advanced physical

CHEMISTRY 464

ADVANCED PHYSICAL CHEMISTRY LABORATORY

LABORATORY MANUAL

P. S. Phillips 2010. All rights reserved.

Plot the graph, then put the data on.

There are no truths, only facts to be manipulated.

Give me six variables and Ill fit an elephant. Give me seven and Ill make its tail wag.

INSTRUCTIONS AND GENERAL INFORMATION. The physical chemistry laboratory is equipped for the following experiments. Students will carry out six of them. Each takes two lab sessions. Which experiments, and when you do them, will be organized the start of term. The experiments are done with partners.

These experiments are suites. Each suite consists of two or three experiments.

ID TOPIC EXPERIMENT

A Activity coefficients Activities of ions by electrochemistry (and some programming )*

E Enzyme kinetics Various approaches. Inhibition. Non-linear fitting.

G Thermodynamics of glycine Properties of a weak acid, pKa and thermochemistry.

H The hydrophobic effect A study of partitioning and co-solvents to illustrate the hydrophobic effect.

K Kinetics A couple of kinetics experiments, mainly to illustrate computer fitting methods.

M Micelles CMC and aggregation numbers by fluorimetry, UV-Vis and conductometry .

P Permeability A simple investigation of permeability and osmosis.

T Transitions in biomolecules Examine the effect of phase temperature and pH on myoglobin. Comparisons with other enzymes.

W Acidity of wines Look at the pH of mixed diprotic acids and cations on buffering in wine and the effect of alcohol on pKa

Some of the material required for the experiments will not have been covered in any of your classes. Background research is an essential part of the experiments.

*May not run this year.

Introduction:2

PERSONAL EXPERIMENT ROSTER

GROUP #

Experiment Date/Time

Partners Name: Phone No. .

Experiment/Group rotation.

Week Group No.Z 1 2 3 4 5 6

1 2 3 4 5 6

Introduction:3

CHEMISTRY LABORATORY SAFETY REGULATIONS A chemical laboratory is a potentially dangerous environment; the most prevalent hazards are fire, chemical burns, cuts, and poisoning. NOTE: Safety rules only work if you obey them and encourage others to do so. Please familiarize yourself with the following regulations. NOTE: These regulations represent a minimum. A member of the lab staff will inform you of variances and other regulations, or supply you with appropriate references. If unsure about anything ask them. NOTE: An eyewash station is available for the treatment of minor accidents. For first aid, phone local 78111 or 807-8111. In an emergency (i.e. one requiring police, fire, ambulance or a hazmat team) phone 911 then phone local 78111 or 807-8111. A member of the lab staff will normally make such calls.

1. Regulations All accidents and incidents (near misses and spills) must be reported immediately to a member of the lab staff. Students are not usually permitted to use the laboratory except during their scheduled laboratory period. No student should attempt unauthorized experiments in the laboratory, or modify any experimental apparatus. No student can work in a laboratory without a supervisor present unless they have completed a WHMIS and the Chemistry Department Safety course, and then only with the supervisors consent. Any student deemed dangerously incompetent or intoxicated will be required to leave the laboratory. An incident report will be filed. Keep walkways clear at all times. Do not leave cupboard doors open. 2. Personal Safety. Many of the chemicals in the laboratory are poisonous, whether taken orally or absorbed through the skin. If any chemical is swallowed, the supervisor should be summoned immediately. Immediately wash off any chemical comes in contact with the skin with plenty of water. Consult the MSDS data sheets for further information. Make sure you know the location of the eyewash station and emergency shower. As a minimum students MUST wear safety glasses at all times. You must provide your own safety glasses. Contact lenses must not be worn. Other protection such as side shields, goggles or face shields may be required. Make sure your goggles are sealed against the face. No food or drink may be bought into the laboratory. Do not chew gum in the laboratory. There must be no smoking in the laboratory. Students should keep their arms, legs and torso covered. Students should keep their arms, legs and torso covered. Wearing 100% cotton lab coats is required. Most chemicals will stain or burn your clothing. You are not permitted to wear open toed shoes in the lab. Long hair should be tied back at all times. Unless otherwise informed assume all unknown samples are dangerous, that is you must wear goggles, gloves and lab coat while handling them. Assume all chemicals are corrosive or toxic by ingestion, and take appropriate precautions. Never handle chemicals with your bare hands. While heating a substance in a vessel with a narrow mouth (e.g. a test tube) ensure that the mouth of the vessel is not pointing at anyone, including yourself. When using compressed gas, vacuum equipment, high temperature or high voltage equipment be especially careful. Ask the laboratory instructor for help if you are uncertain of any procedure. Strongly corrosive or toxic materials should only be handled in the fume hood, with the sash down, and suitable gloves on. Do no kneel to look under

the sash! It defeats the whole point of the sash. Do not wipe your face or eyes with gloves on! Water play or squirting wash bottles will not be tolerated. Do not kneel or sit when preparing hazardous samples. If theres a

spill, you must be able to move fast and keep your face out of the way. Use the center shelf of the lab bench to fill volumetric flasks to the line.

Be sure you have received proper instruction in: Boiling of liquids (use boiling chips!) Use of separatory funnels (dont point them at anyone) Use of any unfamiliar equipment or chemicals

Insertion into or removal of, glass tubing (rods, thermometers, pipettes etc.) rubber or plastic items (tubing, pipette bulbs, bungs etc.) e.g. see figure for the correct way of inserting a pipette into a dispensing bulb.

Do not wave pipettes around (especially Pasteur pipettes) around or you will spray residual any material around.

Do not crush materials with stirring rods.

2. Fire Students should be aware of the location and use of the fire extinguishers in the laboratory. In case of fire, the flames should be extinguished with one of the extinguishers and the supervisor notified immediately. If a student's clothing or hair catch fire, use the emergency shower (make sure you know its location). If this is not possible, smother the flames immediately with a laboratory coat or a fire blanket (know the location of the latter)

3. Breakages, spillages and fumes Immediately report all breakages and minor spills of chemicals to the supervisor or technician. A spill kit is available if needed. Know its location. Spills will normally be dealt with by the laboratory staff. Failure to report mercury spills may result in a reprimand. Remember that broken class is the sharpest material known. Any experiment involving the evolution of toxic materials, or pungent or unpleasant odors or fumes, must be carried out in the fume hood. Dont wear aftershave or cologne in the labs. It interferes with our ability to detect fumes (they are fumes) and can mess up analyses.

4. Disposal Make sure all broken glass or sharps are disposed of in the appropriate

container. Make sure all materials are disposed of in the appropriate container

(solids, organics, halogenated solvents or inorganics)

Beware of materials that hydrolyze rapidly, e.g. SOCl2 and acid anhydrides; they cannot be disposed of in containers containing alcohol or water.

Never add hydrogen peroxide, nitric acid or any other oxidants to organic materials (unless instructed) or into the organic disposal container. Acetone and alcohol are a particular problem.

Introduction:4

MARKING POLICIESAllocation of marks will be different from usual labs. and may vary with the experiment. As above, you start with one or two marks less than the maximum otherwise an unrealistic spread of marks will occur. The follow items will be considered when marking labs.

TECHNIQUE. This mark will be assigned for things like speed (finishing time), sloppiness, preparation (did you read the lab. before arrival?), contribution to the laboratory discussions, breakages, record of original data in your lab. notebook, attentiveness and attitude, as well as pipetting, weighing, titrating and other standard lab. skills. These experiments are all straightforward and in many cases you have done them before so odd factors like attitude will weigh heavily quietly sitting in a corner, noisily sitting in a corner, or slipping out for a pint will be viewed dimly. Also, you will be marked on how you solved the problem and the computer techniques used; e.g. proper choice of data ranges, checking convergence criteria, use of special features rather than brute force (e.g. for Excel; in-cell iterations versus huge tables or use of named ranges).

RESULTS. Clearly, the better your technique, the better your results should be, but since some of these experiments are designed to produce bad results so you can use fancy techniques to fix them up a bit, this mark is a little odd. However, the experiments do often have built in checks, there are certain errors that can only be achieved by incompetence this mark will address these.

PRESENTATION. This will be an opportunity to demonstrate your word-processing skills, HAND-WRITTEN REPORTS WILL NOT BE ACCEPTED for any lab. reports. You will be marked on presentation e.g. clear format, data properly identified, clear use of labels, proper choice of axes on graphs, clear comments (in Maple). Proper placement and sizing of titles, captions and graphs. You MUST record your original data and any in lab. notes in an approved lab. book. Do not submit this book with your reports, but I will wish to see it and assign marks according to its clarity and completeness. (The supplementary questions may involve a lot of pictures or derivation in that case it may be acceptable to hand-write that section check with the instructor).

