Practical - Experiment z - experimental error analysis PDF

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Experiment Z - Experimental Error Analysis...

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Chemistry Laboratory: Year 1

Experiment Z: Experimental Error Analysis Name:

UCL email:

Home Department: Course (please circle):

CHEM1602

/

CHEM1603

Lab Group: Date and Time of Experiment: Date due / submitted:

/

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Aims    

to learn how to read and record errors in raw data to learn how to propagate and combine errors to learn how to plot experimental data, including error bars to learn how graphically to extract information concerning the intercept and gradient of linear trends in data

Introduction Every measurement made in a laboratory will have an associated “error”. No matter how carefully one designs the experiment, nor how skillfully that design is executed, the experimenter can never eliminate all sources of uncertainty in their measurements. These errors may be systematic or random. Systematic errors affect all measurements equally and if the source of a systematic error can be identified, that error can be compensated for and so will not affect the final result of the experiment. Random errors on the other hand are… random. They do not affect all measurements equally, either in terms of magnitude or direction (i.e. random errors may increase the value of some measurements and decrease the value of others). The theory of recording, propagating and analyzing errors can be very complicated. There are, however, a number of key rules to be followed, together with some important simplifications that are perfectly adequate for the treatment of errors in this laboratory course. A comprehensive overview of the treatment of errors can be found on the Moodle Course. You are strongly advised to read through this guide to assist you with the calculations in this “experiment”.

Questions Recording Errors 1. Read the scales in the pictures below. Record the values indicated and their associated errors in the table below. Don’t forget to include units.

(a) / (b)

Picture (a) (b) (c)

(c)

Value

Absolute Error

Relative error

Propagating errors: single variable 2. Complete the table below. In the first two columns you are told the value of some measured quantity x and the associated error in this value,  x . In the third column you are told the definition of some derived quantity z in terms of x . Your task is to calculate an expression for the absolute error z in z in terms of

x

 x and (put that in the fourth column) then to use this expression and the value of both x from column 1 to determine a numerical value for z . You should include this together with the numerical value of z in column 5. z  f (x ) x  z (function) z  z

3.44 g

± 0.02 g

z  2x

17.65

± 0.01

z  ln x

2.6

± 0.3

1.8 V

± 0.1 V

z 5/ x

7.66 m

± 0.15 m

z  x3

z  exp( x )

 z  2 x

z  (6.88  0.04)g

Propagating errors: combining errors in multiple variables 3. Use the data in the table below to calculate values for z and its associated error. In the first columns you are told the values and absolute errors in two independent quantities x and y . In the fifth column you are told the definition of some new quantity z , expressed in terms of x and y . As in question 2, your task is to calculate z together with an expression for the absolute error z in z in terms of

 x and  y then to use this expression and the values of x and  y from columns 2 and 4 to determine a numerical value for  z . Do not forget that you are supposed to calculate the absolute error in z , so be careful when multiplying and dividing to convert from relative to absolute errors. Because of limitation on space, you should do all your calculations and quote your results, in the empty space below the table. x

y

y

(a) 2.03 m

± 0.01 m

5.25 m

± 0.02 m

z x y

(b) 104 m

±2m

17.3 s

± 0.5 s

z x / y

x

z  f (x , y )

(c) 1.0055 g ± 0.0001 g 2.7882 g ± 0.0001 g z  y  x (d) 2.8 N

± 0.2 N

4.07 m

± 0.02 m

z  y *x

(e) 5.55 J

± 0.01 J

6J

±1J

z  2x  5y

(a)

z



(b)

z



z



(c)

(d)

z



z



(e)

4. It is very common to have to calculate the error in the concentration of a standard solution made up by dissolving an accurately weighed mass of some substance in an accurately measured volume of solvent. Consider the following scenario: a. A weighing boat containing iron alum is weighed on an analytical balance and found to have a mass of (0.7342 ± 0.0001) g. b. After transferring the solid to a beaker, the empty weighing boat is weighed and found to have a mass of (0.0021 ± 0.0001) g. c. The solid is now dissolved in a solution of concentrated sulfuric acid and this solution is added to an excess of water. The resulting solution is transferred to a 1000 cm3 volumetric flask and diluted to the mark. The flask has a tolerance of ± 0.02 cm3. Assuming that the only sources of error come from the precision of the balance and the tolerance of the volumetric flask, what is the concentration of the iron ions in the solution and what is the error in this concentration? Useful information: (i) You may assume a molar mass of 392.14 g mol-1 for iron alum. (ii) Iron alum has the formula Fe(NH4)2(SO4)2.6H2O.

5. A student records the following values for the absorbance of an unknown solution at a wavelength of 420 nm: Measurement number 1 2 3

Absorbance 0.872 0.853 0.901

Determine the average value of the absorbance, A at this wavelength, together with an estimate of the error in your value.

A



6. The graph attached to back of this script shows an Arrhenius plot for an unspecified bimolecular reaction. Call the gradient m and the intercept c . What are the values of m and c and what are their associated errors m and  c ? You will need to plot lines of maximum and minimum slope.

m



c



7. Use the following data to plot a graph oflog(k / k 0 ) against I 1/ 2 . I 1/ 2 0.071 0.1 0.122 0.141 0.158 0.173

log(k / k 0 ) -0.14 -0.23 -0.25 -0.30 -0.32 -0.34

 log( k / k 0 ) 0.01 0.02 0.01 0.02 0.02 0.01

Include error bars and plot lines of maximum and minimum slope to fit the data. Use your graph to determine values for the gradient m , intercept c and their associated errors. The data in this graph relates to the “kinetic salt effect” and the gradient is given by the equation m  1.02 zB where z B is the charge on the cation forming the activated complex. Given that the charge must be an integer, what is the value of zB ?

m c z  B

 ...