Chapter 11 Measurement and Data Processing 11 from the IB SL Chemistry Curriculum ... 11.1.4 State random uncertainty as an uncertainty ... Chemistry provides us with a deep understanding

Chapter 11 – Measurement and Data Processing Page 1

Students are to read and complete any part that requires answers and will submit this assignment on the first day of class. You may use internet sources as well to answer and complete the document. I addition you are given a copy of the internal assessment lab activity that must be completed during the school year. You are to review this document and come up with at least three research questions that will be discussed in the first two weeks of school. If you have any questions you can contact me at the following email address. I will be checking my email weekly. Mr. Agnew

Chapter 11 – Measurement and Data Processing Topic 11 from the IB SL Chemistry Curriculum

11.1 Uncertainty and error in measurement (1 Hour) Assessment Statement Obj Teacher’s Notes

11.1.1 Describe and give examples of random

uncertainties and systematic errors. 2

11.1.2 Distinguish between precision and accuracy. 2 It is possible for a measurement to have great precision yet be inaccurate

(for example, if the top of a meniscus is read in a pipette or a measuring

cylinder).

11.1.3 Describe how the effects of random

uncertainties may be reduced. 2 Students should be aware that random uncertainties, but not systematic

errors, are reduced by repeating readings.

11.1.4 State random uncertainty as an uncertainty

range (+). 1

11.1.5 State the results of calculations to the

appropriate number of significant figures 1 The number of significant figures in any answer should reflect the number of

significant figures in the given data.

11.2 Uncertainties in calculated results (0.5 Hour) Assessment Statement Obj Teacher’s Notes

11.2.1 State uncertainties as absolute and percentage

uncertainties. 1

11.2.2 Determine the uncertainties in results. 3 Only a simple treatment is required. For functions such as addition and

subtraction, absolute uncertainties can be added. For multiplication,

division and powers, percentage uncertainties can be added. If one

uncertainty is much larger than others, the approximate uncertainty in the

calculated result can be taken as due to that quantity alone.

11.3 Graphical techniques (0.5 Hour) Assessment Statement Obj Teacher’s Notes

11.3.1 Sketch graphs to represent dependences and

interpret graph behavior. 3 Students should be able to give a qualitative physical interpretation of a

particular graph, for example, the variables are proportional or inversely

proportional.

11.3.2 Construct graphs from experimental data. 3 This involves the choice of axes and scale, and the plotting of points.

Aim 7: Software graphing packages could be used.

11.3.3 Draw best-fit lines through data points on a

graph. 1 These can be curves or straight lines.

Chapter 11 – Measurement and Data Processing Page 2 11.3.4 Determine the values of physical quantities from

graphs. 3 Include measuring and interpreting the slope (gradient), and stating the

units for these quantites.


Science is a communal activity and it is important that information is shared openly and honestly. An essential part of

this process is the way the international scientific community subjects the findings of scientists to intense critical

scrutiny through the repetition of experiments and the peer review of results in journals and at conferences. All

measurements have uncertainties and it is important these are reported when data is exchanged, as these limit the

conclusions that can be legitimately drawn. Science has progressed and is one of the most successful enterprises in our

culture because these inherent uncertainties are recognized. Chemistry provides us with a deep understanding of the

material world, but it does not offer absolute certainty.

Data collected from investigations is often presented in graphical form. A graph is a useful tool as it shows relationships

between variables and identifies data points which do not fit the general trend and so gives another measure of the

reliability of the data.

Uncertainty in measurement

Measurement is an important part of chemistry. In the laboratory you will use different measuring apparatus and there

will be times when you have to select the instrument that is most appropriate for your task from a range of possibilities.

Suppose, for example, you wanted 25 cm3 of water. You could choose from measuring cylinders, pipettes, burettes,

volumetric flasks of different sizes, or even an analytical balance if you know the density. All of these could be used to

measure a volume of 25 cm3, but with different levels of uncertainty.