REPORT. See the appropriate section for a description of the format of your report. Most importantly, do example calculations for one sample and tabulate the final, and some intermediate data, for the rest of the samples. Be sure to

tabulate your original data points (that is working data point; for instance there is no need to reproduce the data from time-runs, only the data points that arises form the time-runs) at the start of the report. Also, at the end, tabulate your calculated results and the literature values. Error discussion is required for each lab. Error analysis may also be required. Make sure you indicate the algorithms used for doing calculations, particularly if you used a computer. Marks are given for extra background research and any insightful comments. (415 is a research based lab. so these are part of your main mark, not an opportunity for brownie points as with most labs.) Marks are deducted for; arithmetic errors, incorrect answers, failure to answer questions, failure to comment on results; incomplete error discussion, lack of literature values, bad organization and extreme untidiness, excessive neatness, and of course handing in the lab. report late. Be sure to answer all questions given. Not all reports are equally easy, marks allocated to a report may vary from 10 to 15 to 20, depending on the length and difficulty of the calculations and questions.

PROBLEMS. Some labs. have lots of questions to answer. In fact, some look like a lab. with a problem set attached to the end. The problems carry significant weight.

SAFETY. Normally, physical chemistry experiments are designed to work with safe chemicals and equipment. For advanced courses this is neither possible nor desirable. Place a small section in your lab. report giving toxicity information and precautions for each (and every one) of the chemicals used (starting material, products and by-products). Look it up, dont guess. If you dont take this seriously I wont let you in the lab. However, remember to use credible sources as some sources tend to over exaggerate the hazards for legal reasons (i.e. beware of the manufacturers literature). The WHIMIS CD-ROM is a good place to start.

PLAGIARISM. See the separate section for details. Basically, if you copy from somebody else or allow somebody else to copy from you, you will get zero for that lab. report. Repeated infractions may get you suspended. Discussion of ideas is permitted, but the write ups must be independent.

Introduction:5

THE FORMAT OF A LABORATORY REPORTIDENTIFICATION. All labs. reports should have a front sheet that gives: Your name, the course number (Chem.309) and section no. Name of partner (if any). Full date of experiment (e.g. Tue. May 10th 2018). Experiment number. Lab. Profs. name.

TITLE. This should give both the substance or system studied and the method of physical measurement made, or of property determined.

OBJECT OR PURPOSE. One sentence; may not be necessary if it just duplicates the title.

PRINCIPLE OF METHOD. Briefly discuss the theory underlying the laboratory and briefly give the principle of the method, and use this as opportunity to write down any equations you will use in results and calculations.

PROCEDURE. You may refer to the lab. manual for this, but you should any modifications to the procedure and indicate possible improvements and sources of errors or other general problems. Draw diagrams of apparatus setup if required.

RESULTS AND CALCULATIONS. Tabulating usually saves a lot of time and space and keeps the prof. happy. Draw graphs using a computer, unless otherwise stated. Embed both the graphs and tables in the text. Be sure to make the graphs readable, say 6x5 in minimum size For repeated calculations, give one example and one only, in full. The example calculation should be embedded in the text, but may be hand written (try doing one by typing to see what a pain it is). This should come more naturally than in the past as you should be using Excel for calculations and will need to explain them. Make sure you always compare your results with the literature values when available. Make sure you indicate the algorithms or programs used for doing calculations.

DISCUSSION. This is a discussion of the results and their errors in the context of the data processing methods used. Benefits and shortcomings of the methods should be discussed.

ERROR DISCUSSION/ANALYSIS. For any quant-ity for which you have found a numerical value, give an estimate of error limits. Do not propagate the errors in your calculations unless an error analysis is required. Estimate random errors from observed scatter of data either visually, or by statistical computation. List possible sources of

systematic error, and estimate their limits (guesswork!). In general, don't make the discussion too detailed, unless you are told that an error analysis is required for that experiment. If you are told to do a full error analysis you need the formula in Table 3 of the Error Analysis section). For most purposes an error analysis using Table 2 of that section will suffice. Error analysis should be done along with your calculations. Discussion is done as a separate section.

CONCLUSIONS. Your major findings in a few lines. This section may not be necessary if you have them clearly set out at the beginning or end of the "discussion" section. Be sure that you have answered all questions in the text. Also, make any comments about the findings and there implications for errors in this and related experiments

QUESTIONS. The question may be in the body of the main body or given in a separate section at the end of the experiment. Clearly identify, separate and answer the questions at the end of your report. Do not embed the questions in the discussion or conclusions or in a long rambling paragraph at the end. Text answers should be typed. Numeric or algebraic questions can be hand-written.

REFERENCES. Place any references, in the standard A.C.S. format, in this section. WEB references, with the exception of the CHEMBOOK site are NOT acceptable. See the section on the Internet elsewhere.

Please feel free to discuss report format and other problems with the instructor, but do not expect detailed explanations or corrections written on your laboratory reports.

See the A.C.S. Authors Guide (in library) for further details on style and format of chemistry documents.

Introduction:6

SOME NOTES ON TYPE SETTING REPORTSAll lab. reports are to be typed using a computer. I recommend Word (and Word on the Mac at a push). Make sure the equation editor is installed (and you know how to use it). If you have last minute printing problems, see me, I may accept a disk copy. For the purposes of proofing your reports I recommend printing them out. I will supply you with a symbol font that contains nearly all the symbols youll need for chemistry, and a Greek font with some modifications suitable for scientific work (the standard symbol font has Greek characters, but they have to be italicized). I usually assign some keystrokes to a macro to turn them on or off. I will also give you some macros you might want to use. I want you to follow some basic typographic conventions for your report (or youll loose marks). They are as follows.

a) The main body of text should be in a proportionally-spaced (e.g. not Courier) western (e.g. not Cyrillic) serif font (e.g. Times New Roman, Garamond, School Book), not a sans serif font (e.g Arial) or any decorative font (e.g. Brush Script) or anything weird (e.g. Tekton). Use a normal face, not italicized, hollow or bold. Type size should be 12, 13 or 14pt. b) Titles should be bold, serif or sans serif, and larger than the main text; dont bother with anything fancy. c) There should be a maximum of three fonts in a report (the main text, titles, symbols). If you want some variation on the title page you can use bold and italic (sparingly). d) Keep it simple, no curly borders or color (except maybe in graphs). Use bold or italic for emphasis, do not underline. e) Use tables for your data, dont rely on tabs etc. Do not box the table, a couple of lines here and there usually does the job. f) Note that stuff like 13 CO3

2+ can only be typed in using the equation editor learn to use it. g) The main body should be double or triple spaced (with a 1 margin all round) so I can insert comments. However, I will tolerate single spaced reports. h) Diagrams can be drawn by hand, but keep them neat. They should be captioned by a line of type. Equations must be typed in, I suggest you reference the manual (e.g. Exp. G eq.3) to reduce this burden. All graphs must be done by computer, but you can manually cut and paste them into your report if you wish (OLE works nicely, but only on fast machines with lots of memory). i) Do not cut and paste lumps of text from your partners reports. I want to see some originality; in both the content and the formatting (Also see the section on plagiarism).

j) Watch for ls (ells) they look like 1s (ones). In sans serif you get ls (ells), 1s (ones) and Is (eyes). k) Number the pages just in case the staple falls out. If you need any help with word processing dont hesitate to come and see me its part of the course. Office 2007 is almost useless for lab. reports get a copy of Office 2003.

Introduction:7

TYPOGRAPHIC CONVENTIONS FOR PAPERSAll journals have conventions that you must adhere to in order to get a paper published in them. These conventions vary from journal to journal, although there are some common conventions to all journals. Below is a set of conventions that you must adhere to for your lab. report to be accepted. (These are over and above the format conventions laid out elsewhere in this manual). The conventions are based on those for A.C.S. journal and are detailed in the A.C.S. style guide (along with a grammatical guide and conventions for hyphenation, abbreviations, capitalization etc., well worth a peek at). Conventions for other journals are usually laid out in the front of the journal.

Fonts and case. Use 12 pt Roman (or similar). Do not used bold or italic except as indicated below. Do not underline ever. Lower case Greek letters are always italicized. Upper

Greek letters case are not. Mathematical variables and constants are italicized. Numbers and operators are not italicized. Vectors and tensors are bold. Math variables are never case sensitive (like in the

moronic C). d and D are the same. Element symbols are not italicized. Use italics for emphasis. Use bold for titles. Increasing

font size is also useful for emphasis. When defining a new word it is common to italicize it. Italicize latinates (actually optional, but I tend to do it). Try to make superscripts and subscripts 10pt (theres a

macro on the disk for this). Symbols and axis labels in graphs should be 12pt.

Abbreviations and units. Units are never capitalized when spelt out. Abbreviations for units should be spaced, e.g. 10 km hr-1

not 10km.hr-1. I tend to ignore this rule as it leads orphaned or widowed units.

