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Uncertainty in Measurement

Uncertainty in Measurement. When recording measurements it’s very important to have the correct number of significant digits. This is determined by the increments on the instrument. The significant digits are all of the numbers that you know with certainty plus one more

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Uncertainty in Measurement

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  1. Uncertainty in Measurement When recording measurements it’s very important to have the correct number of significant digits. This is determined by the increments on the instrument The significant digits are all of the numbers that you know with certainty plus one more that you estimate.

  2. 4.8 cm 4.83 cm 4.830 cm Whatever the increments are, estimate one more place, but no more! For digital instruments, record exactly what appears on the display.

  3. Rules for Significant Figures 1. All nonzero digits are significant figures 2. Leading zeros are never significant Ex: 0.00025 has 2 sig figs 3. Trailing zeros are significant figures if a decimal point is present Ex: 0.0002500 has 4 sig figs If no decimal point is present, then the zeros are not sig figs Ex: 25000 has 2 sig figs

  4. Significant Figures in Calculations multiplication& division The number of significant figures in the answer is determined by the measurement having the fewest significant figures 10.75 x 103 Ex: (2.500 x 10-4)(4.3 x 107) = 10.75 x 103 =11 x 103 =1.1 x 104

  5. addition & subtraction the measurement with the least accurate place value determines the place value in the answer. Ex: 19.5 + 200.060 + 0.25 = 219.81 =219.8 Ex: 250 – 12 = 238 = 240 Combination Problems: Ex: (80.75) (4.18) (32.8- 24.5) 8.3 =2801.5405 =2800

  6. Uncertainty Whenever you record a measurement you must state the uncertainty in the value; i.e. the range of possible values (above or below your estimate) analytical balance centigram balance Ex: 123.08 g +/- 0.01 g Ex: 123.0835 g +/- 0.0001 g Digital: +/- 1 of last digit !

  7. Non-electronic instruments 52.9 mL +/- 0.2 mL 87.4◦ C +/- 0.3 ◦ +/- 0.1- 0.5

  8. buret 21.30 mL +/- 0.02 mL +/- 0.02 – 0.05 mL Acceptable Uncertainties? 103.25 mL +/- 0.005 no 15.38 g +/- 0.1 no 24.7762 g +/- 0.001 no 3.61 mL +/- 0.05 yes 32.95 ◦ +/- 0.01 yes

  9. Propagating Uncertainty Calculated answers should have the right number of significant figures and the uncertainty Not the same as percent error: % error = |Δaccepted and experimental value| accepted value 100 Ex: Actual melting pt. = 84.8 ◦ C Experimental m.p. = 86.5◦C % error = |86.5 – 84.8 | 84.8 100 = 2.0 %

  10. answers derived from addition or subtraction: Just add the uncertainties in the measurements to get the uncertainty in the answer Ex: mass of crucible and compound 24.31 g +/-0.01 mass of empty crucible 19.94 g +/- 0.01 mass of compound 24.31 – 19.94 = 4.37 g +/- 0.02 If getting an average of several values, the uncertainty remains the same Ex: 4.5 mL +/- 0.2 + 4.7 mL +/- 0.2 avg= 4.6 mL +/- 0.2 not +/- 0.4

  11. multi-step problems involving mult. & division: 1. Convert the absolute uncertainties in all the measurements into a percentage 2. Add all the percentages 3. Convert the percentage uncertainty back into an absolute uncertainty in the final answer

  12. Ex: An experiment is performed to determine the enthalpy change, ΔH, for the reaction between HCl and magnesium metal. If 2.00 g of Mg are added to 50.08 g of HCl and the temperature of the solution rises from 23.8◦ C to 38.2◦ C, calculate ∆H and specify the uncertainty. Specific heat of the solution = 4.18 J/g ◦ C balance uncertainty = +/- 0.01 g thermometer uncertainty = +/- 0.2 ΔH = m C ∆T ΔH = (50.08 +2.00 g) (4.18 J/g◦C)(38.2 – 23.8 ◦ C) ΔH = 3,134.7993 J 3.1347993 KJ 3.13 KJ

  13. uncertainty in the answer: mass of solution: 0.01 + 0.01 = 0.02 % = 0.02 x 100 52.08 = 0.0384 % temp. of solution: 0.2 + 0.2 = 0.4 % = 0.4 x 100 14.4 = 2.777 % Total = 0.0384 + 2.777 = 2.815 % 3.13 KJ (3.13) (0.02815) =0.0881 3.13 KJ +/- 0.09

  14. Ex: 25.00 mL of an acid measured from a pipet (+/- 0.05) is titrated with sodium hydroxide solution measured from a buret. The initial buret reading at the beginning of the titration is 0.82 mL (+/-0.02). After the titration the final buret reading is 33.87 mL (0.02 ). If the molarity of the NaOH is 0.25 M (+/- 0.01 ), what is the molarity of the acid with the uncertainty? M1V1 = M2V2 acid base M1 (25.00 mL) = (0.25 M )(33.87 – 0.82 mL) 25.00 mL 25.00 mL = 0.33 M M1= 0.3305

  15. uncertainty in the answer: volume of NaOH: 0.02 + 0.02 = 0.04 % = 0.04 x 100 33.05 = 0.121 % volume of acid: 0.05 = 0.2 % % = 0.05 x 100 25.00 molarity of NaOH: 0.05 = 4 % % = 0.01 x 100 0.25 Total= 0.121 + 0.2 + 4 = 4.321 %

  16. Total= 0.121 + 0.2 + 4 = 4.321 % (0.33) (0.04321) = 0.01426 = 0.33 M +/- 0.01

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