GENERAL INSTRUCTIONS FOR WRITING THE LABORATORY REPORTS

Transcription

1 AALTO UIVERSITY School of Chemcal Technology Department of Chemstry Physcal Chemstry Laboratory GEERAL ISTRUCTIOS FOR WRITIG THE LABORATORY REPORTS 06

2 Aalto Unversty School of Chemcal Technology Department of Chemstry Physcal Chemstry Laboratory TABLE OF COTETS Structure of the Report... 3 The Format of the Report General Unts and Equatons Fgures and Tables Graphs Attachments Data Handlng and Error Estmatons Error Types Mttauksssa esntyvät vrheet Estmaton of the Error of Measured Values Error Estmaton for Calculated Quanttes Calculatng Error Estmatons by Total Dfferental Calculaton of Error Estmaton by Logarthmc Dervatve Studyng the Dependence between the Quanttes Least Squares Method (LSM) Makng References... 5 References... ATTACHMETS Attachment. Laboratory notebook

3 Structure of the Report ame Page The frst page of the report s the name page whch s flled approprately. The name pages can be downloaded from the webpages of the course or they can be collected from the laboratory. Introducton The background of the lab work, the work tself as well as the purpose of the lab work are brefly ntroduced n the ntroducton (max. page). Ths secton does not nclude any equatons. Theory The most approprate equatons -and especally the ones that are used later n the report- are shown n ths secton. There must also be short explanatons of the equatons. The mportant equatons wth respect to the experments and results are derved from generally known theory. More detaled dervatons of the equatons, whch are shown n work nstructons or n books, are not demanded. It s enough to gve the approprate equatons wth references. Expermental Equpment and Methods The equpment, expermental methods and materals used are ntroduced n ths secton. Results All the results from the laboratory notebook are collected and shown, for example n the form of a table. All the calculatons ncludng the ntermedate steps whch are needed to reach the fnal result are shown. Addtonally, an example calculaton wth proper values and unts must be shown for every equaton.

4 4 When usng an equaton, the Theory secton must be referred to or the orgn of the equaton has to be shown by other means. All the results (and possble ntermedate results) are shown, for example n tables and/or fgures. Error Estmaton An estmaton of the relablty of the results s added to all the results and the bggest sources of errors must be clearly explaned. Dscusson and Conclusons The most mportant results wth ther error lmts are shown n ths secton. Also, the results are compared to the lterature values f avalable. All the lterature values must be referenced n the usual way. Results are dscussed: Are the results n good accordance wth lterature values? If not, what s the reason? Answers to the Questons The answers to the homework questons are shown n ths secton. References The references are lsted n the order of appearance n the scrpt or n alphabetcal order. All the lsted references must be mentoned n the report text. Attachments All the addtonal pages are added as attachments: the page of orgnal laboratory notebook, prnted data sheets from the computer, graphs from the graphc plotter, etc. Correctons All the correctons for the report are made on separate pages and attached to the back of the orgnal report by staplng.

5 5 The Format of the Report. General The report s prnted sngle sded on A4 paper. Text s wrtten n a sngle column wth lne spacng.5 and the text s justfed. The left margn s 3.5cm and all the other margns are 3 cm. Page numbers are placed n the centre ether at the top or bottom of the page. The name of the lab work s wrtten n font sze 6, bold, usng captal letters and t s centred. Man headngs are wrtten n font sze 4 and bold, the subheadngs are wrtten n font sze and bold, the text tself s font sze. The fonts of fgure and table captons are dentcal to the body text. Before all the headngs, there are two empty lnes. Between the paragraphs there s an empty lne and text starts from the begnnng of the row. The readablty of the text s mportant. Text must be short, concse and grammatcally correct. All the sentences must be whole sentences. The passve and mperfect tenses are always used n the text. SI unts are used whenever possble. It s not permssble to use the format 5E-6m, nether n the text nor n tables, the formats m or 5 mm should be used. Decmal ponts are used n numbers and they are rounded to reasonable levels, dependng on the errors n the measurements.. Unts and Equatons Equatons and reactons are numbered by contnuous numberng and a number s placed n parentheses on the rght sde of the page. Equatons are wrtten by equaton edtor (.e. usng approprate symbols, not letters).