Uncertainty in analog instruments

An uncertainty range applies to any experimental value. Some pieces of apparatus state the degree of uncertainty. In other cases, you will have to make a judgement. Suppose you are asked to measure the volume of water in the measuring cylinder shown on the right. The bottom of the meniscus of a liquid usually lies between two graduations and so the final figure of the reading has to be estimated. The smallest

division in the measuring cylinder is 4 cm3 so we should

report the volume as 62 ± 2 cm3. The same considerations apply to other equipment such as burets and alcohol thermometers that have analog scales. The uncertainty of an analog scale is ± half the smallest division.

Figure 02.01 – The volume reading should be taken from the bottom of the meniscus. You could report

the volume as 62 cm3

but this is not an exact value.

Uncertainty in digital instruments

Consider a top pan balance that has a digital scale. The mass of the sample of water shown on the digital display is

100.00 g but the last digit is uncertain. The degree of uncertainty is ± 0.01 g – the smallest scale division. The

uncertainty of a digital scale is ± the smallest scale division.


Measurements Significant figures

0.45 mol dm−3 2

4.5 × 10−1 mol dm−3 2

4.50 × 10−1 mol dm−3 3

4.500 × 10−1 mol dm−3 4

4.5000 × 10−1 mol dm−3 5

Other sources of uncertainty

Chemists are interested in measuring how properties change during a reaction and this can lead to additional sources of

uncertainty. When time measurements are taken for example, the reaction time of the experimenter should be

considered.

Similarly, there are uncertainties in judging, for example, the point that an indicator changes color when measuring the

end-point of a titration, or what is the temperature at a particular time during an exothermic reaction, or what is the

voltage of an electrochemical cell. These extra uncertainties should be noted even if they are not actually quantified

when data are collected in experimental work.

Exercise: A reward is given for a missing diamond, which has a reported mass of 9.92 ± 0.05 g. You find a diamond and

measure its mass as 10.1 ± 0.2 g. Could this be the missing diamond? Explain your answer.

Significant figures in measurements

The digits in the measurement up to and including the first uncertain digit are the significant figures of the measurement.

There are two significant figures, for example, in 62 cm3 and five in 100.00 g. The zeroes are significant here as they

signify that the uncertainty range is ± 0.01 g. The number of significant figures may not always be clear. If a time

measurement is 1000 s, for example, are there one, two, three, or four significant figures? As it is not clear, it is useful to

use scientific notation to avoid any confusion with one non-zero digit on the left of the decimal point. 0.98 for

example, is written as 9.8 × 10−1.

Measurements Significant figures

1000 s unspecified 1 × 103 s 1 1.0 × 103 s 2 1.00 × 103 s 3 1.000 × 103 s 4

Exercise: Express the following in scientific notation:

(a) 0.04 g

(b) 222 cm3

(c) 0.030 g

(d) 30 °C

Chempocalypse Now! Chapter 02 – Measurement and Data Processing Page 4

Exercise: What is the number of significant figures in each of the following?

(a) 15.50 cm3

(b) 150 s

(c) 0.0123 g

(d) 150.0 g

The experimental error in a result is the difference between the recorded value and the generally accepted or literature

value. Errors can be categorized as random or systematic.

Random errors

When an experimenter approximates a reading, there is an equal probability of being too high or too low. This is a

random error. Random errors are caused by:

The readability of the measuring instrument

The effects of changes in the surroundings such as temperature variations and air currents

Insufficient data

The observer misinterpreting the reading

As they are random, the errors can be reduced through repeated measurements. This is why it is good practice to

duplicate experiments when designing experiments. If the same person duplicates the experiment with the same result

the results are repeatable. If several experimenters duplicate the results they are reproducible.

Suppose the mass of a piece of magnesium ribbon is measured several times and the following results obtained:

0.1234 g, 0.1232 g, 0.1233 g, 0.1234 g, 0.1235 g, and 0.1236 g

The average value = (0.1234 + 0.1232 + 0.1233 + 0.1234 + 0.1235 + 0.1236) g

= 0.1234 g 6

The mass is reported at 0.1234 ± 0.0002 g as it is in the range 0.1232−0.1236 g.

