Topic 9 Retake this Wed during Homeroom You must meet w/ Ms. Clewett to review Topic 9 by Wednesday…even if you have already met with me once, come remind me, please. * Topic 10 Exam redo next Tuesday, April 11th in class (all students).

Measurement and Data Processing Topic 11 Measurement and Data Processing

Topic 11.1: Uncertainties and errors in measurement and results Measured uncertainties, A (also called absolute uncertainties) Digital measurements (usually +/- 1 in the last digit) 1.2 g implies uncertainty (m) of +/- 0.1 g 1.213 g implies +/- 0.001 g 1.2127 g implies +/- 0.0001 g Glassware measurements (frequently read to half of a tick mark) Burettes have tick marks every 0.1 mL; thus, measurement must be to the hundredths of a mL & uncertainty (V) ~0.05 mL (or V = 0.02 mL also used; both are acceptable) Thermometers have tick marks every 1 degree; thus, uncertainty ~0.5◦C

Absolute Uncertainty, Relative Uncertainty, and Percent Relative Uncertainty What is the absolute uncertainty (V) of a 50 cm3 graduated cylinder with tick marks at every 1 cm3 when used to measure 25.0 cm3 of water? (Assume uncertainty = half of a tick mark.) What is the relative uncertainty (= V/V) for the above? What is the percent relative uncertainty (= V/V x 100%) for the above? Repeat all three for a measurement of 2.924 g …and for 2.9 g.

Size of the uncertainty ☛ Instrument Precision Measured value of 1.0 g (= low precision, cheap balance!) Note that true values ranging from 0.950 g to 1.049 g would all be reported as 1.0 g on this low precision scale due to place value limitation at the tenths place. Uncertainty, m, = 0.1 g Relative uncertainty, m/m, = 0.1 g/1.0 g = 0.1 Percentage relative uncertainty, m/m x 100%, = 0.1 g/1.0 g x 100% = 10% Measured value of 1.0000 g (= high precision, costly analytical balance!) You deduce the corollaries! Precision = Degree to which data is repeatable Agreement between Trial 1 value & Trial 2 value (etc.) High precision = relatively low random errors (which cause scatter—equal likelihood of too high & too low of values)

Closeness to true value = Accuracy Measured value of 1.0 g may be highly accurate, even if it is achieved through a relatively imprecise instrument  Frequently reported using Percentage error % Error = |experimental value – true value| true value (true value  literature value) Find percentage error if measurement is 0.92 V and true value = 1.10 V.

Evaluate precision & accuracy (given true value = 20.0 g) Set 1 (g,  0.1 g) 19.8 17.2 18.3 20.1 18.4 Set 2 (g,  0.1 g) 19.2 19.1 19.3 Set 3 (g,  0.1 g) 20.0 19.9

Types of errors Random errors Example(s): A/C turns on and off during mass measurements (scale fluctuates) Reaction time Main Effect: Reduces precision May or may not affect accuracy Reduced via: Larger number of measurements Systematic errors Read to top/side of meniscus; Instrument not calibrated correctly; Calorimeter loses heat to environment Reduces accuracy (measured value is consistently too big or too small) Fixing a systemic problem w/ procedure

Classify the type of error: You measure gas production using a rubber tube that has some cracking/leakage. You over-titrate and estimate the reading in a region of glassware that does not have any tick marks. Your density experiment is conducted on a volatile liquid; the volume is measured after the mass. You rinse your buret with distilled water but not with the titrant solution. NoS: Can you ever completely eliminate random errors?

Significant Figures Rules for counting sf’s Count all 1-9 NEVER count leading zeroes: 0.00542 g = 3 sf’s ALWAYS count sandwich zeroes: 1003 = 4 sf’s ONLY count trailing zeroes IF there is a decimal: 2,500 = 2 sf’s but 2,500. = 4 sf’s and 2,500.0 = 5 sf’s Rules for calculating with sf’s Answer to least # of sf’s when multiplying and dividing: 1.250 g / 2.5 cm3 = 0.50 g cm-3 (limited to 2 sf’s) Answer to least significant place value when adding and subtracting: 148.45 g – 147.95 g = 0.50 g ( 2 sf’s) and 134.65 m + 150 m = 280 m (2 sf’s b/c limited to tens place)

Solve: A piece of metallic indium with a mass of 15.456 g was placed in 49.7 cm3 of ethanol in a graduated cylinder (IB calls these measuring cylinders, btw). The ethanol level was observed to rise to 51.8 cm3. From these data, the best value one can report for the density of indium is 7.360 g cm-3 7.4 g cm-3 1.359 x 10-1 g cm-3 32.4 g cm-3

Exact numbers have infinite sf’s Some numbers are defined exactly; these numbers will never restrict the sf’s of a final answer. Coefficients of balanced equations (mole ratios) Subscripts of chemical formulas (exactly 2 atoms of N in a mole of nitrogen gas) Metric conversions (1000 J = 1 kJ) 22.7 dm3 mol-1 has 3 sf’s; 1.01 g mol-1 has 3 sf’s; these non-metric conversions are rounded; elsewhere, they may be found reported to more sf’s. (A few oddball conversions are defined exactly, but these are beyond the scope of the IB syllabus.)

Propagating Uncertainty If adding or subtracting to calculate a value (Atotal = A1 + A2 or Atotal = A1 - A2 ) Add the absolute uncertainties: Atotal = A1 + A2 Volume of base added while titrating: v-initial: 0.00 cm3  0.02 cm3 and v-final: 21.35 cm3  0.02 cm3 Mass of a substance: mass of beaker = 146.25 g  0.01 g and mass of beaker plus substance = 146.25 g  0.01 g If multiplying or dividing to calculate a value (X = Y * Z or X = Y / Z) Add the relative (or percentage) uncertainties (X/X) = Y/Y + Z/Z Show a complete density calculation if mass = 6.500 g (deduce uncertainty) and volume is 2.41 cm3  0.02 cm3. Would there be any benefit to using a more precise scale, in terms of reducing the final uncertainty? Try this: Calculate the moles delivered when a 50.0 cm3 pipette with an uncertainty of 0.1 cm3 is used to measure 50.0 cm3 of 1.00  0.01 mol dm-3 sodium hydroxide solution.

Graphing variables Temperature v. Pressure, or “Dependence of temperature on Pressure” for a given volume of an ideal gas What is the Dependent Variable? & What is the Independent Variable? How are they graphed? What are several ways the relationship may be described?

Graphing Must Do’s If you are asked to sketch or draw a graph: You MUST label axes with variable name & include units Be sure you place the IV on the x-axis & DV on the y-axis For drawn graphs (only) you must have axis values noted to scale (use a ruler!)

Interpreting a Graph Interpolating v. Extrapolating Evaluating the slope Determining Intercepts Units of the slope

Lines: y = mx + b (m = slope or gradient) When is it appropriate to say “directly proportional”? … “inversely proportional”? Lines: y = mx + b (m = slope or gradient) If b = 0, then y is directly proportional to x, shown as y  x or, equivalently, y = k x. What do you call a relationship where m = -1? Inversely proportional is y = k/x or y  1/x.