A list of approved abbreviations are in the A.C.S. Guide. All abbreviations, except as listed below, must be defined

at their point of first use. a) The symbols for the elements. b) the latinates (i.e., e.g., etc.). c) at. wt., w/w, w/v, v/v, vol.

Layout. Consult the Format section for other details. If the report is double-spaced I can insert comments

easily, but single space is OK. I use 14pt exactly. Use one inch margins all round. That leaves some space

for comments. Use a fly page. Insert diagrams, graphs and tables into the text rather

than on separate pages. (Use cut and paste). Number the pages. Do not start sentences with numerals (use the word). Breaks and indenting for paragraphs are your choice. Number graphs, figures and tables clearly. You should use the equation editor for equations.

Calculations can be inserted into the text in hand writing though (they are a real pain to type).

Equations, Tables & Figures. Equation numbers should be at the left margin. The

equation should be centered. Refer to the equation as eq.n. Label figures and tables underneath. The caption should

not exceed the with of the figure or table and should be centered. Refer to them as Figure.n or Table.n.

References. Any consistent referencing scheme will do, but the following is recommended. References should preferably be numbers and grouped at the end of the report. Do not use MS Words Endnote facility. Placing references in footnotes is ok though. References should be referred to by just a number in parenthesis or an author and a number.

Journals. 1) For sequentially numbered volumes : Name, M. Y.; Other, A.N. , Journal Abbrv. Year, volume, pages. . 2) For individually number volumes: Same, M. Y. , Journal Abbrv. Year, volume(issue), pages. Books. 3) No editors: Name, M. Y.; Other, A.N. Title of the Book; Publisher: City, Year, Chap. or page refs. 4) Editor only: Title of the Book; Ed. Name; Publisher; City, Year, Chap. or page refs. 5) Author and editor: Author. A. N. In Title of the Book; Editor Name, Ed.; Publisher; City, Year, Chap. or page refs.

Web pages. You should not take reference material from the web as it is not peer reviewed and not a permanent medium. Do look around the page to see if it has a formal reference (most government pages will) or has a reference from which the material is taken. If not, note down the URL of the page and the date you obtained the information.

For reference to other types of materials see the A.C.S. style guide. (For which you have no reference, notice how irritating that is) its in the library). Abbreviations for journal are also in the A.C.S. guide.

Introduction:8

TYPE SETTING TECHNICAL DOCUMENTS Introduction. For hints on typesetting in general, see Robin Williams book The Non-Designers Design Book. She also has a number of useful tips at www.eyewire.com/magazine/columns/robin. Some more information can also be found under www.microsoft.com/typography/ . The hints below refer specifically to type setting lab. reports. or lab. manuals. Ive attempted to set up this page according to these hints, even if the rest of the manual is not setup that way (do as I say not as I do).

Emphasis and Titling. Do not use underline or ALL CAPS for emphasis or titling, they are a hangover from the typewriter days. However, I still find it useful to use a liberal sprinkling of DO NOTs in lab. manuals.

For emphasis in text use italic, bold italic or bold. Occasionally, using a small font surrounded by white space works well as does using a larger font. For titling use a larger font, normal, bold or italic. If your main text is serif (as it should be) then a larger Sans Serif font is good. Placing a line across the page (as above) also works well in moderation. The lines may be various weights or doubled.

Typeface. If you are preparing a long document readability is very important. In print readability is best achieved by using a classic serif font such as Times Roman, Caslon, Garamond, Baskerville or similar (This manuals text is set in a slightly narrow Roman font Adobe Minion). Dont use anything fancy. If you are type setting a Web page or have a lousy printer then a san-serif font such as Arial may work better. I avoid Arial because of the confusion between I (eye) l (el) and 1 (one). Check for appearance on screen vs. the printer, the fonts named above (except Minion) look quite different on screen than they do in this text. Also, check the numbers for a given font in Times you get 1234567890 with Bulmer you get 1234567890, which is a little ugly (especially on screen).

In general, one should not use more than two or three typefaces per page, typically one for body text, one for titles, one for symbols. One may add one more font to represent computer or instrument input, Courier is usually used for that purpose.

Font Size. As can be seen in the section above that different fonts have different widths, heights, weights and spread, even though they have the same point size. Times is large and open and Caslon small and cramped so Times works well at 10pt, Caslon does not. On the other hand Caslon works fine at 14pt, and so does Times. I use 12pt for lab. manuals since they are usually read standing up (and Im a little short sighted). For reports 10pt is ok if you have a laser printer, for inkjets 12pt is better.

Spacing. Spacing between lines is usually set to single (which is actually variable). For plain text this is ok, but is poor when youre using super/subscripts or symbols. I find spacing at exactly 14pt (for 12pt text) works well if the dont center exact height lines option in Word is off. If youre writing a draft, double or triple spacing can be useful to allow for annotations.

Spacing between words is set by using left justification (no extra spacing) or by using full justification (spacing filled so line exactly fits the line). Full justification is fine as long as you dont use narrow columns (a fine example of what can go wrong is in the opening paragraph).

Misc. Lines of text should not be more than (on average) eight-nine words long. For large format pages that means two or more columns. Large margins can also help. Setting up double columns can be a little awkward and slows down the computer. Some of the lab. manual is not set that way so I dont expect it in lab. reports.

Introduction:9

PLOTTING GRAPHS. Most graphics packages give half decent graphs, however Excels defaults settings are setup for Business slides (i.e. for appearance not content.) However, the plots can be readily customized. Below are a set of refinements which can also be used to as general guidelines for plotting.

The top graph is Excels default scatter plot. The scales are wrong, the annotation too large and the plot cluttered. It is almost useless. You should setup a custom graph in a proper format and use that as the default..

For the first step we get rid of the grid lines (only leave grid lines if you need to read data of the graph) and the gray background which reduces contrast. The legend is also deleted. Again, legends are useful when working off the graphs, but for reports the legend should be below the graph along with other information. Note how the graph rescales when the legend is removed. Here we have removed the remaining background color and the joining lines. There is no measured data between the points so we cant put a line there. We have also just changed the annotation to normal typeface. If you intend to reduce the figure you should scale the annotation accordingly. For reports annotation should match your text size (10-12pt). If you are making slides, overheads or reducing graphs, it will need to be bigger (18-24 pt often works). The title has been moved to the bottom and numbered. Most of these changes are accessible by left clicking on the graph, then right clicking to access the various properties.

The final step is to remove both figure and axis boxes, they serve no purpose. The title has been removed and replaced by text from Word because you cant do anything fancy in Excel - in this case simply to put the legend in. Note that it is centered below the graph. The axes have been tidied up, the span has been corrected, decimal places reduced and tick marks added. The least square fit lines have been added. The data points have been changed to black and made easily distinguishable.

These graphs have been cut and paste in. If you do that, make sure they are a decent size, including the annotation. (see exp.P for some bad examples. Graphs can be presented on a separate page if necessary.

Figure 1. A plot of Cv vs. Vent time for the Clement Desormes. Dots at 20C, squares at 30C

Example Graph

0.0020.0040.00

-4.00 1.00 6.00

Time(s)

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Example Graph

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Figure 1. Example Graph

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Introduction:10

PRESENTING TABLES.

Time(s) P1 (cmHg) Cv 4.03 76.49 28.04099 3.25 76.67 26.34766 3.32 76.57 26.65888 3.06 76.73 26.85892 2.6 76.63 26 2.28 76.52 25.06601 1.75 76.6 25.01056 1.31 76.58 24.84336 0.96 76.6 24.65 0.66 76.51 24.25748

Time(s)

P1 (cmHg) Cv

4.03 76.49 28.04 3.25 76.67 26.35 3.32 76.57 26.66 3.06 76.73 26.86 2.60 76.63 26.00 2.28 76.52 25.07 1.75 76.60 25.01 1.31 76.58 24.84 0.96 76.60 24.65 0.66 76.51 24.26

Vent

Time(s) Pressure, P1

(cmHg) Cv

J/mol-1 4.03 76.49 28.0 3.25 76.67 26.4 3.32 76.57 26.7 3.06 76.73 26.9 2.60 76.63 26.0 2.28 76.52 25.1 1.75 76.60 25.0 1.31 76.58 24.8 0.96 76.60 24.7 0.66 76.51 24.3

Table 2. Vent times, initial pressure, P1, and calculated specific heat, Cv , for the Clement-Desormes experiment.

Excels default tables are nearly as poor as the graphs. Here we will illustrate how to tidy them up.

Generally you should cut and paste in from Excel in-line to the text, but if the table is very large it should be presented on a separate page, in landscape mode if neccessary. Either way the default cut & paste table is poorly laid out with lots of clutter and white space. Center the text and the table. The number of decimals are made uniform to clean up the appearance and allow a more meaningful column alignment and spacing. The heavy grid has also been removed. The grids can serve a purpose in large multicolumn data look-up table, but generally they just clutter the table.