Systematic errors

Systematic errors occur as a result of poor experimental design or procedure. The cannot be reduced by repeating the

experiments. Suppose the top pan balance was incorrectly zeroed in the previous example and the following results

were obtained:

0.1236 g, 0.1234 g, 0.1235 g, 0.1236 g, 0.1237 g, and 0.1238 g

All values are too high by 0.0002 g.

The average value = (0.1236 + 0.1234 + 0.1235 + 0.1236 + 0.1237 + 0.1238) g

= 0.1236 g 6


The mass is reported at 0.1234 ± 0.0002 g as it is in the range 0.1232−0.1236 g.

Examples of systematic errors:

Measuring the volume of water from the top of the meniscus rather than the bottom will lead to volumes which

are too high.

Overshooting the volume of a liquid delivered in a titration will lead to volumes which are too high.

Heat losses in an exothermic reaction will lead to smaller temperature changes.

Systematic errors can be reduced by careful experimental design. When evaluating investigations, it is crucial that the

experimenter distinguishes between systematic and random errors.

Accuracy and precision

The smaller the systematic error, the greater will be the accuracy. The smaller the random uncertainties, the greater

will be the precision. The masses of magnesium in the earlier example are measured to the same precision but the first

set of values is more accurate.

Precise measurements have small random errors and are reproducible in repeated trials. Accurate measurements have

small systematic errors and give a result close to the accepted value.

Figure 02.02 – The set of readings on the left are for high accuracy and low

precision. The readings on the right are for low accuracy and high precision.

Exercise: Repeated measurements of a quantity can reduce the effects of:

I random errors

II systematic errors

A. I only

B. II only

C. I and II

D. neither I or II


Uncertainties in the raw data lead to uncertainties in processed data and it is important that these are propagated in a

consistent way.

Multiplication and division

Consider a sample of sodium chloride with a mass of 5.00 ± 0.01 g and a volume of 2.3 ± 0.1 cm3. What is its density?

Using a calculator:

density ρ = mass

= 5.00

= 2.173913043 g cm−3

volume 2.3

Can we claim to know the density to such precision when the value is based on less precise raw data?

The value is misleading as the mass lies in the range 4.99−5.01 g and the volume is between 2.2−2.4 cm3. The best we

can do is to give a range of values for the density.

The maximum value is obtained when the maximum value for the mass is combined with the minimum value of the

volume.

ρmax = mass

= 5.01

= 2.277273 g cm−3 volume 2.2

and the minimum value is obtained by combining the minimum mass with a maximum value for the volume.

ρmin = mass

= 4.99

= 2.079167 g cm−3 volume 2.4

The density falls in the range between the maximum and minimum value.

The second significant figure is uncertain and the reported value must be reported to this precision as 2.2 g cm−3. The

precision of the density is limited by the volume measurement as this is the least precise.

This leads to a simple rule. Whenever you multiply or divide data, the answer should be quoted to the same number of

significant figures as the least precise data.

Addition and subtraction

When values are added or subtracted, the number of decimal places determines the precision of the calculated value.

Suppose we need the total mass of two pieces of zinc of mass 1.21 g and 0.56 g.

The total mass = 1.77 g can be given to two decimal places as the balance was precise to ± 0.01 in both cases.

Similarly, when calculating a temperature increase from 25.2 °C to 35.2 °C,

the temperature increase = 35.2−25.2 °C = 10.0 °C


Worked example: Report the total mass of solution prepared by adding 50 g of water to 1.00 g of sugar. Would the use

of a more precise balance for the mass of sugar result in a more precise total mass?

Solution:

Total mass = 50 + 1.00 g = 51 g

The precision of the total is limited by the precision of the mass of the water. Using a more precise

balance for the mass of sugar would have not improved the precision.

When evaluating procedures, you should discuss the precision and accuracy of the measurements. You should

specifically look at the procedure and use of equipment.

Percentage uncertainties and errors

An uncertainty of 1 s is more significant for time measurements of 10 s than it is for 100 s. It is helpful to express the

uncertainty using absolute, fractional, or percentage values.

The fractional uncertainty = absolute uncertainty/measured value

This can be expressed as a percentage.