6 6 Equatons must be explaned n the text pror to ncluson. The number of the equaton s separated n text usng the parenthess,.e. Equaton (). Equatons can be shown ether as equatons between quanttes or dmensonless equatons wth quanttes. Dmensonless equatons wth quanttes must be used always when the equaton s not homogenc wth dmensons,.e. the unts are not cancellng themselves out, as shown for example n Equaton (). An example of the equaton between quanttes: q P nc P, m DT () where q P n heat transferred molarty C P,m molar heat capacty DT temperature dfference An example of the dmensonless equaton wth quanttes: C P, m H,g) 3 T 7 9,064-0, , ( T () J K mol K K where C H,g) the molar heat capacty of hydrogen gas at constant pressure P,m ( T temperature An example calculaton must be shown for each equaton frst wth measured or other values and dmensons. The workngs of the calculaton have to be shown n such a way that the fnal value wth the unts can be checked and corrected when necessary. When usng a lterature value, a reference must be mentoned. An example calculaton:

7 7 Equaton () gves q P kj 75,3 555mol K mol ( 33-83) K 670 kj.3 Fgures and Tables There s a separate numberng for fgures and tables throughout the text. All the fgures and tables must be referred to n the text before showng them for the frst tme. Fgure and table captons have to be understood wthout the report text. The font of the fgure or table capton s exactly the same as the body text. The fgure capton s placed under the fgure and the table capton s placed above the table. The ttle and unts for every row and column must be mentoned n the tables. The rows must be vertcally algned wth respect to a decmal pont and too many dgts should be avoded. An example of a fgure: Fgure. Couplng of calormeter for the determnaton of heat capacty. V s a voltmeter, A an ammeter and E the voltage source.

8 8 An example of a table (accordng to IUPAC [] recommendatons): Table. The dependence of n-propanol vapour pressure P on temperature T. Experment T / K 0 3 K / T P / kpa ln (P/ kpa) 333,5 3,007 9,6,98 343,5,94 3,3 3, ,5,837 50, 3,9.4 Graphs Fgure numbers must be shown for all the graphs and the fgure capton s placed under the graph. Measured or calculated data ponts (x, y) are llustrated graphcally as shown n Fgure. The error lmts Dx and/or Dy are added to the data ponts by horzontal and/or vertcal lnes for every data pont (x, y) f needed. 30 lämpötla/k aka/mn Fgure. The ncrease of temperature as a functon of tme n experment XX.

9 9 The axes and the scale of the graph are chosen approprately. An approprate scale s such that the graph flls the paper as wdely as possble and for example, n the case of a straght lne, the plot s approxmately at angle of 45 when compared to the X axs. o background colour or a box around the fgure s allowed. Axes are labelled and the unts are also shown. When addng data ponts to the graph, each data pont must be presented clearly. All the data ponts must be clearly vsble after drawng the graph. If there are several data ponts at dfferent condtons, one must use clear markers whch are easly dstngushable from each other. Good markers are for example crcles, trangles, squares, etc., as shown n Fgure höyrynpane/kpa lämpötla/k Fgure 3. The dependence of the vapour pressure of a pure substance wth temperature. dethyl ether acetone

10 0 The curve whch llustrates the best relaton between the values s shown n graphs. When fttng the curve, obvous error ponts can be excluded from the fttng but these data ponts must be stll clearly seen n the graph. As measured results always contan some error, data ponts rarely st exactly on the ftted curve. Thus, data ponts are not allowed to be connected to each other wth a fracton lne. If data ponts are ftted to gve a straght lne, t s a good dea to draw error lnes nto the graph (broken lnes n Fgure ). Error lnes cross each other at the focal pont of the set of data ponts. If the measurements are performed at regular ntervals and the error lmts are approxmately the same, the focal pont s more or less n the centre of all the data ponts. It s also mportant to remember that the error lnes should be wthn the error lmts of each of the data ponts as often as possble. Error lnes can also be drawn on the fgure by hand..5 Attachments Attachments are added to the end of the report. Page numbers do not contnue to the attachments but every attachment has ts own number and ttle. Attachment number s wrtten on the top rght corner usng captal letters (ATTACHMET, ATTACHMET, etc.). If one attachment contans several pages, t can be shown lke the followng: ATTACHMET (/3). The note n parenthess ndcates the frst page of three pages. All the attachments must be referred to n the report text. The fgures and tables shown n the attachments are formatted accordng to Chapters.3 and.4. Each attachment s an ndependent entty,.e. the numberng starts from n each attachment. Attachment shows an example of the laboratory notebook. Laboratory notes can be made by hand usng a ballpont pen.

