Check List - Data Analysis

Transcription

1 Chem 360 Check List - Data Analysis Reference: Taylor, J. R. An Introduction to Error Analysis; University Science Books, 2nd Ed., (Oxford University Press): Mill Valley, CA,1997 Listed below are data analysis concepts that a B.S. chemist should know about. These are all addressed in Taylor, and many are contained inside the front and back covers. You are responsible for correctly applying these concepts in your Chem 360 lab work and reports. You may want to use this list as a check list as you read Taylor. Error vs. blunder Definition vs. measurement uncertainty Experimental discrimination Interpolation of scale readings Systematic vs. random error Report best value ± uncertainty Round uncertainty to 1 significant digit Matching last significant figure with order of magnitude of uncertainty Significant discrepancy vs. insignificant discrepancy Accepted value vs. true value Checking proportionality with graph or ratio Error bars on graph Relative or fractional uncertainty Significant digits - rounding off, arithmetic operations Estimating uncertainties in direct measurements repeated measurements resolution of measuring devices Propagation of uncertainty in q(x,y,...) - general formula (3.47) Role of quadrature in propagation of error for normal (Gaussian) errors Propagation through several experimental steps Mean x Standard deviation σ, rms deviation, variance σ 2 (5 or more measurements required) ±σ and 68% confidence Standard deviation of mean (SDOM) σ x = σ x / N

2 Distribution functions: f(x) dx - probability of finding x in dx b " f (x)dx for probability of finding x between a,b a x = # $ "# f (x)dx " x # $ 2 = (x "x ) 2 f (x)dx "# 1 Normalized Gaussian distribution f x,σ (x) = (" 2# ) exp($(x $ x )2 /2" 2 ) ± 1.96σ and 95% confidence Confidence and reasonable outcomes Rejecting data - need for criteria Chauvenet's criterion Chi squared criterion, χ 2 Linear least square regression - line of maximum likelihood minimize χ 2 = [y i - (A + Bx i )] 2 /σ 2 y in normal distribution Least square equations for A, B, σ 2 y, σ 2 A, σ 2 B Extrapolation dangers

3 Physical Measurements Data Analysis Assignment 1 Problems for Chapters 1 & 2 of Taylor, An Introduction to Error Analysis Warning: Most of these questions have no exact right or wrong answers. Answer the following questions on 8.5x11 inch paper (i.e., NOT IN YOUR LAB NOTEBBOK!). 1. (10 pts) Below are listed physical properties that have been measured by an experienced scientist, who gives us no explicit indication of the uncertainty in each measurement. Estimate a reasonable range of values for each, using Taylor's notation of page 15. mass of crystal = 1.64 g boiling point = C height of GC peak = 8.3 inches Explain any assumptions you make about the measurement tools used in each experiment. (Taylor, Sec. 1.5, 1.6, 2.1) 2. (10 pts) Criticize the following ways of reporting an experimental result. Rewrite each in a more sensible form. a) Our measurements determined R to be 8.3 ± J/(K mol) b) Our measurements determined R to be ±.1 J/(K mol) (Taylor, Sec. 2.2) 3. (10 pts) Mary and Joe measured R in the lab with the following results: Mary's R: 8.3 ±.2 J/(K mol) Joe's R: 8.5 ±.2 J/(K mol) Would you judge the discrepancy in their results to be significant or insignificant? Explain. (Taylor, Sec. 2.3) 4. (20 pts) When Mg metal is treated with HCl to generate hydrogen gas (the Chem 150 experiment), the ideal gas law predicts that : V Hydrogen = RT GAW Mg P g Mg

4 Does the data listed below verify this relation between V Hydrogen and g Mg? Use a graphical or ratio argument to support your answer. You may wish to include estimated uncertainties into your argument. V Hydrogen (ml) g Mg (Taylor, Sec. 2.6)

5 Physical Measurements Data Analysis Assignment 2 Elementary Propagation (50 pts) Apply the general uncertainty propagation equation (Taylor, 3.47, p.75) to the measurement situations listed below to develop specific explicit uncertainty expressions. Answer the following questions on 8.5x11 inch paper (i.e., NOT IN YOUR LAB NOTEBBOK!). Example: Absorbance = molar absorbtivity x path length x molarity A = a x b x M Uncertainty in A δa, etc. δa = A 2 a δa 2 + A 2 b δb 2 + A 2 M bm am ab = ( b 2 M 2 δa 2 + a 2 M 2 δb 2 + a 2 b 2 δm 2 ) 1/2 δm 2 1/2 Do All: v rms = 3 RT M, R constant P = RT V-b - a V 2, R, a, b constants E = n 2 h 2, n, h constants 2 8 ma q r = 2 I kt σ h 2, k, σ, h constant q tr = 2 π m kt 3/2 h 2 V, k, h constant