Percentage uncertainty = absolute uncertainty

× 100% measured value

Percentage uncertainty should not be confused with percentage error. Percentage error is a measure of how close the

experimental value is to the literature or accepted value.

Percentage error = accepted value – experimental value

× 100% accepted value


Propagation of uncertainties: Addition and subtraction

Consider two buret readings:

Initial reading/±0.05 cm3 = 15.05

Final reading /±0.05 cm3 = 37.20

What value should be reported for the volume delivered?

The initial reading is in the range: 15.00−15.10

The final reading is in the range: 37.15−37.25

The maximum volume is formed by combining the maximum final reading with the minimum initial reading:

volmax = 37.25−15.00 = 22.25 cm3

The minimum volume is formed by combining the minimum final volume with the maximum initial reading:

volmin = 37.15−15.10 = 22.05 cm3 therefore vol = 22.15 ± 0.1 cm3

The volume depends on two measurements and the uncertainty is the sum of the two absolute uncertainties. This

result can be generalized: When adding or subtracting measurements, the uncertainty is the sum of the absolute

uncertainties.

Propagation of uncertainties: Multiplication and division

Working out the uncertainty in calculated values can be a time-consuming process. Consider the density calculation:

Value Absolute uncertainty % Uncertainty

Mass/g

24.0

±0.5 = 0.5

× 100% = 2% 24.0

Volume/cm3

2.0

±0.1 = 0.1

× 100% = 5% 2.0

Value Maximum value Minimum value

density/g cm−3 = 24.0

= 12.00 2.0

= 24.5

= 12.89 1.9

= 23.5

= 11.19 2.1

Value Absolute uncertainty % Uncertainty

density/g cm−3

12

= 12.89 – 12.00 = ±0.89 = 0.89

× 100% = 7.4% 12.00

As discussed earlier, the density should only be given to two significant figures given the uncertainty in the mass and

volume values. The uncertainty in the calculated value of the density is 7% (given to one significant figure). This is equal

to the sum of the uncertainties in the mass and volume values: (5 + 2% to the same level of accuracy). This approximate

result provides us with a simple treatment of propagating uncertainties when multiplying and dividing measurements.

When multiplying or dividing measurements, the total percentage uncertainty is the sum of the individual percentage

uncertainties. The absolute uncertainty can then be calculated from the percentage uncertainty.


Worked example: The lengths of the sides of a wooden block are measured and the diagram below shows the

measured values with their uncertainties.

What is the percentage uncertainty in the calculated area of the block?

Solution:

Area = 40.0 × 20.0 mm2 = 800 mm2 (area is given to three significant figures)

(% uncertainty of area) = (% uncertainty of length + % uncertainty of breadth)

% uncertainty of length = (0.5/40.0) × 100% = 1.25%

% uncertainty of breadth = (0.5/20.0) × 100% = 2.5%

% uncertainty of area = 1.25 + 2.5 = 3.75 ≈ 4%

Absolute uncertainty = (3.75/100) × 800 mm2 = 30 mm2

Area = 800 ±30 mm2

Exercise: The concentration of a solution of hydrochloric acid = 1.00 ±0.05 mol dm−3 and the volume = 10.0 ±0.1 cm−3.

Calculate the number of moles and give the absolute uncertainty.


Discussing errors and uncertainties

An experimental conclusion must take into account any systematic errors and random uncertainties. You should

recognize when the uncertainty of one of the measurements is much greater than the others as this will then have the

major effect on the uncertainty of the final result. The approximate uncertainty can be taken as being due to that

quantity alone. In thermometric experiments, for example, the thermometer often produces the most uncertain results,

particularly for reactions which produce small temperature differences.

Can the difference between the experimental and literature value be explained in terms of the uncertainties of the

measurements or were other systematic errors involved? This question needs to be answered when evaluating an

experimental procedure. Heat loss to the surroundings, for example, accounts for experimental enthalpy changes for

exothermic reactions being lower than literature values. Suggested modifications, such as improved insulation to

reduce heat exchange between the system and the surroundings, should attempt to reduce these errors.

Exercise: What is the main source of error in experiments carried out to determine enthalpy changes in a school

laboratory?

