value = 4.00 ±6% expt - correct %Error = ( )´100% correct %Error = ( 4 - 3.63 )´100% =10% 3.63 Mole = (4.00 ± 6%) % Random Error %Random Error = 6% %Systematic Error = 4%
  • Random and Systematic Error Measuring moles using dropper and volumetric flask 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 %Dn = 1% + 5% = 6% Mole = (4.00 ± 6%) %Dc = Treatment of Uncertainty Mole, n = Conc ´Vol Multiplying or dividing measured quantities % uncertainty = sum of % uncertainty of individual quantity 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 AbsoluteDn = Mole = (4.00 ± 0.24) Mole = (4.0 ± 0.2) Dropper, volumetri c flask 6 ´ 4.00 = 0.24 100 %Percentage Error = 10% Random and Systematic Error Correct value = 3.63 Expt value = 4.00 ±6% expt - correct %Error = ( )´100% correct %Error = ( 4 - 3.63 )´100% =10% 3.63 Mole = (4.00 ± 6%) % Random Error %Random Error = 6% %Systematic Error = 4% % error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – improve on method/steps used. Ways to reduce error Random error (6%) More precise instrument -pipette Systematic error (4%) Calibration of instrument
  • Random and Systematic Error Measuring density using mass and measuring cylinder Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3 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% = 2.1% Density = (1.87 ± 2.1%) %Dm = Treatment of Uncertainty Mass Multiplying or dividing measured quantities Density, D = 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 AbsoluteDD = 2.1 ´1.87 = 0.04 100 Density = (1.87 ± 0.04)
  • Random and Systematic Error Measuring density using mass and measuring cylinder Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3 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% = 2.1% Density = (1.87 ± 2.1%) %Dm = Treatment of Uncertainty Mass Multiplying or dividing measured quantities Density, D = 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 AbsoluteDD = 2.1 ´1.87 = 0.04 100 Density = (1.87 ± 0.04) %Percentage Error = 5% Random and Systematic Error Correct value = 1.78 Expt value = 1.87 ±2.1% expt - correct %Error = ( )´100% correct 1.87 -1.78 %Error = ( )´100% = 5% 1.78 Density = (1.87 ± 2.1%) % Random Error %Random Error = 2.1% %Systematic Error = 2.9%
  • Random and Systematic Error Measuring density using mass and measuring cylinder Mass, m = (482.63 ±1)g Volume, v = (258 ±5)cm3 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% = 2.1% Density = (1.87 ± 2.1%) %Dm = Treatment of Uncertainty Mass Multiplying or dividing measured quantities Density, D = 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 AbsoluteDD = 2.1 ´1.87 = 0.04 100 Density = (1.87 ± 0.04) %Percentage Error = 5% Random and Systematic Error Correct value = 1.78 Expt value = 1.87 ±2.1% %Random Error = 2.1% expt - correct %Error = ( )´100% correct %Systematic Error = 2.9% % error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occurs 1.87 -1.78 %Error = ( )´100% = 5% 1.78 Ways to reduce error Density = (1.87 ± 2.1%) Random error (6%) Precise instrument mass balance % Random Error Precise balance Systematic error (4%) Use different method like displacement can Displacement can
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer 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 % 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 = m ´ c ´ DT 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 %DH = 1% + 20% = 21% Enthalpy = (16.72 ± 21%) %Dm = AbsoluteDH = 21 ´16.72 = 3.51 100 Enthalpy = (16.72 ± 3.51) Enthalpy = (17 ± 4)
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer 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 % 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 Random and Systematic Error Correct value = 33.44 Expt value = 16.72 ±21% %Error = ( expt - correct )´100% correct 16.72 - 33.44 %Error = ( )´100% = 50% 33.44 Enthalpy = (16.72 ± 21%) % Random Error = m ´ c ´ DT 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 %DH = 1% + 20% = 21% Enthalpy = (16.72 ± 21%) %Dm = AbsoluteDH = 21 ´16.72 = 3.51 100 Enthalpy = (16.72 ± 3.51) Enthalpy = (17 ± 4) %Percentage Error = 50% %Random Error =21% %Systematic Error = 29%