The final step is to tidy up the table headings (centered vertically and horizontally) and add subscripts. Use the equation editor if necessary. A caption has also been added along with a light grid. Number formatting has been changed to reflect the correct number of significant figure. You may want to change the font type and size to match your text as well. A final note: Try to avoid E or exponential notation if necessary rescale your data to M or whatever. e.g. use 0.234 not 2.34E-01 or 2.34x10-1 (the latter is the lesser of two evils). Use 0.345M, not 3.45x10-7M and certainly not 0.000000345M. Also remember to limit the number of decimal places, Excel defaults to something too large.

Introduction:11

USING FIGURESFigures are rarely prepared in situ they are usually prepared separately then cut and paste (manually or electronically) into the document. The only real exceptions are sketches in your lab. notebook. Regardless of the type of figure they all need a figure number and caption placed beneath it. The caption should be no wider than the figure. There are a number of ways of preparing figures for formal reports, each with advantages and disadvantages.

a) Hand drawn b) Drafted c) Computer drafted. d) Scanning.

Hand-drawn figures are not usually used in formal reports, but they are fast and convenient. Drafting involves special pens, templates, drawing boards etc. The usual approach is to draw an oversize figure and Xerox reduce it (that cleans it up a lot). You then manually cut and paste it into a prepared gap in the report and Xerox that page. This process is very labor intensive (the chem. dept at UBC has two graphic artists for this purpose), but it is the only way to do some types of pictures. Computer drafting moves all the drafting accessories onto the computer, however it can be difficult to do chemical drafting on the computer and the results are often unsatisfactory. It is most satisfactory if you have prepared templates of flasks etc. (CorelDraw has a good selection) or of rings and bonds (ChemDraw). Drawing pictures from scratch can be a real pain. Be sure to distinguish between CAD type programs (CorelDraw) used for drafting, and paint type programs (CorelPaint). The latter is good for pictures and tends to be poor for equipment figures. Computer figures can be printed out and manually cut and paste in or they can be electronically cut and paste in using OLE. OLE is the best way to go, you dont loose resolution, but it sucks up computer resources. Also Corel has never mastered OLE so you cant always use it. You can cut and paste in bitmap files from just about any program, but be careful as their resolution is limited (or they become so big they crash the computer). You need at least 600dpi resolution to reproduce line drawings properly.

Scanning is great when it works. You use somebody elses pictures and just cut and paste them electronically into you report. There are three problems though, one is copyright, they are, after all, somebody elses pictures. Resolution is another, and color is the last. Copyright is usually not a problem for one-off lab. reports, but for anything you sell (such as lab. manuals) you have to get copyright permission, which may of may not be free. Resolution is a serious problem. Grayscale pictures may give acceptable results at 300dpi (check by printing a rough copy out), but even then they will be huge (on disk). Line drawings need at least 600dpi or you get jaggies. Scaling down often exacerbates the problem. Color also creates a problem. Most reports are in black and white with perhaps grayscale pictures. Color does not always converts to grayscale well, and almost never convert to good line drawings (black and white). Basically any picture with a colored background will be unusable. As an exercise go through this manual and see if you can determine which figures are hand drafted and scanned, computer drafted, scanned from other sources, bitmaps from other programs or OLE links.

Introduction:12

KEEPING A LABORATORY NOTEBOOKBasically, you need to record what, with what, when, where, how and how long, all in relation to your laboratory activities. An outline of the information you should record (by category) is given below (you dont need to put such titles in the book). The laboratory book should have numbered duplicate pages. The duplicates should be removed and stored in a separate place from the laboratory book. If you have a laboratory, the laboratory book should be locked up in the laboratory. Keep the duplicates at home or in your office. Increasing amounts of data are kept on computers, be sure to backup the disks and record file names in your laboratory book. The lab. Manual contains some NOTES sheets, they are an artifact of production. They are useful as scratch sheets, but anything of significance should be written in the lab. notebook, only. Incidentally, the examples given are real, not made up for your entertainment.

General a) Each page should be dated and titled. Start each new

date on a new page, unless the entries are less than of a page. The title should be short, but descriptive and placed in an index at the front of the book.

b) Entries should be neat and in water insoluble pen. c) Entries should be organized start new topics on a new

line. d) Mistakes should be deleted with one line (leaving the

mistake clear it might not be a mistake). Do not delete or change entries after the fact.

e) Place any literature references against appropriate entries.

f) Note any modifications to any standard procedures. g) Keep your notes legible. Dalton may have been a great

chemist, but his lab. notes werent that great as seen below.

Pre-Laboratory Preparation. a) Write in any directions, calculations, diagrams, flow

charts, molecular weights, cautions, weights and volumes.

Instruments a) Record instrument settings (scan time, resolution, scan

width, temp. etc. etc. anything that can be varied). Anything that doesnt have a value, indicate any changes. E.g. lamp height was optimized for the standard sample.

b) State sample cells used (try and get your own). If they break record that (different cells give different results).

c) If there are any power glitches or the signal seems very noisy, indicate those along with the time. E.g. at UBC it used to be impossible to work at 4.30, when everybody was turning their instruments off. There was also trouble with big lasers firing; recording times will help you track these things down.

d) Record the computer file names that any data is held in. Make them useful, have a convention. If the OS supports long file names use them. I find it useful to use a convention like XXYYnnnn.mmm where XX is a two letter month code, YY is the day (number), nnnn is a code for the experiment/sample type, mmm is the extension, usually you wont have a choice about that. E.g. JL09dx12.spc might mean a spectrum (spc) taken July 09, of a doxyl-12 spin probe. Most computers are pretty good about keeping the file date correct so you may want to just use an 8-letter code instead. However, its often easier to search for a file name than for a file date. I like to use an extension nnn where nnn was 000, 001, 002 etc. for a sequence of files. It can be very difficult to locate three month old data, especially if you generate 100 files a day.

Preparation a) Record the chemicals, grades, weights and volumes

used. b) Record the order in which the chemicals were added. I

once did a preparation with three chemicals, which gives three possible orders (assuming the order of a given pair is irrelevant). Not only did two of the combinations give the wrong product, the reactions were explosive. The order was not given in the original instructions.

c) Draw, or otherwise indicate the equipment used. (Some reactions depend on the size and shape of the flasks). Something a little more auspicious that the efforts of the Greek alchemist shown below, but drafting is not required.

Introduction:13

d) If any of the bottles have old labels, broken caps, or discolored contents, record that. Some chemicals just dont work when old. Hydrogen peroxide is really bad for that, check the bottle date.

e) Record any changes to the preparation procedures. E.g. 200mL flask used instead of recommended 50mL flask.

f) Note the course of reaction and other details. E.g. 1) reaction proceeded smoothly and was complete in 10mins. 2) Reaction proceed rapidly and blew a bung out (replaced rapidly). 3) Some silicone grease dripped into the reaction vessel. 4) Reaction turned green after 30 min (instead of yellow at 45min.). Mixture detonated after 32min. (The student who noted the latter dropped chemistry and became a lawyer). 5) Initial preparation showed traces of NO2. Reaction vessel was flushed with nitrogen again to remove oxygen. 6) Foul smell, sample stored in fume hood.

Times Time is more important than you think. One preparation in my laboratory, at UBC, didnt work on sunny winter days, but worked fine on sunny summer days. This was because the sun would shine into the laboratory at 4pm, in winter, just when the silver salts were added to the reaction (it was an 8hr prep.). In the summer, the sun was to high to shine in. Record reaction times, flushing times, reflux times, start times, end times, just about any time you can think of.

Results a) Record basic physical data; total mass, state, % yield,

m.p.t. etc. b) Record any other data including program and version

number, if the result is produced by a computer. Be sure to record any file names.

Miscellany Record anything else you notice. e.g. 1) samples were yellow when removed from freezer, but turned a transient red

(almost a flash) then went pale yellow, the color before freezing. (In this case the pale yellow turned out to be iron contamination from the cheap stainless steel syringe needles used to deliver into reactants to the sample). 2) Fine white needles formed in the acetone wash bath after two days of use. (They turned out to be acetone peroxide, a contact explosive, even when wet).

Computers. REMEMBER TO BACK UP ALL YOUR COMPUTER DATA IMMEDIATELY. Computers generate huge amounts of information and its silly to reproduce it in a notebook. There are some exceptions, X-ray data and spectra typically find their way into special libraries. For your work, you can restrict yourself to the following items. a) Record filenames used in a given experiment. Record

any data that the instrument doesnt put into its files. b) If doing simulations setup a table of filenames, input

parameters and comments on success or whatever. c) Computers are finicky, record any difficult to remember

procedures (say for importing data) into your book. d) Paste in things like ASCII tables or program procedures

into the lab. book. e) Record the computer type and program version nos.