A. Uncertain volume measurements

B. Heat exchange with the surroundings

C. Uncertainties in the concentrations of the solutions

D. Impurities in the reagents

Graphical techniques

A graph is often the best method of presenting and analyzing data. It shows the relationship between the independent

variable plotted on the horizontal axis and the dependent variable on the vertical axis and gives an indication of the

reliability of the measurements.

Plotting graphs

When you draw a graph, you should:

Give the graph a title.

Label the axes with both quantities and units.

Use the available space as effectively as possible.

Use sensible linear scales – there should be no uneven jumps.

Plot all the points correctly.

A line of best fit should be drawn smoothly and clearly. It does not have to go through all of the points but should

show the overall trend.

Identify any points which do not agree with the general trend.

Think carefully about the inclusion of the origin. The point (0,0) can be the most accurate data point or it can be

completely irrelevant.


T he “ best-fit” straight line

In many cases, the best procedure is to find a way of plotting the data to produce a straight line. The “best-fit” line

passes as near to as many of the points as possible. For example, a straight line through the origin is the most

appropriate way to join the set of points in the following figure.

Figure 02.03 – A straight line graph which passes through the origin shows

that the dependent variable is proportional to the independent variable.

The best fit line does not necessarily pass through any of the points plotted. Two properties of a straight line are

particularly useful: the gradient and the intercept.

Finding the gradient and the intercept

The equation for a straight line is y = mx + c.

x is the independent variable, y is the dependent variable, m is the gradient and c is the intercept on the vertical axis.

The gradient of a straight line is the increase in the dependent variable divided by the increase in the independent

variable. The triangle used to calculate the gradient should be as large as possible.

Figure 02.04 – For a straight line, m = Δy/Δx


The gradient of a straight line has units; the units of the vertical axis divided by the units of the horizontal axis.

Sometimes a line has to be extended beyond the range of measurements of the graph. This is called extrapolation.

Absolute zero, for example, can be found by extrapolating the volume/temperature graph for an ideal gas.

Figure 02.05 – Extrapolation to determine a value for absolute zero.

The process of assuming that the trend line applies between two points is called interpolation. The gradient of a curve

at any point is the gradient of the tangent to the curve at that point.

Figure 02.06 – This graph shows how the concentration of a reactant decreases with time. The

gradient of a slope is given by the gradient of the tangent at that point. The equation of the tangent

was calculated by computer software. The rate at the point shown is −0.11 mol dm−3

min−1

. The

negative value shows that reactant concentration is decreasing with increasing time.

Errors and graphs

Systematic errors and random uncertainties can often be recognized from a graph. A graph combines the results of

many measurements and so minimizes the effects of random uncertainties in the measurements.


Figure 02.07 – A systematic error produces a displaced straight line. Random uncertainties lead to points

on both sides of the perfect straight line. (Note: This graphic might make more sense if viewed in color.)

Choosing what to plot to produce a straight line

In many cases, the best way to analyze measurements is to find a way of plotting the data to produce a straight line. For

example, the ideal gas equation:

PV = nRT

can be rearranged to give a straight line graph:

P = nRT(1/V)

The pressure is inversely proportional to the volume. This relationship is clearly seen when a graph of 1/V against P

gives a straight line passing through the origin at constant temperature.

Figure 02.08 – This straight line graph shows that the pressure is inversely proportional to the volume.


Using spreadsheets to plot graphs

There are many software packages which allow graphs to be plotted and analyzed; the equation of the best fit line can

be given and other properties calculated. For example, the tangent to the curve in Figure 02.06 has the equation:

y = −0.1109x + 0.3818

so the gradient of the tangent at that point = −0.11 mol dm−3 min−1. Care should, however, be taken when using these

packages.

Figure 02.09 – An equation which produces a “perfect fit” is not necessarily the best description of the

relationship between the variables. (Note: This graphic might make more sense if viewed in color.)