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer 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 % 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 = 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 %DH = 1% + 20% = 21% Enthalpy = (16.72 ± 21%) %Dm = AbsoluteDH = 21 ´16.72 = 3.51 100 Enthalpy = (16.72 ± 3.51) Enthalpy = (17 ± 4) Random and Systematic Error %Percentage Error = 50% Correct value = 33.44 Expt value = 16.72 ±21% %Error = ( Enthalpy, H = m ´ c ´ DT expt - correct )´100% correct %Random Error =21% 16.72 - 33.44 %Error = ( )´100% = 50% 33.44 Enthalpy = (16.72 ± 21%) %Systematic Error = 29% % error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occurs – reduce this error Ways to reduce error Random error (21%) Precise Temp sensor % Random Error Temp sensor Systematic error (29%) Reduce heat loss using styrofoam cup
  • Random and Systematic Error Measuring speed change using stopwatch Recording measurement using uncertainty of equipment G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0) Treatment of Uncertainty Multiplying or dividing measured quantities Speed, s = ✔ Addition add absolute uncertainty (G + H ) 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 Speed, s = G+H = (36 ± 1) Z = (106 ± 1.0) (G + H ) Z (20 +16) = 0.339 106 1.0 %D(G + H ) = ´100% = 2.77% 36 1.0 %DZ = ´100% = 0.94% 106 Speed, s = %DS = %D(G + H) +%DZ %DS = 2.77%+ 0.94% = 3.7% Speed, s = (0.339 ± 3.7%) AbsoluteDS = 3.7 ´ 0.339 = 0.012 100 Speed, s = (0.339 ± 0.012)
  • Random and Systematic Error Measuring speed change using stopwatch Recording measurement using uncertainty of equipment G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0) Treatment of Uncertainty Multiplying or dividing measured quantities Speed, s = ✔ Addition add absolute uncertainty (G + H ) 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 Random and Systematic Error Speed, s = G+H = (36 ± 1) Z = (106 ± 1.0) (G + H ) Z (20 +16) = 0.339 106 1.0 %D(G + H ) = ´100% = 2.77% 36 1.0 %DZ = ´100% = 0.94% 106 Speed, s = %DS = %D(G + H) +%DZ %DS = 2.77%+ 0.94% = 3.7% Speed, s = (0.339 ± 3.7%) AbsoluteDS = Speed, s = (0.339 ± 0.012) %Percentage Error = 3% Correct value = 0.330 Expt value = 0.339 ±3.7% %Error = ( %Error = ( expt - correct )´100% correct 0.339 - 0.330 )´100% = 3% 0.330 Speed = (0.339 ± 3.7%) % Random Error 3.7 ´ 0.339 = 0.012 100 %Random Error = 3.7%
  • Random and Systematic Error Measuring speed change using stopwatch Recording measurement using uncertainty of equipment G = (20 ± 0.5) H = (16 ± 0.5) Z = (106 ± 1.0) Treatment of Uncertainty Multiplying or dividing measured quantities Speed, s = ✔ Addition add absolute uncertainty Speed, s = G+H = (36 ± 1) Z = (106 ± 1.0) (G + H ) 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 (G + H ) Z (20 +16) = 0.339 106 1.0 %D(G + H ) = ´100% = 2.77% 36 1.0 %DZ = ´100% = 0.94% 106 Speed, s = %DS = %D(G + H) +%DZ %DS = 2.77%+ 0.94% = 3.7% Speed, s = (0.339 ± 3.7%) AbsoluteDS = *Adding or subtracting- Max absolute uncertainty is the SUM of individual uncertainties Random and Systematic Error 3.7 ´ 0.339 = 0.012 100 Speed, s = (0.339 ± 0.012) %Percentage Error = 3% Correct value = 0.330 Expt value = 0.339 ±3.7% %Error = ( %Error = ( expt - correct )´100% correct %Random Error = 3.7% % error fall within the % uncertainty (%Random error) • Little/No systematic error • Result is reliable – need to reduce random error 0.339 - 0.330 )´100% = 3% 0.330 Ways to reduce error Speed = (0.339 ± 3.7%) Random error (3.7%) Precise time sensor % Random Error precise time sensor No systematic error Steps/method are reliable.
