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by: Guiseppe Bednar


Guiseppe Bednar

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Class Notes
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This 14 page Class Notes was uploaded by Guiseppe Bednar on Friday October 30, 2015. The Class Notes belongs to CHEM 4581 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/232187/chem-4581-university-of-colorado-at-boulder in Chemistry at University of Colorado at Boulder.




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Date Created: 10/30/15
Introduction to Error Analysis for the Physical Chemistry Laboratory January 22 2007 1 Introduction In the physical chemistry laboratory you will make a variety of measurements and then manipulate them to arrive at a numerical value for a physical property However without an estimate of the error of these numbers they are largely useless Some published numbers in physics and chemistry are accurate to 10 signi cant gures or more while others are only accurate to an order of magnitude no signi cant gures The estimated error of a published number is a crucial piece of information that must be calculated This handout provides a practical introduction to the error analysis required in a typical physical chemistry lab Error in an experiment is classi ed into two types random error and systematic error Random error arises from uncer tainty in measurement devices and is the subject of this handout Systematic error includes all other sources of error including simpli cations in your model biased instrumentation impure reagents etc Random error measures the pre cision of the experiment or the reproducibility of a given result Systematic error measures the accuracy of a result or how close a result is to the true value The distinction between accuracy and precision is illustrated schematically in gure 1 2 What is uncertainty Most chemists have an intuitive idea of what uncertainty is but it is instructive to give a more rigorous de nition Suppose you perform an experiment to determine the boiling point of a liquid and you measure 32 degrees How con dent are you in this number This question could in principle be answered by repeating the experiment many times and collecting the results If you took this large collection of results and counted all of the values that lie within speci ed intervals eg between 30 and 301 301 and 302 etc you could make a histogram plot gure 2 You can see that the results are spread over a range of values The width of this spread is a measure of the uncertainty in Figure 1 Schematic illustration of accuracy and precision The lefthand target represents a high precision but low accuracy experiment The righthand target represents a low precision but high accuracy experiment your initial measurement The scatter can be quanti ed using the standard deviation 7 of the distribution which is de ned as where f are the results of your individual experiments f is the average of your results and N is the number of trials performed The standard deviation gives limits above and below a measured value in which subsequent experimental results will probably lie with 70 certainty 3 Error propagation Most often in the physical chemistry lab you will only perform an experiment one or two times In this case it is not possible to calculate the standard deviation of a large number of trials directly Instead it must be estimated by propagating the errors in the individual measurements that lead to your nal result First of all we need to discuss the uncertainties in individual measurements Sometimes an instrument such as a volumetric ask will state its uncertainty in its technical speci cations If not you can make an educated estimate of the uncertainty If an instrument gives a digital reading you can generally take the uncertainty to be half of the last decimal place For example if a digital thermometer reads 254 degrees the uncertainty is 005 degrees For analog instruments rst read the measurement to as many signi cant gures as there are marks on the gauge and then estimate one more signi cant gure The uncertainty should then be estimated based on how con dent you are in the estimated signi cant gure For example if a mercury thermometer has marks at every degree you would read the number of degrees and estimate the tenths of a degree The uncertainty might be plus or minus 02 degrees eg 254 i 02 degrees 70 i i i i i i i 60 m Counts Boiling point degrees C Figure 2 Distribution of a series of 1000 boiling point experiment results The uncertainty in a single trial is related to the width of the distribution and is called the standard deviation of the distribution a To begin our discussion of error propagation consider an experiment that measures some quantity as The result we are looking for is some function The measurement of x is subject to some uncertainty bounds and the most general case does not assume symmetric uncertainties above and below x In this case the measured value 950 is within the range x0707ltx0ltx00 2 where 950 is the measured value of 95 7 is the uncertainty above 950 and a is the uncertainty below 950 The desired property f is then within the range we 7 cu lt mo lt me 0 3 If the uncertainty in x is assumed to be small the uncertainty in f becomes df i x 4 0 dx 0 0 where 71 is the uncertainty in x which we now take to be symmetric If we have a function of many variables and if the errors are both small and independent the uncertainty is 5 We can derive some special cases from equation 5 If a function only contains addition and subtraction operations the uncertainty is af4o o 6 If a function only contains multiplication and division operations the uncer taintyis o 2 o 2 PW quot39 7 Now that we are equipped with these formulas we can proceed to propagate our individual uncertainties We will illustrate this procedure with an example Suppose you want to measure the molar heat