The set of data points can either be joined by a best fit straight line which does not pass through any point except the

origin:

y = 1.6255x (R2 = 0.9527)

or a polynomial which gives a perfect fit as indicated by the R2 value of 1.

y = −0.0183x5 + 0.2667x4 – 1.2083x3 + 1.7333x2 + 1.4267x (R2 = 1)

The polynomial equation is unlikely, however, to be physically significant as any series of random points can fit a

polynomial of sufficient length, just as any two points define a straight line.


Practice Questions

1. The volume V, pressure P, and temperature T, and number of moles of an ideal gas are related by the ideal gas

equation: PV = nRT. If the relationship between pressure and volume at constant temperature of a fixed amount of

gas is investigated experimentally, which one of the following plots would produce a linear graph?

A. P against V

B. P against 1/V

C. 1/P against 1/P

D. No plot can produce a straight line.

2. The mass of an object is measured as 1.652 g and its volume 1.1 cm3. If the density (mass per unit volume) is

calculated from these values, to how many significant figures should it be expressed?

A. 1

B. 2

C. 3

D. 4

3. The time for a 2.00 cm sample of magnesium ribbon to react completely with 20.0 cm3 of 1.00 mol dm−3

hydrochloric acid is measured four times by a student. The readings lie between 48.8 and 49.2 s. This measurement

is best recorded as:

A. 48.8 ± 0.2 s

B. 48.8 ± 0.4 s

C. 49.0 ± 0.2 s

D. 49.0 ± 0.4 s

4. A student measures the volume of water incorrectly by reading the top instead of the bottom of the meniscus. This

error will affect:

A. neither the precision nor the accuracy of the readings

B. only the accuracy of the readings

C. only the precision of the readings

D. both the precision and the accuracy of the readings

5. A known volume of sodium hydroxide solution is added to a conical flask using a pipet. A buret is used to measure

the volume of hydrochloric acid needed to neutralize the sodium hydroxide. Which of the following would lead to a

systematic error in the results?

I. The use of a wet buret

II. The use of a wet pipet

III. The use of a wet conical flask

A. I and II only

B. I and III only

C. II and III only

D. I, II and III


6. The number of significant figures that should be reported for the mass increase which is obtained by taking the

difference between readings of 11.6235 g and 10.5805 g is:

A. 3

B. 4

C. 5

D. 6

7. A 0.266 g sample of zinc added to hydrochloric acid. 0.186 g of zinc is later recovered from the acid. What is the

percentage mass loss of the zinc to the correct number of significant figures?

A. 30%

B. 30.1%

C. 30.07%

D. 30.08%

8. Which type of errors can cancel when differences in quantities are calculated? I.

II.

Random errors

Systematic errors

A.

B.

C.

D.

I only

II only

I and II

Neither I or II

9.

Th

e enthalpy change of the reaction:

CuSO4(aq) + Zn(s) → ZnSO4(aq) + Cu(s)

was determined experimentally, using a calorimetry setup.

Assuming that zinc is in excess and that all of the heat of reaction passes into the water, the molar enthalpy change

can be calculated from the temperature change of the solution using the following expression:

ΔH = −cH2O × (Tfinal – Tinitial)

−1 kJ mol

[CuSO4]

Where cH2O is the specific heat capacity of water, Tinitial is the temperature of the copper sulfate before zinc was

added and Tfinal is the maximum temperature of the copper sulfate solution after the zinc was added.

The following results were recorded:

Tinitial ± 0.1/°C Tfinal ± 0.1/°C 21.2 43.2

[CuSO4] = 0.500 mol dm−3


a. Calculate the temperature change during the reaction and give the absolute uncertainty.

b. Calculate the percentage uncertainty of this temperature change.

c. Calculate the molar enthalpy change of reaction.

d. Assuming the uncertainties in any other measurements are negligible, determine the percentage uncertainty in

the experimental value of the enthalpy change.

e. Calculate the absolute uncertainty.

f. The literature value for the standard enthalpy change of reaction = − 217 kJ mol−1. Comment on any differences

between the experimental and literature values.

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Chapter 11 Measurement and Data Processing 11 from the IB SL Chemistry Curriculum ... 11.1.4 State random uncertainty as an uncertainty ... Chemistry provides us with a deep understanding