  • Random and Systematic Error Recording measurement using uncertainty of equipment Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Temp, t = (4.52 ± 0.02) Treatment of Uncertainty Multiplying or dividing measured quantities Energy, E = % uncertainty = sum of % uncertainty of individual quantity Time, t = (4.52 ±0.02) %uncertainty temp (%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 x %ΔI + ½ x %ΔV t ´ I2 v1/2 %DE = ( Energy, E = t ´ I2 v1/2 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 0.02 0.6 1 0.2 ´100% ) + ( 2 ´ ´100% ) + ( ´ ´100% ) = 45% 4.52 3.0 2 2.0 Energy, E = (28.638± 45%) AbsoluteDE = Energy, E = (29 ±13) 45 ´ 28.638 =13 100
  • Random and Systematic Error Recording measurement using uncertainty of equipment Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Temp, 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 quantity Time, t = (4.52 ±0.02) %uncertainty temp (%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 x %ΔI + ½ x %ΔV %DE = ( Energy, E = t ´ I2 v1/2 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 0.02 0.6 1 0.2 ´100% ) + ( 2 ´ ´100% ) + ( ´ ´100% ) = 45% 4.52 3.0 2 2.0 Energy, E = (28.638± 45%) AbsoluteDE = Energy, E = (29 ±13) 45 ´ 28.638 =13 100 Random and Systematic Error %Percentage Error = 50% Correct value = 19.092 Expt value = 28.638 ±45% %Error = ( %Error = ( expt - correct )´100% correct 28.638 -19.092 )´100% = 50% 19.092 Energy, E = (28.638± 45%) % Random Error %Random Error = 45% %Systematic Error = 5%
  • Random and Systematic Error Recording measurement using uncertainty of equipment Energy, E = t ´ I2 v1/2 % uncertainty = sum of % uncertainty of individual quantity Time, t = (4.52 ±0.02) %uncertainty temp (%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 x %ΔI + ½ x %ΔV %DE = ( t ´ I2 v1/2 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 Volt, v = (2.0 ± 0.2) Current, I = ( 3.0 ± 0.6) Temp, t = (4.52 ± 0.02) Treatment of Uncertainty Multiplying or dividing measured quantities Energy, E = 1 %DE = %Dt + 2 ´%I + ´%Dv 2 0.02 0.6 1 0.2 ´100% ) + ( 2 ´ ´100% ) + ( ´ ´100% ) = 45% 4.52 3.0 2 2.0 Energy, E = (28.638± 45%) AbsoluteDE = Energy, E = (29 ±13) 45 ´ 28.638 =13 100 Random and Systematic Error %Percentage Error = 50% Correct value = 19.092 Expt value = 28.638 ±45% %Error = ( expt - correct )´100% correct %Random Error = 45% % error fall outside> than % uncertainty (%Random error) • Random error cannot account for % error • Systematic error occur – small compared to random error 28.638 -19.092 %Error = ( )´100% = 50% 19.092 Energy, E = (28.638± 45%) % Random Error %Systematic Error = 5% Reduce random error – HUGE (45%) Precise instrument. Temp sensor
  • Expt on enthalpy change of displacement between Zinc and copper sulphate 25 ml (1M) (0.025mole) CuSO4 solution added to cup. Initial Temp, T1 taken. Excess zinc powder was added. Final Temp T2 was taken. Calculate ΔH for reaction. Treatment of uncertainty Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + CuSO4 M2 = (45.00 ±0.01)g Initial Temp, T1 = (20.0 ±0.2)C Final Temp, T2 = (70.6 ±0.2)C Mass CuSO4 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 CuSO4 m = (45.00 –20.00) = 25.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass CuSO4 m = (25.00 ±0.02)g Diff Temp ΔT = (70.6 –20.0) = 50.6 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (50.6 ±0.4) Mass CuSO4 m = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C
  • Expt on enthalpy change of displacement between Zinc and copper sulphate 25 ml (1M) (0.025mole) CuSO4 solution added to cup. Initial Temp, T1 taken. Excess zinc powder was added. Final Temp T2 was taken. Calculate ΔH for reaction. Treatment of uncertainty Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Addition/Subtraction/Multiply/Divide Multiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertainty Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + CuSO4 M2 = (45.00 ±0.01)g Initial Temp, T1 = (20.0 ±0.2)C Final Temp, T2 = (70.6 ±0.2)C Mass CuSO4 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 CuSO4 m = (45.00 –20.00) = 25.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass CuSO4 m = (25.00 ±0.02)g Diff Temp ΔT = (70.6 –20.0) = 50.6 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (50.6 ±0.4) Mass CuSO4 m = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C Enthalpy, H = (M2-M1) x c x (T2-T1) Multiplication Add % uncertainty