of solvation of LiCl in water This involves 1 weighing an amount of LiCl 2 measuring a volume of water and 3 measuring the temperature change when the reagent is dissolved We will rst calculate the heat of solvation itself which is expressed in terms of our three measurements C X V X T 8 mM Here in is the mass of LiCl M is the molecular weight C is the heat capacity of water per unit volume V is the volume of water and T is the temperature H change Note that the function H only contains multiplicationdivision opera tions so we can use the error propagation rule for multiplication and division equation 7 The variables we need to consider are m V and T We do not include M and C in our list of variables because they are assumed to be known to much higher relative precision than m V and T If they weren t we would have to include them in our error analysis even if we didn t measure them Plugging our variables into equation 7 we have a 2 7V 2 0T 2 H 4 l l 9 0H l l m V T lt gt Problem 1 Calculate the heat of solvation of LiCl and its associated uncer tainty as discussed above if the mass is 21 i 005 g the molecular weight is 42394 i00005 gmol the volume is 01 i 002 L the temperature change is 4 i 05 K and the heat capacity per volume is exactly 4184 kJLK Answer H 73379 kJmol 7H 801 kJmol so you would report the heat of solvation as 7338 i 8 kJmol Problem 2 Perform the same calculation as in the last problem but include the molecular weight in your list of error propagation variables Answer You should get the same answer as before to at least 6 signi cant gures This is why it is often possible to ignore variables known to high preci sion in your error propagation Sometimes a function may contain both addition subtraction and multipli cationdivision in which case the two rules can be combined The easiest way to do this is to break the calculation into stepsl Going back to the heat of solvation experiment suppose that two separate masses were weighed and then both masses were added to the solvents Now the total mass is m m1 m2 and the equation for the heat of solvation becomes H 7 C X V X T 10 7711 m2M The rst step is to calculate the uncertainty in m m1 m using the error propagation rule for additionsubtraction ilel om 4031 072 11 Then simply use the total mass m and its calculated uncertainty and proceed as in Prob em 1 Problem 3 Calculate the uncertaintyfor the previous example the two weights were 05 i 005 g and 11 i 005 g the volume is again 01 i 002 L and the temperature change is 5 i 05 K Answer om 007 g H 3326 kJmol and 7H 8 78 kJmol Almost all of the error propagation you will do in the physical chemistry lab will only require the rules for addition subtraction and multiplicationdivisions However occasionally you might come across a more complicated function in which case we need to use equation 5 For example suppose you have de termined AG for a reaction and are interested in calculating the equilibrium constant AG K exp 12 Assuming that R and T are known to high precision we only need to calculate the partial derivative of K with respect to AC 8K 71 AG TAG We 13 and using equation 5 the uncertainty in K is 71 AG 7K W exp lt7WgtUAG 14 In many experiments you will be required to calculate the slope or intercept of a linear functions For example the rate of a unimolecular reaction obeys an exponential rate law Ct Aexp7kt 15 Figure 3 How to determine the uncertainty in the slope and intercept of two data points The bars indicate the uncertainty in the y variable and the dashed lines give upper and lower bounds for the line where c is the concentration of reagent k is the rate constant t is time and A is a constant If you want to calculate k you will need values for c at di erent times Taking the natural logarithm of equation 15 gives lnc7ktlnA 16 which is a linear equation with the familiar form y mac b In this case what interests us is the slope as a function of t which is our desired rate constant Using two data points x1 y1 and x2 yg the slope and intercept can be calculated directly m 31231 17 952 7951 by17mx1 18 and the uncertainties can be calculated using error propagation If the error in x is negligible compared to the error in y you can calculate the maximum and minimum values for the slope and intercept as illustrated in gure 3 Drawing a line through the upper bound of y1 and the lower bound for y gives a lower limit to the slope and an upper limit to the intercept Similarly drawing a line through the lower bound of y1 and the upper bound of y gives an upper limit to the slope and a lower limit for the intercept If you have many data points the average slope and intercept can be calculated using linear regression In practice this is always calculated using a computer program so the details of the procedure do not concern us You simply need to become acquainted With a program capable of performing a linear regression With error analysis The result of the calculation Will give a value and an uncertainty for the slope and intercept Further reading 1 Pl Bevinton and Di Kl Robinson Data Reduction and Error Analysis for the Physical Sciences McGraWHill 2002i 2 E B Wilson Jrl An Introduction to Scienti c Research Dover 1990i 3 l R Taylor An Introduction to Error Analysis The Study of Uncertainties in Physical Measurements University Science Books 199 l 4 Di Pl Shoemaker Cl Wt Garland and l Wt Nibler Experiments in Physical Chemistry McGraWHill 1996i Binary SolidLiquid Phase Diagram Tn hi iah The binary phase composiu39ons L L 439 L mixture as well diphenylanune are determined For r L a given pressure 39L39L 39 A nha e whicha 39 L he i nae nu diagram 39 L The txiple point ofa 39 A in Figure 1 39 uiul 39 quot39L39 quotL point B in Figure 1 is the point at which a superenu39eal uidforms A supercriu39eal uid is a state in which gas and liquid cannot be distinguished from one another Figure 1 General phase diagram for apure substance Solid