Bugs are often undetected for a long time, this data will help you decide their relevance.

f) Make sure you indicate the algorithms used for doing calculations, particularly if you used a computer.

INTRODUCTION:14 P.S.Phillips 2011 October 31, 2011

PLAGIARISM

You cheat, Youre dead meat.

For a detailed discussion of plagiarism see the OUC Guide to Plagiarism available from the bookstore. The policy is basically one of zero tolerance. The penalty for plagiarism is usually a zero for the work involved and a letter of reprimand. Penalties for theft of papers or repeated plagiarism may be expulsion. Avoiding plagiarism in chemistry labs. is quite simple; dont copy other peoples work ! That is, do not copy or read the whole or sections of another person laboratory report, draft or final.

There are cases where it is acceptable to copy other peoples work, in which case you should note the following:

a) If you wish to copy blocks of text from some source, put it in quotes and reference the source. You should not do this a lot or you will be penalized for poor style. The source of material must not be another students lab. report or from a tutoring service.

b) If you use any source to assist in writing your labs. make sure you reference those sources in your write up. If you paraphrase the text from these sources, try to do it as loosely as possible (i.e. keep it close to your personal style). If the paraphrasing is too similar to the original you may be open yourself to a plagiarism charge. I find it best to read two or more sources, close those sources and then write out what I understand of what Ive just read. Again, the sources must not be other students lab. reports, profs. notes etc.

c) You may discuss labs. with other students and you may compare original data and graphs. At a pinch, you can compare calculations to track down errors. Under no circumstances should you read or copy other students lab. reports (final or draft), that is considered plagiarism. If you read another students report (but not necessarily copy) it will taint the style of your write up. It is quite shocking how easy that is to detect, particularly is small classes so dont do it!

d) Do not copy other students graphs or tables either electronically or by Xeroxing. Producing your own graphs and tables is an important part of this course. (This may be acceptable in other courses, but not this one).

See the supplementary information on others marks penalties.

HOUSE KEEPING Before starting an experiment, wash, and if necessary dry, any glassware, before use. Remember, the last person to use the glassware was a student.

At the end of the experiment be sure to clean and wash any glassware. If you found it on the bench, in the cupboard, the draws, the outside and return it there. If it came from a stock cupboard , put it on the drying rack. Make sure you leave the balances clean. If any equipment was disassembled when you got it, disassemble it again and return to where you found it. Leave any equipment or chemicals that the instructor/technician gave you on the bench. Clean up any spills, paper towels, or any other mess you make. Report any breakages or depleted reagents so we can make replacements. Failure to do this housekeeping may result in marks being deducted.

P.S.Phillips October 31, 2011 ERROR ANALYSIS:1

INTRODUCTION TO ERROR ANALYSIS INTRODUCTION. There is a hierarchy of error analysis. Each step being successively more rigorous.

i) sigs figs ii) simple addition of errors ii) addition of squares of errors iv) series expansions to estimate errors v) calculus

Although each step is more rigorous, they are less conservative. i.e. sig. figs always overestimates the errors. Note that each case requires a prior estimate of the error. If you have to estimate the error from your data set (typical in the earth, life and social sciences) then you must resort to statistics. All quantitative chemistry experiments should incorporate a check using significant figures. For error analysis usually simple addition of errors will suffice, although one may have to resort to series expansion of functions to do that.

I. SHORT CUTS IN ERROR ANALYSIS: A NOTE ON SIGNIFICANT FIGURES. Error analysis is a pain so if we assume all the errors are +1 in the last reported figure of the data, then the error analysis is greatly simplified: One just counts significant figures or decimal places and propagates those according to the rules familiar to you from first year chemistry and physics. At this stage of life, you should do significant figures instinctively. However, the glowing display on your calculators seems to interfere with this process so here are a couple of rules to stop you going too far wrong.

1) Never, ever write down all 10-12 figures off the calculator, but feel free to carry that number in your calculations.

2) Computers/calculators often do not do calculations to more than 5-6 sig. figs. for functions Even though they may print out more.

Never quote errors to more than two sig. figs. One sig. fig. will often suffice. e.g. 56.74+1.32 becomes 57+1, 1032.4+1.3% becomes 1.03x103+1%. Also you must get the sig. figs. right. An error is a + quantity so follows the rules for addition/subtraction; that is the number of decimal places in your error must match those of the error.

3) If your using volumes in calculations you will never have more than four sig. figs. Usually three after subtractions.

4) If your using weights (in our labs.) you will never have more than six sig. figs. Usually only five after subtractions.

5) Atomic masses are often only good to three sig. figs. due to variations in isotopic composition.

6) If you worry about how many sig. figs. to carry in calculations, carry six, although four is often adequate. (In general, carry one more sig. fig. than you know the answer will generate).

7) Significant figures underestimate the true error so beware that reasonable significant figures may hide a gross error. If youre down to 2 sig. figs. you should consider doing an error analysis to see if the data actually means anything.

8) Remember sig. figs. only apply to simple math operations. If functions are involved you will have to resort to proper error analysis, at least for that step in the calculation. Watch polynomials in x, in particular, if x1 then the high order coefficients need more sig. figs. Both Excel and Origin have a bad habit of reporting polynomial coefficients with insufficient significant figures. Work out how to reset that.

9) One notable exception to sig. figs, of functions are logs. The leading number carries the exponent so has now error. Thus, the log, 3.13 has two sig. figs., the 3 just tells you to multiply by 1000.

10) Soooooo, unless you have a very good reason (e.g. you are measuring frequencies) never write numbers down to more than 6 sig. figs. In addition, you never write a final answer down to more than four sig. figs, in fact three will do in many cases. IN ADDITION, do the proper sig. figs, dont just blindly write down three or four sig. figs at each step and dont give errors more than two (preferably one) sig. fig.

A short cut, but one that requires a fair amount of competence and experience in error analysis, is to break the calculations down into small steps and try to identify where the largest error occurs (Actually its always a good idea to do this before the experiment then you know where to focus your attention). Often this error will be so much larger than all the other errors that you can ignore them, i.e. you only need propagate the one error. The most common example of this is where you mix volumes and weights in a calculation. Weights are 2-3 orders of magnitude more precise than volume so you often only need propagate the volume errors.

II. ERRORS. All measurements are associated with some kind of error or uncertainty. We never know exactly how big

ERROR ANALYSIS: 2 P.S.Phillips October 31, 2011

the error is, nor even, exactly, what range a quantity we are trying to measure lies in. Much error analysis consists of educated guesses. Statistical analysis and formal computational procedures can help, but to use them properly we must understand that they do not remove uncertainty, but merely let us express, more clearly, what the uncertainty is. For instance the statement

[Mn2+] = (1.56 +0.03) x 104 g/ml

does not mean that [Mn2+] lies, for sure, between 1.53 and 1.59 x 10-4 g/ml. It may mean several things, according to the convention used by the particular writer for the meaning of +0.03. To avoid ambiguity, state your convention along with the numerical result, for example,

[Mn2+] = (1.56+0.03) x 10-4 g/ml (95% conf. limits)

This means: "I think there are 19 chances out of 20 that [Mn2+] lies between 1.53 and 1.59 x 10-4 g/ml". The various conventions are discussed later.

III. SYSTEMATIC AND RANDOM ERRORS. Errors are usually divided into two categories. Systematic errors and random errors.

RANDOM error is just that, random - that is its origin and magnitude are unknown, but are equally likely to be +ve or ve and for a given measure, average to zero. (Note though in practice errors cannot add in a simple fashion and thus do not cancel to zero). The probability of a given error arising is usually assumed to follow a Gaussian distribution. Usually random errors are estimated from the standard deviation of a set repeat tests. This is discussed below..

SYSTEMATIC errors can arise in many ways, some of which may not come to mind very easily. For example, the volume of a given flask might not be exactly l00mL or the balance may not be exactly zero. Systematic errors occur in a fixed direction only (e.g. the flask is too big, or balance not zeroed) and for a given source do not average to zero, ever. We cope with systematic error by using common sense and scientific experience to list the likely sources of it and devise procedures to eliminate as many as possible. For what sources remain, we have to make educated guesses although they and will show up as apparent random errors in a statistical analysis. Experiments should always be checked with a known sample to help identify such errors, but often these errors are assumed negligible as an act of faith (aka from experience).