  • Expt on enthalpy change of displacement between Zinc and copper sulphate 25 ml (1M) (0.025mole) CuSO4 solution added to cup. Initial Temp, T1 taken. Excess zinc powder was added. Final Temp T2 was taken. Calculate ΔH for reaction. Treatment of uncertainty Adding or subtracting Max absolute uncertainty is the SUM of individual uncertainties Addition/Subtraction/Multiply/Divide Multiplying or dividing Max %uncertainty is the SUM of individual %uncertainties Addition/Subtraction Add absolute uncertainty Initial mass beaker, M1 = (20.00 ±0.01) g Final mass beaker + CuSO4 M2 = (45.00 ±0.01)g Mass CuSO4 m = (M2 –M1) Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Initial Temp, T1 = (20.0 ±0.2)C Final Temp, T2 = (70.6 ±0.2)C Diff Temp ΔT = (T2 –T1) Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Enthalpy, H = (M2-M1) x c x (T2-T1) Enthalpy, H = m ´ c ´ DT Multiplication Add % uncertainty Enthalpy, H = 25.00 ´ 4.18´ 50.6 = 5.29 Mass CuSO4 m = (45.00 –20.00) = 25.00 Absolute uncertainty, Δm = (0.01 + 0.01) = 0.02 Mass CuSO4 m = (25.00 ±0.02)g Mass CuSO4 m = (25.00 ±0.02)g Diff Temp ΔT = (70.6 –20.0) = 50.6 Absolute uncertainty, ΔT = (0.2 + 0.2) = 0.4 Diff Temp, ΔT = (50.6 ±0.4) ΔTemp = (50.6 ±0.4) C Treatment of Uncertainty Multiplying or dividing measured quantities Enthalpy, H = m ´ c ´ DT % uncertainty = sum of % uncertainty of individual quantities Mass, m = (25.00 ±0.02) %uncertainty mass (%Δm) = 0.02 x 100 = 0.08% 25.00 ΔTemp = (50.6 ±0.4) %uncertainty temp (%ΔT) = 0.4 x 100 = 0.8% 50.6 % uncertainty H = % uncertainty mass + % uncertainty temp % ΔH = % Δm + %ΔT 0.025moleCuSO4 = 5.29 1moleCuSO4 = 5.29 ´ 1 = 212 0.025 0.02 ´100% = 0.08% 25.00 0.4 %DT = ´100% = 0.8% 50.6 %DH = %Dm + %DT %DH = 0.08% + 0.8% = 0.88% Enthalpy = (212 ± 0.88%) %Dm = AbsoluteDH = Enthalpy = (212 ±1.8) Enthalpy = (212 ± 2) 0.88 ´ 212 =1.86 100 Continue  next slide
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer Recording measurement using uncertainty of equipment Mass CuSO4 = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C Enthalpy = (212 ± 0.88%) %Percentage Error = 15% Random and Systematic Error Correct value = 250 Expt value = 212 ±0.8% %Random Error =0.88% %Error = ( %Error = ( expt - correct )´100% correct 212 - 250 )´100% =15% 250 Enthalpy = (212 ± 0.88%) % Random Error %Systematic Error = 14.1%
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer Recording measurement using uncertainty of equipment Mass CuSO4 = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C Enthalpy = (212 ± 0.88%) %Percentage Error = 15% Random and Systematic Error Correct value = 250 Expt value = 212 ±0.8% %Random Error =0.88% %Error = ( %Error = ( expt - correct )´100% correct 212 - 250 )´100% =15% 250 %Systematic Error = 14.1% % error fall outside> than % uncertainty (%Random error) • Small random error cannot account for % error • Systematic error occurs – reduce this error Ways to reduce error Enthalpy = (212 ± 0.88%) % Random Error Systematic error (14.2%) Small random error Equipments OK
  • Random and Systematic Error Measuring Enthalpy change using calorimeter/thermometer Recording measurement using uncertainty of equipment Mass CuSO4 = (25.00 ±0.02)g ΔTemp = (50.6 ±0.4) C Enthalpy = (212 ± 0.88%) %Percentage Error = 15% Random and Systematic Error Correct value = 250 Expt value = 212 ±0.8% %Random Error =0.88% %Error = ( %Error = ( expt - correct )´100% correct %Systematic Error = 14.1% % error fall outside> than % uncertainty (%Random error) • Small random error cannot account for % error • Systematic error occurs – reduce this error 212 - 250 )´100% =15% 250 Ways to reduce error Enthalpy = (212 ± 0.88%) % Random Error Reduce heat loss use styrofoam cup Small random error Equipments OK Systematic error (14.2%) Extrapolate to higher temp (Temp correction) Stir the solution to distribute heat stirrer • • • • Assumption wrong Heat capacity cup is significant Specific heat capacity CuSO4 is not 4.18 Thermometer has measurable heat capacity Density solution not 1.00g/dm3 ✗
  • 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