liquid and gas phases are separated by PointA D n A twocomponent system can have a much more complex phase diagram In this case the stability of a certain phase depends on the pressure temperature and the composition of the solution The binary solidliquid phase diagram in Figure 2 shows the stability of different phases as a function of temperature and composition at a given pressure This particular example shows a case where the solid components are partially miscible In this gure 0Ls represents a solid state mixture predominantly composed of substance A with B present as an impurity and Bs represents the opposite case where A is an impurity When a substance is dissolved in a liquid and the freezing point of the liquid is lowered this is called freezing point depression The shape of the phase boundaries between the or liquid region and B liquid region the liquidus curves describes the freezing point depression for this mixture The equation for the liquidus curves can be derived from the ClausiusClaperyon equation under the assumption that the solution behaves ideally TXA TfA lnXA RT AzAHA TA 7 1 xA 1 xA22 RT AzAHA 2 TfyA is the freezing point of compound A and is also shown in Figure 2 AHA is the heat of fusion for compound A and XA is the mole fraction of compound A An analogous equation can be written for compound B The two liquidus curves intersect at the eutectic point C Below the liquidus curves are two regions or liquid and B liquid The x liquid region represents a mixture of 0c dissolved in liquid A and B while the B liquid region is a mixture of solid B dissolved in liquid A and B The bottommost region at s B s is a solid solution of 0c and B The regions at the edges ofthe phase diagram denoted at s and B s are present because the solids are partly miscible For compounds that are less miscible these regions are smaller whereas for solids that are completely immiscible these regions would not exist In the latter case the regions of the phase diagram corresponding to the solid states at and B would be replaced by regions corresponding to pure A and pure B h md T u represents the freenng pnnt uf pure A whue T represents the freenng pnnt uf pure E 05 represents a campased predammm y uf E wrth A present as an rrnpunty The phase dAagram ean be eonstrueted from observed ehanges m slope of the temperature versus trrne pro le or a eoohng curve In the absenee ofaphase ehange the rate ofchange ofthe ternperature obeys Newton s Law of eoohng whreh pred cts an m t of m V mo r Mr remams eonstant whreh rs eaned atherrna1 arrest 1n atworcomponent system as the freezrng pornt For thrs reason the rate ofcoolmg rs not eonstant butrs dAfferent from L Mmth fur known as athermal break When the remammg hqurdreaehes a eertarn ratro of the two known as the eutectic point A common example of a eutectic mixture is solder which is the eutectic mixture of tin 67 and lead 33 which melts at 183 C I u m w x T T Y 2 time gt gtltA Figure 3 Example cooling curves and a corresponding phase diagram W Y and Z denote thermal arrests and X denotes a thermal break EXPERIMENTAL The binary system will be naphthalenediphenylamine The temperature measurements will be made with a thermocouple system A thermocouple system produces a voltage proportional to the temperature difference between the two junctions One junction is kept at 0 C by immersing it in a slushy mixture of distilled water and ice The thermocouple system is attached to a chart recorder to plot voltage as a function of time to see the thermal arrests and breaks Charts will be available in the lab to convert the voltage reading to temperature A diagram of the apparatus is attached to the end of this document Make up the rst mixture in as speci ed in Table l in a small test tube Make sure to record exactly how much of each compound you add to the tube Table 1 Mxture details Here N stands for naphthalene and D stands for diphenylamine Next it is necessary to obtain a cooling curve To do this one must first heat the mixture in a beaker of water until it is completely liqui ed Remove the test tube from the hot water bath and insert a thermocouple It is possible that some of the liquid mixture may have solidi ed on the tip of the thermocouple upon its insertion If this is the case return the test tube to the hot water bath until the solid mixture is completely remelted Turn the chart recorder on The small test tube can then be transferred to rest inside a beaker and allowed to slowly cool While it is cooling the mixture must be agitated When the mixture has completely solidi ed the chart recorder can be turned off You should see at the least a single arrest on the cha1t Continue to add to the mixture as speci ed in Table 1 After adding a large amount of impurity to your original sample it may be necessary to have an ice bath to cool the mixture CALCULATIONS o On each cooling curve determine the break andor arrest temperatures using the thermocouple chart Calculate mole fractions Plot brealdarrest temperatures vs mole fraction Find the liquidus curves by tting with a polynomial t trendline Calculate the melting points of both naphthalene and diphenylamine from the phase diagram Find the eutectic mixture and eutectic temperature from the intersection of the trendlines for the liquidus curves Estimate the error References Atkins P Physical Chemist 5Lh edition New York WH Freeman and Company 1994 Levine I N Physical Chemist 5Lh edition New York McGrawHill 2002 Shoemaker DP Garland CW Nibler JW Experiments in Physical Chemist 7Lh edition New York McGrawHill 2003 White MA Properties of Materials New York Oxford UP 1999 PHASE D AGPAM APPARATUS tn rcnrar and rernc K nrnncnun cuhuvembe thevmucuumg Wrm sampre assemmy wam urwamr me ham cuuhngjacket Wrm sampre mm suppun and mnmmmgwmmm to recorder and reference thermocoupre to battery box hqmd eve Water or water we rm


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