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# 680 Class Note for CHEM C1260 at Purdue

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This 24 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 30 views.

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Date Created: 02/06/15

Properties of Gaussian curves 6 x 02 y IV 27 Can also be written as Where z represents the deviation of a result om the population mean relative to the population standard deviation z is a dimensionless quantity commonly used in statistical tables and graphs 04 Relative frequency 0 0 N b 0 Deviation from mean x p E 2004 Thomson 7 BrooksCole Gaussian curves plots of relative frequency of measurements versus deviations from the mean The mean occurs at the central point maximum frequency They are symmetrical There is an exponential decrease in frequency as the magnitude of Xu increases The standard deviation indicates precision ie 63 2 therefore precision associated with curve B is half that of curve A 04 o m l Relative frequency 0 m 01 g 2004 Tnomsnn BrooksCole When relative frequency is plotted versus 2 the curves are normalized Wit 6 X aXis shows deviation from mean in units of 0 Therefore all Gaussian curves plotted in this way are identical The probability that a measured value will fall within a pair of limits is given by the area under the Gaussian curve 2 z 4 area 2 dz Z V27 for z 00 to 00 area 1000 for z l to 1 area 0683 05 04 03 02 01 2004 Thomson BrooksCole for z 2 to 2 area 0954 05 04 03 02 01 2004 Thomson BrooksCole for z 3 to 3 area 0997 05 04 03 02 01 2004 Thomson BrooksCole Alternate Expression of Standard Deviation The common expressions for precision Standard deviation see above Coef cient of Variation CV or Percent Relative Standard Deviation RSD A statement of the standard deviation in relative terms rather than as an absolute value is referred to as the coef cient of variation or the relative standard deviation RSD i RSD 3 x sample population CV i x 100 x sample CV 3x 100 population Variance Another commonly used measure of precision is called the variance Sample Variance S N 1 Population Variance N Variances have the advantage of being additive Calculation of Pooled Standard Deviations For a series of similar samples Where the same sources of random errors might be expected eg the NaOH standardization data being collected in Expt 2 the results can be pooled to yield a better measure of the standard deviation of the method Note the averages of the data sets need not be the same spooled is only an estimate of 6 not u N1 N2 N3 2 2xi x1z2xj x2z2xk xg S 11 jl k1 Id W N1N2N3 Nt where N is the number of pooled data sets S s12N1 ls22N2 ls N3 l pmquot N1N2N3 Nt see example 62 in your text Signi cant Figures quick review Signi cant gures are all the digits in a number known with certainty plus the rst uncertain digit 2004 Thomson Bron 302 is certain 3024 or 3025 should be reported Therefore there are 4 signi cant gures Zeroes amid other gures are always signi cant Zeroes after a decimal point and before another digit are not signi cant 00034 has two signi cant gures same as 34X10393 Zeroes after a digit and before an implied decimal point cause ambiguity 23400 3 4 or 5 sig gs 234x104 3 4 5 2340x104 23400x104 Use of scienti c notation removes this ambiguity Signi cant gures in calculations Addition subtraction last signi cant gure determined by the number With uncertainty in the highest decimal place 34 0020 731 10730 report 107 Multiplicationdivision rule of thumb is to retain in the result the same number of gures as the factor With the least number of sig gs Usually this is OK A safer approach is to determine the last signi cant gure om the factor With the lowest relative certainty 6g 24 x 4521000 108 24 x 4021000 0965 124 X 108 0045 l24 X 0965 0040 uncertainty in 2nd decimal place for both report 108 and 096 Logs same gures in mantissa as in number 6g 16g 6536 x102 28153 Propagation of errors in Arithmetic Calculations Fory fa b c sy fsa sh so 2 3 Si y sj 361 TABLE 6 4 ayZ 6b 2 Sb 322 sf ac Error Propagation in Arithmetic Calculations Type of Calculation Example Standard Deviation ofy39i Aleilion or subtraction 39 u b i r39 s 3 52 l Mullipliculion or division Av u X 11 2 Exponcnlinlion 39 ul 3 quotii Logzirnhm 39 log I Aquot z 0434 4 u 5 Antilogarithm 39 antilogmn 230 rd 3 0 Ir and r39 are cxperimenlzil vnrlnhlv Willi slundzlrd aleviulinnx ol39x A and Aquot respectively quotThese relationships are derived in Appendix 9T1 Values for XJ arr nhsolule Villll il r ix 1 negative