Suppose, for example, that one had a couple of dozen volumetric flasks, and took a different one off the shelf for each repeat of the experiment. Is the error in the volume of

the flask now systematic or random? This depends on what was going on in the factory where the flasks were made. The manufacturer may or may not have done something which made all the flasks consistently too big. One can only guess about this kind of thing unless one has a lot of information about how equipment was made and calibrated, right back to the primary standards of mass, length, time, temperature, etc. Fortunately, when experiments are being done to about three-figure accuracy, the expected range of error in calibrations of volumetric equipment, or of the weights built into analytical balances is usually very small in comparison to other sources of error (table 1). The purpose of the above example was not to get you locked in to worrying about volumes, but to encourage you to be wide-ranging in your thoughts about what could have gone wrong in your attempt to measure something quantitatively. A large part of an error analysis should consist of your assessment, in words, of the most likely sources of large error.

IV. ACCURACY, PRECISION, ACCEPTED VALUE, TRUE VALUE. One never knows the true value of any quantity; but for many quantities there is an "accepted value", which is the result obtained in the experiment, which is generally judged to have been the best performed to determine this quantity. Alternatively, the result may be compared to a standard an item that is defined as having an accepted value. The ACCURACY of any determination means the closeness with which the determination matches the accepted value, or, the closeness you think it has to the true value. This may again be expressed in terms of "19 chances out of 20...", etc., but it includes the effects of both random and systematic errors. The PRECISION, also called reproducibility, relates only to random errors. The better the precision the lower the random error (and, probably, the better the experimenter).

Process Error Total titer by burette (50mL) +0.03mL Total titer by burette (10ml) +0.01mL End point detection +0.01mL Volumetric flask (50mL) +0.02mL Volumetric flask (100mL) +0.08mL Volumetric flask (200mL) +0.10mL Volumetric flask (250mL) +0.10mL Pipetting (10mL) +0.01mL Pipetting (25mL) +0.02mL Weighing on an analytical balance +0.0002g

Table 1. Examples of Reasonable Error Limits


V. USING OF STANDARD DEVIATION AS THE ERROR. The most common way of characterizing random error in a data set is the standard deviation. The true standard deviation for a population is of n observations of a variable x is

22 1lim ( )x i xn

xn

s m

= - where x is the true or population mean. This is sometimes written

22 1 ( )x ix xNs = -

where its understood that N (as opposed to n) is the true population size and not the sample set size, and x is the mean (equals the population mean as N is the population size). In practice, x is unknown and is calculated for the data; the limit on n ensures that x approaches x, the true (population) mean. If n is small, then the standard deviation and average are no longer independent as, given n-1 data points and x , the nth data point can be calculated, we have lost a degree of freedom. To account for this we must reduce the sample size by one so

22

1

( )

1

ni

xi

x xsd

n=

-=-

Here n is the sample size, x is the mean of the sample set (an estimate of x) and sd the standard deviation (an estimate of x). If n is large and we know x then any difference between x and x can be attributed to a systematic error. If n is small then x is inherently unreliable. We can estimate the reliability of the mean by using the standard error on the mean, m, which is given by

m = / n

m tends to zero for large n. i.e. we can thus improve our estimate of the average by collecting more data points. Do not confuse the standard error on the mean with the standard deviation. The former is a measure of the error of estimating of the mean (i.e. accounts for the fact that the sample size and population size differ); the latter is a measure of the spread (or error) of the data

CALCULATING STANDARD DEVIATION. For the purpose of calculation, its convenient to make the following assignments

2( )xx iS x x= - 1xx

xS

Vn

=-

and x xsd V=

However, its inconvenient to calculate Sxx using the above formula so its often calculated using the mathematically equivalent formula

( )22 1xx i iS x xn= - While this is algebraically equivalent, its not computationally equivalent. The former method requires lots of storage memory so the latter method tends to be used on calculators. However, the latter method is subject to round-off errors so you can get quite wildly different vales for the standard deviation with different calculators so be warned. An interesting variation in calculating the standard deviation is as follows

( )1

221

1

1

2( 1)

n

i ixi

x xsdn

-)

=-=

-

In the absence of systematic errors (such as drift or rolling baseline), this reduces to the normal value for the sd. However, in the presence of drift (low frequency noise) it gives the sd. associated with high frequency noise. (i.e. it is robust wrt. to drift). This is useful for identifying noise in the presence of drift. Furthermore, this is a running measure, that is, it can be easily calculated as the data is acquired. The usual sd. requires you to constantly recalculate the mean.

CONFIDENCE LIMITS. If the data is normally distributed, then the standard deviation is just the half-width of the normal curve at half-height. While this is a convenient measure of the normal curve, much of the data lays in the tails of the curve. Roughly, one in three points lay outside the +1sd limit of the curve or to put it another way, theres only a 2 in 3 chance that the true mean lies within the quoted limits. For some reason, this bothers some people and they prefer to use wider error limits, for instance +2sd. That is, limits such there are 19 chances out of 20 that the true value of the mean lies within the limits. Occasionally you will see +3sd, or 99 chances out of hundred. Because of this, it is necessary to state the so-called confidence limits for your error so

3.5+1.0 (68% confidence limits)

means 3.5 with a standard deviation of 1.0






For our purposes, we will just use first method, i.e. one standard deviation and the limits not explicitly stated. A more rigorous approach is recognize that for small data sets the errors are not normally distributed, they are distributed according to the Student-t distribution. The confidence limits should then be written cd xx t sd where

cdt is the t

value for a confidence limit of c (0.1 for 90%, 0.05 for 95% etc.) and d degrees of freedom (number of data points 2, which you should quote in your results).

OUTLIERS. Sometimes you obtain data thats way outside the expected range of values. Such data points are called outliers. There are three reasons for spurious data i) It s real: While the probability of a data point beyond the five s.d. limit is small, it is still finite so occasionally you can expect an odd point. ii) It is real: The data distribution is not normal. For example, the data may be distributed with a Poisson or Lorentzian distribution, both of which have long tails. Under such circumstances, outliers can be quite common. iii) It is due an error, e.g. contamination, power bumps. The last case is the most common cause of outliers and for that reason outliers are often just thrown out. The problem is that it is not always clear that the data is a glitch or that it is even an outlier. For highly scattered data, one should always do an outlier test before rejecting a point. Also, remember that it may be real data. Although it may be necessary to throw the point out to keep your statistical analysis valid, that data point may contain information. For instance in soil sampling that point may represent an ore deposit (in fact ore deposit are detected by looking for outliers). In medicine, the data point may be due to a diseased specimen, in which case your analysis may provide a diagnostic for that disease.

My personal preference is to leave outliers in and use robust estimation methods, which are, by definition, insensitive to outliers. If your analysis methods are automated in any way you should use robust methods, because it is very difficult to automate outlier rejection in noisy data.

VI. ERROR ANALYSIS. Although the following is a more rigorous approach to error analysis than you will see in first year chemistry texts and will suffice for our needs, you should be aware that there are still a number of approximations made. Briefly, they are as follows: It is assumed that the errors are random and distributed in a normal fashion (this will usually be true if outliers are rejected). It is assumed that the errors are small ~1% (and certainly


The + implies that the errors could mysteriously cancel out which is not the case so its usual to ignore the signs of the errors and write

y xw

y x

d dd )

This overestimates the error, but will be dealt with more rigorously below. For more complex functions we need to incorporate the derivative, this will also be dealt with below. For basic error propagation the formula below suffice.

Operations and Errors Addition or Subtraction Z = A + B + C+.

Z =| A| + |B| + |C|+.

Multiplication or Division Z = A.B.C.

|Z/Z| = |A/A| + |B/B| + |C/C|+.

General Z = f(A,B,C.)

|Z| = |A.df/dA| + |B.df/dB| + |C.df/dC|+.

ERROR PROPAGATION FOR MULTIDIMENSIONAL FUNCTIONS. We could proceed as above, but it slightly clearer to use the chain rule. To calculate the effect of a small error we need to examine the effect of the error on (i.e. small changes in) the function, which is the same questions as what is the size of the derivative of the function with respect to each of its variables. For a function w=f(x,y,z) we have that

, ,,

( , , ) ( , , ) ( , , )

y z y xx z

f x y z f x y z f x y zdw dx dy dz

x y z

= ) )

If the errors are small, we can approximate dx to x etc. This is equivalent to dropping the high order terms in the Taylor expansion. Its thus straightforward to get the error for such function. Also, if we recognize that the product, xy, is a multidimensional function, f(x,y), its easy to see how to get the error for a product.

