Questions on measurement, uncertainty and significant figures in chemistry

Question 1

Evaluate, in terms of accuracy and precision, the results of two students who titrate portions of the same solution, and suggest the kind of error each set shows. Their accurate titres are given in the table; an experienced analyst using the same solutions finds the true titre to be 23.95 cm³.

Accepted Answer

Indicative content

student A's titres all agree within 0.10 cm³ of one another, so they are concordant and A's results are precise
A's mean titre is 24.12 cm³, 0.17 cm³ above the true value and outside the ±0.10 cm³ of a titre, so A's results are not accurate
A's results show a systematic error that moved every titre up by a similar amount, such as a burette rinsed with water that diluted the titrant
student B's titres spread over 0.80 cm³ and no two agree within 0.10 cm³, so B's results are not precise and none are concordant
B's mean of 23.97 cm³ lies close to the true value only by chance, because a mean of scattered, non-concordant titres cannot be relied on
B's results show random error, such as judging the end point inconsistently or reading the meniscus from different heights; B should repeat until titres agree

Levels

Level 3 (5–6 marks) both students are judged on both accuracy and precision against the true titre, with the concordance and the means used as evidence, the kind of error each set shows is named with a plausible cause, and the answer says what each student should do; precision and accuracy are kept apart throughout
Level 2 (3–4 marks) both sets are correctly called precise or accurate and one kind of error is named with a cause; the other kind is not identified, or the means are never set against 23.95 cm³
Level 1 (1–2 marks) one student's results are called accurate because they are close together, with precision and accuracy swapped and no kind of error named

Question 2

Calculate the titre and its percentage uncertainty when a student's initial burette reading is 1.20 cm³ and her final reading is 23.70 cm³, each reading being uncertain by ±0.05 cm³.

Accepted Answer

M1 titre = 23.70 − 1.20 = 22.50 cm³ M1 the titre is the difference of two readings, so its uncertainty is 2 × 0.05 = ±0.10 cm³ M1 percentage uncertainty = 0.10 ÷ 22.50 × 100; error carried forward from an undoubled ±0.05 cm³ A1 percentage uncertainty = 0.44%; allow 0.444%

Question 3

Calculate the total percentage uncertainty in a concentration of 0.0915 mol dm⁻³, found by multiplying and dividing the three measurements given in the table, and give the concentration with its absolute uncertainty.

Accepted Answer

M1 the separate percentage uncertainties are 0.24% for the pipette, 0.55% for the titre and 0.12% for the flask M1 the quantities are multiplied and divided, so the percentage uncertainties are added, giving 0.91% M1 absolute uncertainty = 0.91% of 0.0915 = 0.00083 mol dm⁻³; error carried forward from the total percentage A1 0.0915 ± 0.0008 mol dm⁻³, the uncertainty to one significant figure

Question 4

Compare random errors with systematic errors, giving an example of each from a titration.

Accepted Answer

B1 random errors make readings scatter unpredictably on both sides of the true value, whereas systematic errors shift every reading the same way B1 random errors reduce precision, whereas systematic errors reduce accuracy B1 the effect of random errors is reduced by repeating and taking a mean, whereas systematic errors are not reduced by repeating B1 a random error is judging the end point a drop early or late; a systematic error is a burette rinsed with water, which dilutes the titrant every time

Question 5

Describe how a burette reading should be taken and recorded, and identify which of these recorded readings is wrong: 0.00 cm³, 12.35 cm³, 24.7 cm³, 36.05 cm³.

Accepted Answer

B1 the eye is level with the meniscus, to avoid a parallax error B1 the reading is taken at the bottom of the meniscus B1 it is recorded to two decimal places, to the nearest 0.05 cm³, so the last figure is 0 or 5 B1 24.7 cm³ is wrong, because it has only one decimal place; it should be written 24.70 cm³

Question 6

Calculate the activation energy, in kJ mol⁻¹, of a reaction whose graph of ln k against 1/T is a straight line through the two points given in the table. The gradient of the line is −Ea/R, and R = 8.31 J K⁻¹ mol⁻¹.

Accepted Answer

M1 gradient = change in ln k ÷ change in 1/T = −2.40 ÷ (2.00 × 10⁻⁴) M1 gradient = −1.20 × 10⁴ K M1 Ea = −gradient × R = 1.20 × 10⁴ × 8.31 = 9.97 × 10⁴ J mol⁻¹; error carried forward from the gradient A1 Ea = 99.7 kJ mol⁻¹, to three significant figures; allow 99 700 J mol⁻¹ only with the unit J mol⁻¹

Question 7

Calculate the mean titre from the titration results given in the table, explaining which results are used.

Accepted Answer

M1 the rough titre is left out, because it was run quickly to find the approximate end point M1 22.45 cm³ is left out, because it does not agree with the others within 0.10 cm³ and is anomalous M1 mean = (22.15 + 22.10 + 22.20) ÷ 3, using the three concordant titres A1 mean titre = 22.15 cm³, to two decimal places

Question 8

Suggest two causes of a titre that is 0.40 cm³ larger than the three concordant titres in its set, and explain why it is left out of the mean rather than averaged with the others.

Accepted Answer

B1 the end point was overshot, with extra drops added after the indicator had changed colour B1 an air bubble in the burette tip escaped during the titration and was counted as titre; allow a misread burette reading B1 the titre lies far outside the agreement of the concordant titres, so it comes from a fault in that run, not from the true end point B1 including it would pull the mean away from the value the concordant titres agree on, making the result less accurate

Question 9

State the uncertainty of a single reading on each of: a 50 cm³ burette marked every 0.10 cm³, a balance whose display reads to 0.01 g, and a thermometer marked every 1 °C.