11 3 Data Handlng and Error Estmatons The purpose of the laboratory work performed n the physcal chemstry practcals s usually to fnd out the value of a certan physcal property by expermental methods. Therefore, the results always contan some error. The sze of the error and the man sources of the error are found by error estmaton. When estmatng the errors, the reasonng behnd the estmaton must always be explaned. It can be, for example, the precson of the equpment or the maxmum error based on the total dfferental. In ths secton a bref summary of the prncples and methods of error estmaton s outlned. More nformaton can be found for example from the books of Vaar [] (n Fnnsh) or Shoemaker et al. [3]. 3. Error Types The errors appearng n the expermental measurements are usually dvded nto rough, systematc and random errors. Rough error s due to the napproprate use of equpment, error n readng, etc. It s assumed to vansh when measurements are done carefully. Systematc error always dstorts the results n the same drecton. Systematc error occurs, for example, when calbraton of the equpment s not done correctly (so called calbraton error) or f analogue scale s read from the wrong angle (so called parallax error). Systematc errors should try to be detected and corrected whenever possble. Random error.e. statstcal error does not contort the result n one drecton. Random error s n every expermental measurement and the exstence can be

12 detected only by repeatng the measurement several tmes. Estmaton of random errors s done usng statstcal tools. If the amount of systematc errors s small, the trueness of the measurement s sad to be good. If the amount of random errors s small n the fnal result, the precson of the result s sad to be good. In ths case, the average of the repeated measurements should be close to the absolute value. The effects of systematc and random errors are llustrated graphcally n Fgures 4 and vrta/a jännte/v Fgure 4. The systematc error s small as the theoretcal lne (broken lne) llustrates the data ponts well, but the random error s bg, as the error lmts of each data pont are large and there s clear scatterng of data ponts.

13 3 Fgure 5. The random error s small because the error lmts of data ponts are small and the data fts well on the same straght lne. Systematc error s large, as the data does not ft on the theoretcal lne (broken lne). 3. Estmaton of the Error of Measured Values The maxmum random error for each measured quantty s estmated qualtatvely (precson of the equpment, fluctuaton of the value measured by the equpment, etc). ote that the error of the measurement s normally constant over the entre measurement range. Hence the error n the measurement s absolute and not relatve. Here the maxmum of the random error for measured values means the absolute range wthn whch the absolute value of the random error s at certan probablty (for example, 95% or 90%). Ths value s an absolute part of the measurement range, for example 0.05 bar, and the measured value can be presented n the format P( ) bar. The error of a pressure s therefore gven n bars and not as a percentage.

14 4 3.3 Error Estmaton for Calculated Quanttes The quanttes to be studed cannot always be measured drectly. In ths case the measurements are performed usng expermentally accessble quanttes whch can delver the desred quantty (by a certan functon). The mathematcal model whch descrbes the phenomenon s assumed to be known and the studed quantty s calculated usng a model and measured values Calculatng Error Estmatons usng the Total Dfferental It s assumed that the quanttes x, x, x3... are measured and the error estmates Dx, Dx, Dx3... have been defned for these measured quanttes. If there s a mathematcal model descrbng the relatonshp between the quantty beng studed and expermentally accessble quanttes, F F(x, x, x3...), then the change n the functon F wth respect to the small change n the parameter x s defned by the dervatve of F wth respect to x. If there s a small change (dx) n all of the measured quanttes, the change n the functon F can be calculated usng the total dfferental: F F F df dx + dx + dx3 x x x (3) Partal dfferentaton s utlsed n Equaton (3),.e. the dervatve of the functon s performed for each parameter separately whle keepng the other parameters constant. When calculatng the fnal error for functon F, the dfferentals n Equaton (3) are replaced by the error lmts (Dx). The drecton of the error s unknown but a maxmum lmt can be found when all the errors are kept postve. Addtonally, as the purpose s to calculate the maxmum of the error, all partal dervates are absolute