VII. USING THE STANDARD DEVIATION AS THE ERROR. The error is not a simple number, it can have any number of values, it is in fact a characteristic parameter for a probability distribution of possible errors, it is not the error itself. Assuming the errors are normally distributed and that sufficient data has been collected, the standard deviation of the data will be width of the normal distribution. Since the errors for each term maybe +ve or -

ve they will tend to cancel, but not completely, as that would require all the errors to have their signs anti-correlated, not a very likely event. To take this into account we proceed as follows: Consider that we want to determine quantity w, that is a function of at least two variables, so that w=f(x,y..). We can determine a value of wi for each measured set of values so that wi=f(xi,yi,). Now we wish to get the standard deviation of w

22 1 ( )w iw wNs = -

xi-x is the error for a given measurement so assuming its small we can expand the error in w via the chain rule so that

( ) ( )i i iw w

w w x x y yx y

- - ) - )

hence 2

2 1 ( ) ( )w i iw w

x x y yN x y

s - ) - )

222 21 ( ) ( )

2( )( )

i i

i i

w wx x y y

N x y

w wx x y y

x y

= - ) - ) - - )

If we now define a covariance between x and y, xy, as

2 1 ( )( )xy i ix x y yNs - -

so we get for the error propagation function

222 2 2

2 2

w x y

xy

w w

x y

w w

x y

s s s

s

) ) ) )

If x and y are uncorrelated then we expect that, on average, the covariance will be zero (there will be an equal number of +ve and ve values in the sum) and we may neglect the last term giving us

222 2 2w x y

w w

x ys s s

) )

Which for the simple case of the product of x and y (w = xy) this reduces to

2 2 2w x ys s s) )

i.e. we sum the squares of the errors, to get the total error.


We thus need to modify the formula above to account for the fact that the errors do not add linearly.

If x and y are correlated, as happens when we do linear regression, then we must consider the covariance. This is discussed in the section below.

ERROR PROPAGATION FOR SOME SPECIFIC CASES. As an exercise, you should derive these formulae. The formula are for a function Z=f(x,y), which a and b are constants. In most cases, the covariance term can be ignored.

Other functions can be dealt with by using the error propagation formula above, but another approach is to expand the functions directly as a series and work from there. e.g. ln(x+x)=ln(x)+ln(1+x/x)~ln(x)+x/x as ln(1+x) = x+.. etc. so for small x the error on ln(x) is +x/x. Similarly, using the binomial expansion, the error on 1/x is also +x/x. Ill leave it as an exercise for the student to show that the error on the nth root is +x/n, i.e. it gets less!. Conversely it rises rapidly with powers of n.

VIII. ERRORS IN LINEAR REGRESSION FOR STRAIGHT LINES. Often, student data is so bad that do a proper error analysis of a linear regression is overkill. Simpler procedures for extracting the data and their errors will suffice. That is discussed here; a full error analysis is given below.

When a graph of y versus x should be a straight line, it is common to calculate its slope m and intercept b in the equation y = mx + b statistically, by the method of least squares. This is a good procedure, but it has some pitfalls, which should always be borne in mind: -

1) Always draw the graph, for two reasons: a) Plotting the least squares line on the graph is a good check against numerical error in putting data into the calculator. The line

should clearly be a very good fit to the points. b) The words "should be a straight line" in the first sentence above don't really mean very much. You may find that the plotted points

clearly fit a curve, and that you shouldn't be looking for a straight line.

2) In some experiments, only part of the data fit a straight line, and there is curvature in some other region. In this case, you make a visual choice from your graph of which points to take into account in drawing the line. There is then no point in using anything other than a visual procedure for drawing the best line.

3) The usual method of doing the least squares calculation assumes no error in x, and random scatter in y. For example, if you are constructing a calibration curve of spectroscopic absorbance against concentration of a colored solute, the absorbance commonly has errors amounting to at least +1%, which the standard solutions can be made up to an accuracy of 0.1% or better. In a plot of absorbance as y versus concentration as x, the usual assumption is legitimate. Hence: a) Always tabulate as y the quantity which has the larger expected random scatter. b) If both x and y have large scatter, recognize that a more complicated statistical analysis is called for. This will not be done in these laboratories.

SLOPE AND INTERPOLATION ERRORS. In the instrument room, computer programs are available which, in addition to giving the slope of a line, will give its standard deviation. Treat this just like the standard deviation of a mean. There is a 68% chance that the slope lies within + one standard deviation of the calculated value. Double those limits for 95% confidence. Often, having calculated the line, one wants to take a particular y value (e.g. absorbance of an unknown solution) and read the corresponding x (concentration) from the graph. The

Operation Z=f(x,y) Error in Z 2zs Relative Error in Z Simple sums and differences

x+a 2xs / /z xz xs s= Weighted sums and differences

ax+by 2 2 2 2 22x y xya b abs s s) -

Multiplication +axy 2 2 2( ) ( ) 2 ( )x y xyay ax xy as s s) ) 2 2 2 2( / ) ( / ) ( / ) 2 /z x y xyz x y xys s s s= ) )

Division +ax/y 2 2 2( ) ( ) 2 ( )x y xyay ax xy as s s) - 2 2 2( / ) ( / ) ( / ) 2 /z x y xyz x y xys s s s= ) -

Powers ax+b - / /z xz xs s= Exponentials ae+bx - / /z xz b xs s= Logarithms aln(bx) 2( / )xa xs -

Table 3. Error propagation formula. It is common to ignore the covariance term if x and y are uncorrelated.


programs again give an error limit, which should be treated as standard deviation and doubled for 95% confidence.

If there are large errors or outliers in the data, it is better to use a robust linear regression (e.g. a least median squares). However, such programs are not widely available. Also, if there is significant error in both x and y, its often simpler to abandon statistical procedures. Instead plot the graph, draw around each point a rectangle representing error limits of both x and y, draw the best line through the points, then draw two other lines, passing through the boxes with maximum and minimum slope. If the line can be clearly drawn through all boxes except one, and cannot be adjusted to pass through that one, then there is some error in that one point which has not been taken into account in drawing the box. Such a point is called an outlier. (See figure.1)

mmax ~-0.70, mmin ~ -0.36, and m~-0.50 uncertainty in m = (0.70 - 0.36)/2 = 0.17

so m = -0.45+0.17, similarly b = 5.0+0.8 Figure 1. Choosing a best line fit and the error.

RIGOROUS ESTIMATION OF ERRORS IN AN L.S.F . A rigorous derivation of an l.s.f. is tedious so we will focus on some of the pitfalls and the correct way to estimate errors. Most l.s.f.s for a fit of y to x make the following assumptions; a) The errors are independent of x and y. b) The errors are all the same. (You can circumvent this by using a weighted linear regression). c) The data contains no outliers. (You can partially circumvent this by using least median squares.) d) The x data contains no errors. (There are routines for allowing for this, but they are difficult to find.) e) The errors are distributed normally. (Outliers violate this, but some errors e.g. counting statistics are distributed as a Poisson curve. There are routine to allow for this, but they are difficult to find.) f) Most importantly, the underlying data is in fact linear!

Given that these criteria are met then we need to worry about how good the fit is, how to compare the fitted parameters with other data sets and what the errors are for values calculated from the parameters

HOW GOOD IS AN L.S.F. The usual criterion is to look at how close the regression coefficient, r, is to one. Regression coefficients are not only misleading they are evil. DO NOT USE THEM. The more normal approach is to use the chi-square parameter, but for our purposes that is usually unnecessary and will not be discussed here. Looking at the errors on the fitted parameters is usually adequate. However, if the data is far from the origin the error on the intercept will be large and one may need to do a chi-square test to reassure oneself that all is ok., although that wont change the fact that the error really is large. Generally, one is more interested in the errors on values calculated from the parameters. (See below). COMPARING FITTED PARAMETERS. Often we need to know whether two fits give lines with the same slope or intercept. We will not discuss that further here other than to say that a t-test on the parameters will usually work.

ERRORS ON CALCULATED VALUES. To properly calculate the errors for values calculated for the parameters one needs the covariance so one should be sure to use a program that returns this value. For a linear fit, y = ax+b, derived from data points each with an individual error of i, the errors on the parameters a and b, a and b respectively, and the covariance of a and b, 2ab are given by

2

2

2

/

/

/

a xx

b

ab x

S

S

S

s

s

s

= D

= D

=- D

where 2( )xx xSS SD= -

and

2

2 2 21 1 1

1

n n ni i

xx xi i ii i i

x xS S S

s s s= = == = =

If all the i are the same and equal , this simplifies to 2

2

2

/

/

/

a xx

b

ab x

S

S

S

s

s

s

= D = D =- D

where


2

2( )xx xnS S

s -D =

and

2

1 1

n n

xx i x ii i

S x S x= =

= =

The error may be estimate from the scatter of the y values, i.e.

22 1 ( )2 i

y yn

s = --

where y is the fitted y value i.e.

i iy ax b= )

The n-2 arises because we use a and b to estimate y . Some programs simply assume is one, in which case the error estimates on a and b (if given) are highly suspect.