num ber 2004 Thomson BrooksCole Example 1 69b om the text Estimate the absolute standard deviation and the coef cient of variation for y 1897i004 00025100001 229i008 212625 sy J0042 000012 0082 0089 CV 0089212625 x 100 042 y 2126i009 Example 2 69d from text 860i2 1673i0006 s 2 2 2 y L i 0006 200058 y 251 860 1673 CV 00058 X 100 058 sy 00058 gtlt 129026 750 y 1290i0008 X 105 y 251i1gtlt Lab 2 You are asked to report the molar concentration of the unknown acid Cu the standard deviation Sn and the 95 con dence interval Assuming no uncertainty associated With CNaOH and Vu the overall precision is determined by the precision associated With the VNaOH measurement Example CNaOH 01114 M Vu 2500 mL 5 I7Na0H CNaOH s 2C1 C2 u CM V V N 1 Unknown VNaOH mL gum 1836 008181 1829 008150 1841 008203 1739Na0H 1835 008178 00818 SC 000027 S SC u 00001600002 50 For lab writeup assume no error in Vu CNaOH in determining Cu Cu 2 00818 i 00002M re1ative standard deviation RSD or coef cient of variation CV 00002 100 024 00818 RSD 100 Con dence interval ZEiZXS true C C 13295 2 C 00818M i 43000016 Int 9 C 00818M i 00007 Int 9 Is it OK to assume no uncertainty in Vu and CNaOH Recall propagation of errors Ifymn 2 2 2 S SmSn y variances are additive and in terms of the standard deviations 2 2 Sy 15 5 Lab 2 data Concentration of KHP primary standard solution Mass ofKHP 48327100001 g Volume of KHP stock solution 2500i0 1 mL mass 1 48327g 1 MW VL 2042gmol 02500L S 2 2 CW 2 00001 01 1 200004 CKHP 48327 2500 SCKHP 00946700004 0000038 z 000004 2 009467M KHP CKHPOO9467iOOOOO4 M uncertain to roughly 1 part per 2500 Standardization of NaOH solution VNaOH mL V V2 2123 00001 2129 00025 2119 00025 VNaOH 2124 2 00051 V F s s Zlt 2 0005120057 S VIM VNaOH N 1 2 VNaOH VNQOH 2124i003 mL Concentration of NaOH VNaOHCNaOH VKHPCKHP VKHP KHP 2500i 001mL009467i 000004M NaOH 01114M VNaOH 2124i 003mL 5cm 000004 001 003 00015 CNaOH 009467 2500 2124 sCNaOH 0111400015 000017 z 00002 CNaOH 01114i00002M Uncertainty essentially the same as uncertainty in VNaOH Roughly 1 part in 650 Concentration of Unknown w RSD Vu 2500mL 1 part in 2500 CNaOH 01114 1 part in 650 VavgNaOH 1835 mL 1 part in 400 Concentration of unknown VuCu VNaOHCNaOH VN IOH CNaOH V SC E 1 2 1 2 1 2 1 Cu 2500 650 400 330 Since the relative uncertainty in CNaOH is actually comparable to the uncertainty in the average NaOH volume measurement it is not strictly justi able to ignore the uncertainty in CNaOH However the uncertainty in Vu is suf ciently small so as to make an inconsequential contribution to the resulting concentration of the unknown 3 Experiment 3 Enzymatic Quantitation of Glucose High level objectives Illustrate a KineticBased Method Show Selectivity 0f Enzymes Detailed objectives 1 Use spectrophotomers to collect absorbance data as a function of time 2 Construct a calibration plot curve for AA 3 Determine the concentration of an unknown 4 Use statistical methods for data analysis The radiant power of a light beam decreases from P0 to P as it passes through an absorbing medium such as a solution with absorbing components Absorbing solution of concentration c P T P0 P0 A logP 2004 Thomson BrooksCole T transmittance A absorbance log PoP log lT Beer s Law A ebc log PoP log UT 8 molar absorptivity b path length c concentration of absorbing species c M We expect a linear relationship between A and c Deviations from linearity may occur if polychromatic radiation is used Spectrophotometers use gratings or prisms to disperse the light and lenses and slits to focus and select particular wavelengths of light om a tungsten or deuterium lamp to approach monochromaticity Glucose oxidase is a protein catalyst or enzyme that speci cally catalyses the oxidation of BDglucose to yield hydrogen peroxide and gluconic acid G O C6H12O6 02 H2O gt H202 C6H12O7 SlOW The enzyme is very speci c for this sugar molecule If a known E0 initial enzyme concentration xed 02 a xed temp and a known reaction time are used the initial concentration of BDglucose can be determined by following the production of H202 H202 is not observed directly Rather the absorbance of 1339 which is formed from H202 3139 2H M HZO 1 fast The rst reaction is rate determining so the appearance of 1339 is actually a measure of the concentration of glucose 1339 absorbance is monitored between xed time intervals At to get AA360 Rate 0C AA360 C2 360 C1 tlme C2 I a A360 1 AA I I I I I I I At I t1 2 tlme C 5 I xquot c 4 A1 360 c3 or rate x C2 c 1 I z z Concentration

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