6 Physical Measurements Data Analysis Assignment 3 A real FLC problem: In Room 235CH some FLC students helped construct a time-of-flight mass spectrometer that we use for a variety of laser-surface interaction studies. It is a very simple device. In an environment where the mean free path of molecules is on the order of meters (high vacuum) ions are created by a laser. They are all accelerated to an energy E by placing voltages on electrodes in the region of ionization. All ions are accelerated to this same energy regardless of their mass. Following this acceleration to E energy they are allowed to "drift" in a field free region of length L to a detector. The time between their creation (time=0) and their detection (time=t) is of course mass dependent. Answer the following questions on 8.5x11 inch paper (i.e., NOT IN YOUR LAB NOTEBBOK!). 1. Using the physics that we have required you to take, derive the time-of-flight, t, for an ion of mass, m, of energy, E, that travels a distance, L. 2. One of the problems in Time-of-Flight Mass Spectrometry is that the resolution of the mass spectrum is low (time or mass 'distance' between ions of similar mass is small). This can be due in part to fluctuating acceleration voltages (AC voltage "ripple" on DC voltage power supplies). This results in an δe uncertainty in the overall energy E. Following Taylor 3.47, derive an expression for the uncertainty in the time-of-flight due to uncertainty in E. To help us see this limitation, use an electronic spreadsheet to evaluate the time-of-flight and its uncertainty for the following ions and conditions: (remember your SI units!) Ion Flight Distance Energy δe Cu meter ev 5 ev (Cu-63 isotope) Cu meter ev 5 ev (Cu-65 isotope) protein(1 + ) meter 20,000. ev 200. ev (50,000 Daltons) 3. Let's put the calculations in a more meaningful form and add one more complication. Derive an expression for the uncertainty in the mass of an ion resulting from the uncertainty in the acceleration energy, δe, as above, but include an uncertainty in measuring the time-of-flight as well, a δt. We need an expression for δm in terms of E, δe, L, t, and δt. Use an electronic spreadsheet to evaluate the δm uncertainty for the Cu- 63 and protein ions above using your calculated times, t, the parameters given above and assume that the uncertainty in our measured time is 10 ns (from our measuring device - an oscilloscope).

7 4. Use your spreadsheet to answer some questions - let's play what if - a) What are the requirements of δe (if any) to produce an uncertainty of 10 Daltons in the molecular weight of the 50,000 Dalton protein (keeping all other parameters constant). b) If the δe is limited to.5% (.5% of 20 kev, 100. ev), can we achieve a δm of 10 Daltons in the 50,000 Dalton protein? If so, what is the greatest allowable value of the uncertainty in time, δt?

8 Physical Measurements Data Analysis Assignment 3 Example Here's an example we'll go over in class- we've used this in the past: Estimate the propagated uncertainty for an experiment from an old Chem 150 Lab Manual (Determination the Molecular Weight of a Volatile Liquid) by reconstructing on paper a plausible experiment (lab handout provided). Assume that you use an analytical balance, graduated cylinder, barometer, and ordinary thermometer. I'll remind you of the procedure. A suggested spreadsheet layout we'll discuss: Propagation of Error Molecular Weight Determination R= (l-atm)/(mol-k) Parameter Value Error V-Converted Error Converted Mass(g) Temp( C) Pressure(mmHg) Volume(mL) MW= 46. g/mol Term ±Error (Partials)^2 Term Value = Err^2*Partial^2 Mass (g/mol)^2 Temp (g/mol)^2 Pressure (g/mol)^2 Volume (g/mol)^2 Sum= (g/mol)^2 Overall Error = (±) g/mol ± 1 g/mol

9 Physical Measurements Data Analysis Assignment 4 Linear Regression Problem Assignment - Required: A. Using the equations from Taylor, Chapter 8, prepare a spreadsheet calculation that computes for the (ethanol vapor pressure) data below, A, B, σ A, and σ B. (This calculation will not use the σ p data.) Then use your spreadsheet results to calculate ΔH vap and σ ΔH for ethanol. (Remember to convert to degrees Kelvin). Equation: ln P = "#H R 1 T + constant y = B x + A (Taylor, (8.2) page 183) Data: T( C) P(torr) σ p B. Perform the same analysis using Excel s regression tools in the Analysis Tool Pak. Assignment Optional Bonus: C. You may add a useful a layer of sophistication by calculating a weighted least squares fit as per problem 8.4 (page 169) using w i = 1/(σ ln P ) 2. Compare the A, B, σ A, and σ B results obtained here with your results from part A. How do you get σ ln p data from σ p data? Turn in a numerical printout and a "formula" printout.