The various sums are often provided by the program along with other data. However, what we want is the error in a

predicted y for a given x. Normally we would simply sum the errors for a and b, i.e. 2 2 2( )y a bx = + , however, in this case we must include the covariance la Table 3 as a and b are derived from the same data set so are correlated (or more explicitly x and y are correlated by the definition of a straight line). The desired error is thus

2 2 2 2( ) 2 ( )y a b abx ab xs s s s= ) ) You should note that the covariance term is generally negative so it may partially cancel the other terms. This is expected as a= (y-b)/x so any shift in a, will also shift b. However, y is usually our measured parameter and we are more interested in the spread of the corresponding x value, in which case the error can be deduced by application of the formula in table 3.

2 2 2 2 2( ) ( ) ( ) 2x a a b aby b a abs s s s s= ) ) -

A more common approach is to use confidence intervals for the fit. This is discussed in your anal. chem. course.


Table 1. Examples of reasonable error limits for various pieces of equipment (check manufacturers manuals for exact numbers). Be sure to distinguish between manufacturer tolerances and precision of measure. Manufacturer tolerances are the accuracy of the calibration of the equipment, which translate to a precision when mixing equipment. It is irrelevant if calibrations are done and standard solutions are used.

If you do repeated measures with a single item of equipment your precision should approach the manufacturers precision, but in general will be more.

Measure Error Measure Error Total titer by burette (50mL) +0.03mL pH meter +0.1 pH units

Total titer by burette (10ml) +0.01mL Thermometer (0.1C graduation) +0.02C

End point detection +0.01mL Thermometer (1C graduation) +0.1C

Volumetric flask (1mL) +0.01mL Thermometer (bomb) +0.01C

Volumetric flask (2mL) +0.015mL Thermometer (digital-absolute) +0.1C-1.0C

Volumetric flask (5mL) +0.02mL Thermometer (digital-relative) +0.001C

Volumetric flask (10mL) +0.02mL Thermometer (thermocouple) +0.01C

Volumetric flask (25mL) +0.03mL Time interval (manual) +0.1s

Volumetric flask (50mL) +0.05mL Time interval (computer) +0.01s

Volumetric flask (100mL) +0.08mL Weighing on a top-loader balance +1 in last figure

Volumetric flask (200mL) +0.10mL Weighing on an analytical balance +0.0002g

Volumetric flask (250mL) +0.12mL Viscometer (Canon-Fenske) +0.2%

Volumetric flask (500mL) +0.20mL UV spectrometer wavelength +1nm

Volumetric flask (1000mL) +0.30mL UV spectrometer absorbance +0.003A or 1%

Volumetric flask (2000mL) +0.50mL Pipetting (micro pipets) all +1% Pipetting (GC syringe) all +1%

Pipetting (adjustable-typical) +0.5% Pipetting (1mL max., calibrated) +0.03mL Pipetting (5mL max., calibrated) +0.05mL

Pipetting (10mL max., calibrated) +0.07mL Pipetting (1mL) +0.008mL

Pipetting (2mL) +0.01mL




Pipetting (10mL) +0.02mL Pipetting (15mL) +0.025mL Pipetting (20mL) +0.03mL Pipetting (25mL) +0.03mL Pipetting (50mL) +0.05mL



NOTES

P.S.Phillips 31/10/2011 Exp. E. Enzyme Kinetics:1

Exp. E. ENZYME CATALYZED REACTION KINETICS INTRODUCTION. Many chemical reactions are very slow at ambient temperature, usually because they have high activation energies. One way we can increase the speed of a reaction is to carry it out at higher temperatures. One serious drawback to this approach, however, is that, aside from intrinsic experimental difficulties that might arise, undesirable competing reactions (such as decomposition) also begin to take place faster at higher temperatures. These complications may also decrease the efficiency of the desired reaction.

Increasing the temperature is obviously not an option for biological reactions under physiological (native) conditions; ~37C and pH 7. However, many reactions that are ordinarily very slow at this temperature are nevertheless found to occur very rapidly in natural systems due to the action of catalysts. A catalyst is an agent that increases the rate of a reaction by effecting a decrease in the activation energy of the process. One example of a catalyzed reaction is the breaking up, or digestion, of a protein into amino acid residues. A protein is a macromolecule consisting of many, often hundreds of, amino acids linked together by peptide, or amide, bonds. In order to be useful to an organism, a protein molecule must be broken down into its amino acid constituents for further, specific polypeptide synthesis. A particular example is -chymotrypsin, which cleaves polypeptides (or esters) at points adjacent to aromatic groups (see reaction below). This enzyme has the particular advantage that its own aromatic groups are buried deep in the structure so it does not digest itself, which would complicate interpretation of experiment.

A molecule that catalyses a biologically relevant reaction is called an enzyme. Enzymes are, themselves, usually large polypeptides having high molecular weights (ca.104-106 g/mol). They are remarkable because of their catalytic power and specificity; only certain, narrowly characterized reactions are catalyzed by a given enzyme, and often only under specific conditions (e.g., ionic strength, pH, and temperature). In some single-celled organisms, as many as 3000 different enzymes can be found. Generally, an enzyme derives its catalytic and reaction-specific properties because of a unique structural property called an active site. The shape of the active site, along with other structural features, orients molecules onto (or into) the site, enabling the reaction to take place. The active site often contains groups that affect the reactant molecule's conformation, thereby speeding up the reaction. In essence, an enzyme can catalyze a reaction by stabilizing the transition state and

may be compared with the catalytic site of specially prepared solid surfaces. In general, the rate of a reaction is equal to the sum of the intrinsic (or uncatalyzed) and the catalyzed rates. This assumes that the two processes occur independently of each other. Thus the observed rate constant, kobs is

kobs = ko + kcat where ko and kcat are the rate constants of the uncatalyzed and catalyzed processes, respectively.

If the molecular structure of an enzyme, i.e., the conformation that creates the active site, is changed (even subtly), catalytic activity is diminished or even lost. This denaturation of the enzyme can be brought about by subjecting it to extremes in temperature, pH, or ionic strength, i.e., non-physiological conditions. Sometimes an enzyme has more than one binding site and binding of an inappropriate species to the second site will distort and deactivate the first site (However, in some cases binding at the second site is requires to activate the first site). Sometimes denaturation is reversible, and enzymatic activity can be restored by returning the system to appropriate conditions. In other instances, enzymatic activity can be permanently lost.

In this experiment, we will study the hydrolysis of an ester catalyzed by the enzyme -chymotrypsin. The reaction is studied in buffered aqueous solution and is an example of homogeneous catalysis because it is a one-phase system; all components are soluble. -chymotrypsin is a well-characterized enzyme (denoted hereafter by E). It is a protein having a molecular weight of 24,800 D and is known to have one active site per molecule. In fact, the determination of this enzyme's purity is based on the fact that it reacts stoichiometrically with certain esters. It is also a particularly convenient enzyme to study because it is inexpensive and can be obtained in high purity; it is generally isolated from bovine pancreas extract.

The reactant that undergoes conversion to a product (or products) in an enzyme-catalyzed reaction is called the substrate. In this experiment, the substrate that reacts with -chymotrypsin is an ester of 4-nitrophenol, specifically, 4-nitrophenyl trimethylacetate (S). The overall reaction can be expressed as follows:

E + S + H2O [ P1 + P2 + E where P1 and P2 are the two products of the ester hydrolysis, in this case, 4-nitrophenol and trimethyl-acetic acid, respectively.

Exp. E. Enzyme Kinetics:2 P.S.Phillips 31/10/2011

Because the reaction will be carried out at a pH 8.5, both products exist in their conjugate base forms, trimethylacetate ion and 4-nitrophenoxide ion. In principle, any ester can be used to study -chymotrypsin catalyzed hydrolysis. By using 4-nitrophenyl trimethylacetate, however, we exploit the facts that a) the bulky alkyl group slows the reaction down to a convenient time scale for simple real-time analysis, and b) the aromatic product absorbs visible light (it has a yellow appearance). The latter property allows the course of reaction to be followed spectrophotometrically. All other species involved in this reaction are colorless.

The kinetics of this reaction are particularly interesting because when the enzyme and substrate are combined, there is an initial "burst" of nitrophenol (P1 formation, which is then followed by a gradual increase in P2 concentration. In an attempt to explain this behavior, we really need to consider several mechanisms of enzyme kinetics.

Mechanism I. The simplest scheme (the so called Michaelis-Menten mechanism) involves the reversible binding of the enzyme to the substrate, followed by the dissociation of this complex to form product:

E + S ES P + E-1

1 2

k

k k] [

This is a three-parameter mechanism (rate constants k1, k-1, and k2). Because we intend to follow the time evolution of product, the relevant differential equation is

d

dtk

[P]= 2[ES] (1)

We must obtain a relationship between the measurable quantity, [P], and [ES]. To do this, we first write the material balance for enzyme, i.e.,

[E] =[E] + [ES]o (2)

where [E]o is the total (or bulk) enzyme molarity. Equation (2) states that the enzyme must be present either in i

advanced physical

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