Accepted Answer

B1 the burette reading is uncertain by ±0.05 cm³, half of the 0.10 cm³ division, because the scale is read by eye B1 the balance reading is uncertain by ±0.01 g, one in the last digit of the display; allow ±0.005 g only where the question states it B1 the thermometer reading is uncertain by ±0.5 °C, half of the 1 °C division

Question 10

Calculate the amount, in mol, of sodium hydroxide in 22.40 cm³ of a 0.125 mol dm⁻³ solution, and give the answer in standard form.

Accepted Answer

M1 volume = 22.40 ÷ 1000 = 0.02240 dm³, converting cm³ to dm³ M1 amount = concentration × volume = 0.125 × 0.02240 A1 2.80 × 10⁻³ mol, in standard form to three significant figures; reject 2.8 × 10⁰ from an unconverted volume

Question 11

Calculate the percentage uncertainty in the mass of solid transferred when a weighing bottle and solid weigh 14.73 g and the bottle weighs 12.28 g after the solid has been tipped out. The balance display reads to 0.01 g.

Accepted Answer

M1 mass transferred = 14.73 − 12.28 = 2.45 g M1 the mass comes from two readings, each ±0.01 g, so its uncertainty is ±0.02 g A1 percentage uncertainty = 0.02 ÷ 2.45 × 100 = 0.82%; allow 0.41% only from a stated ±0.005 g a reading

Question 12

Calculate the percentage uncertainty in a temperature rise when the starting temperature is 21.0 °C and the highest temperature is 27.5 °C, both read on a thermometer with an uncertainty of ±0.5 °C per reading.

Accepted Answer

M1 temperature rise = 27.5 − 21.0 = 6.5 °C M1 the rise comes from two readings, so its uncertainty is ±1.0 °C A1 percentage uncertainty = 1.0 ÷ 6.5 × 100 = 15%

Question 13

Explain why repeating a titration several times does not remove a systematic error, using the example of a burette rinsed with water instead of the acid it holds.

Accepted Answer

B1 a systematic error shifts every reading in the same direction by a similar amount B1 water left in the burette dilutes the acid every time, so every titre is too large B1 repeats all carry the same error, so they agree with each other and the mean is still too large; the error is removed only by fixing its cause

Question 14

Explain why a titration designed to give a titre of about 25 cm³ gives a more reliable result than one giving a titre of about 10 cm³.

Accepted Answer

B1 the absolute uncertainty of the titre is ±0.10 cm³ whatever its size, because it comes from the burette B1 on 10 cm³ that is a percentage uncertainty of 1.0%, but on 25 cm³ it is 0.4% B1 the larger titre therefore has the smaller percentage uncertainty, so the calculated result is known more precisely

Question 15

Explain how many significant figures a concentration should be given to when a calculator shows 0.0827519 mol dm⁻³ from a mass of 1.06 g, a titre of 24.15 cm³ and a 25.0 cm³ pipette, and give the answer.

Accepted Answer

B1 a result cannot be more precise than the least precise data used to calculate it B1 the mass and the pipette volume have three significant figures, fewer than the four of the titre B1 the answer is 0.0828 mol dm⁻³; reject 0.08275 and reject 0.083

Question 16

Suggest two changes that would reduce the percentage uncertainty in the mass of a 0.12 g piece of magnesium ribbon weighed by difference on a balance whose display reads to 0.01 g, and show how large that uncertainty is at present.

Accepted Answer

B1 the mass is uncertain by ±0.02 g from two readings, which is 0.02 ÷ 0.12 × 100 = 17% of the mass B1 use a larger mass of magnesium, scaling up the acid and the gas collection to match, so the same ±0.02 g is a smaller fraction of the mass B1 use a balance with a finer resolution, such as one reading to 0.001 g, so a mass by difference is uncertain by ±0.002 g; ignore 'repeat and take a mean'

Question 17

Show that, in a titration using a 25.0 cm³ pipette with an uncertainty of ±0.06 cm³ and a burette read to ±0.05 cm³, a titre of 20.00 cm³ carries a larger percentage uncertainty than the pipette volume does.

Accepted Answer

M1 pipette: (0.06 ÷ 25.0) × 100 = 0.24% M1 the titre is the difference between two burette readings, so its uncertainty is 2 × 0.05 = ±0.10 cm³ A1 titre: (0.10 ÷ 20.00) × 100 = 0.50%, which is larger than 0.24%; the burette contributes about twice the uncertainty of the pipette

Question 18

Suggest what effect a thermometer that reads 1.5 °C too high at every temperature has on a measured temperature rise, and why.

Accepted Answer

B1 the temperature rise is not affected by the faulty thermometer B1 the rise is the difference of two readings, and the same 1.5 °C offset in each reading cancels when one is subtracted from the other

Question 19

Give two sources of random error in a titration.

Accepted Answer

B1 judging the exact drop at which the indicator changes colour B1 reading the meniscus slightly differently from one reading to the next

Student	Titre 1 / cm³	Titre 2 / cm³	Titre 3 / cm³
A	24.10	24.15	24.10
B	23.60	24.40	23.90

Measurement	Value	Uncertainty
Pipette volume	25.0 cm³	±0.06 cm³
Mean titre	18.20 cm³	±0.10 cm³
Volumetric flask	250.0 cm³	±0.3 cm³

Point	1/T / K⁻¹	ln k
First	3.200 × 10⁻³	−6.90
Second	3.400 × 10⁻³	−9.30