15 5 values. If ths was not done, opposte terms mght cancel each other out. Error estmaton for the calculated value can be then calculated by the equaton: DF F x Dx + F x Dx + F x 3 Dx (4) For example: Calculate the error estmaton for the functon xy F x + y F F F y x D Dx + Dy Dx + x y ( x + y) ( x + y) Dy (5) 3.3. Calculaton of Error Estmaton by Logarthmc Dervatve If the mathematcal model contans only multplcaton, dvson and nvoluton, the error estmaton can be done by determnng the logarthmc dervatve. It usually decreases the number of calculatons n error estmaton. Mathematcally usng the logarthmc dervatve and total dervatve yelds dentcal results and therefore, both gve exactly the same value for the error estmaton. Wth the logarthmc dervatve, the logarthm of the functon s taken frst and then the dervatve. For example: Calculate the error estmaton for the functon F ka a y b where k, a, b and c are postve or negatve real numbers.

16 6 Frst, take the logarthm of both sdes: a b ( kx y ) ln k + a ln x b ln y ln F ln + Then the dervatve of both sdes of the functon are determned and the terms are placed nsde absolute sgns: DF F Dx a x Dy + b y (6) Expresson 6 gves the relatve error DF/F wth respect to relatve errors of x and y. The absolute error DF s found by multplyng the relatve error wth the fnal result. 3.4 Studyng the Dependence between the Quanttes The best and most commonly used method to fnd out f the results follow the behavour predcted by the model, s to nterpret the data graphcally. Wth graphc llustraton t s often easy to study the range n whch the model s functonng, detect systematc and rough errors, calculate the values of model parameters and estmate ther errors. If possble, t s desrable to try to formulate the dependence between the quanttes n the form of a straght lne so that the desred quantty s n the slope. The slope and the constant term of the straght lne can be determned by fttng a lne to the set of data ponts ether graphcally or usng the least squares method. From a graphcal llustraton whch s drawn n the format of straght lne t s also relatvely easy to see f the results correspond wth the theoretcal model. If, for example, the dependency y ax s studed and the constant a s to be determned, quantty y s drawn as a functon of x. In ths case the data ponts should le on a straght lne n (x, y) co-ordnates. The slope of the straght lne s then the desred quantty a. Error estmaton (Dx and Dy) s performed accordng to chapter

17 7 3. and the error lmt for the functon x s obtaned by the method explaned n Chapter 3.3, D(x ) xdx Determnng the Slope and Intersect and Estmatng Ther Errors from a Graphcal Illustraton Measured or calculated data ponts (x, y) are plotted on a graph as shown n Fgure 6. The error lmts Dx and/or Dy are added to data ponts by horzontal or vertcal lnes. 30 lämpötla/k aka/mn Fgure 6. Illustraton of how to determne the slope, ntercept and ther error estmatons from a graphcal llustraton. When the straght lne whch fts the data ponts optmally s drawn, the slope k can be determned graphcally. For the determnaton of the slope two ponts whch are relatvely far away from each other and are not measured data ponts are selected.

18 8 The value of the slope s usually determned by nsertng the values and unts of the selected data ponts nto the equaton: k y x - y - x (7) When error lnes (broken lnes) are drawn nto the graph accordng to the nstructons n page 0, the error of the slope can be calculated. The maxmum kmax and mnmum kmn of the slope can be obtaned from the slopes of the error lnes. The error of the slope Dk can be estmated as a maxmum error: Dk k max - k mn (8) Occasonally t s valuable to determne the value of the constant b for a straght lne y kx + b. The constant b can be obtaned from the ntercept of the y-axs and the straght lne. The error lmts can be determned by the ntercepts of the error lnes and y-axs. These error lmts can easly become rather large, especally when the measurements are done far away from the y-axs. If the constant term must be determned from the straght lne, t s good to ensure that the measurements are done close enough to the orgn. As a result, the slope and ntercept can be obtaned wth ther error lmts, k Dk and b Db. If the data ponts le on a straght lne n the approprate co-ordnates, ths shows that the tested model s vald over the measurement condtons Least Squares Method (LSM) To ft a straght lne (or generally, any knd of functon) to a set of data ponts there are several dfferent types of algorthms. One of the most commonly used algorthms

19 9 s the Least Squares Method (LSM) whch can be found n several computer programs and calculators (for example, Excel uses LSM). LSM s a purely mathematcal algorthm, whch works exactly n the same way whether the graph has been drawn or not. Usng LSM a straght lne can be ftted to any knd of set of data ponts! Graphcal llustraton s necessary, to observe systematc and rough errors as well as to fnd out f the measured dataponts are n the range n whch the theory s vald. If fttng a functon to measured dataponts one must be sure that the model really descrbes the data ponts. Ths can be ensured frst by makng a graphcal llustraton. A curve y F(x, a, a, a3,... ) s ftted nto the set of data ponts (x, y); a, a, a3... are the parameters of the curve. At the pont x the resdual (.e. the vertcal dfference between the curve and data pont) s y F(x, a, a, a3...). Usng LSM, the best ft s found by mnmsng the sum S of the squares of all the resduals: S ( y - F x, a, a, a ) 3,...) ( (9) The mnmum of the functon S s found from the zero ponts of the dervatve so that the constants a, a, a3... can be obtaned from the equatons: S a 0 S a 0 S a (0) The most smple case for LSM s fttng a straght lne nto a set of data ponts n whch there are no error lmts for the ponts or they are all equal. In ths case the ftted functon s F(x, k, b) kx + b and the sum S to be mnmsed s S ( y - kx + b) ) ( y - kx - b) ( ()

20 0 Usng Equaton (0), the dervatve of the sum S of the squares wth respect to parameters k and b s determned and then the parameters k and b are solved from two equatons. The followng s then obtaned: ł Ł - y x y x D k () ł Ł - y x x y x D b (3) where the parameter D s obtaned from the expresson ł Ł - x x D (4) The statstcally calculated values of the standard devatons Dk and Db for parameters k and b can then be found: D k s D (5) s D x D b (6) where s s calculated as shown below ( ) s b kx y (7) LSM fttng gves the slope and the ntercept of the straght lne, k Dk and b Db, usng statstcal mathematcs.

21 4 Makng References Every fact, whch cannot be assumed to be general knowledge for a chemcal engneer, must have a reference. In the Physcal Chemstry laboratory works a numbered reference lst s favoured A reference s nserted nto the text whch refers to the orgnal source. The reference s marked usng a number of the reference n square brackets. It may be worth addng the name of the author n the text before the reference number. If there many authors, the notaton frst author s name et al. s used. Example of references n the text: Accordng to Chrstan [], the concentraton of hydrogen ons s calculated as the follows. Corsaro [] measured the velocty factor In the reference lst, the references are presented as a numbered lst and n the order of appearance n the text. A complete lst of authors must be gven. An example of the reference lst:. Chrstan, G.D., Analytcal Chemstry, 4th Ed., John Wley & Sons, ew York 986, p Corsaro, G., A Colormetrc Chemcal Knetcs Experment, J. Chem. Ed. 48 (964)

22 5 References. IUPAC, Quanttes, Unts and Symbols n Physcal Chemstry, edtor I. Mlls, T. Cvtas, K. Homann,. Kallay and K. Kuchtsu, Blackwell Scentfc Publcatons, Oxford 988, p.3.. Vaar, J, Fyskan laboratorotyöt, Suomen Fyyskkoseuran julkasuja 4, 3. edton, SFS Fyskan Kustannus Oy, Jyväskylä 998, pp Shoemaker, D.P., Garland, C.W. and bler, J.W., Experments n Physcal Chemstry, 5th Ed., McGraw-Hll, Sngapore 989, pp. 9-8.

23 ATTACHMET LABORATORY OTEBOOK LAB WORK 50. DETERMIIG THE TRASFERECE UMBER Date:.9.06 Lab group: Ernest Engneer and Sally Student Author: Ernest Engneer Table. Measured values V / ml t / s 0, , , 690 0, ,8 0 0, 0 0, , ,30 704