What every lab question builds on
Each lesson in 4.4 covers one kind of AP lab. Whatever the lab, FRQs test three things: choosing the right equipment, recording data to the equipment’s precision, and calculating a result from the data and judging it. This lesson practices all three.
Equipment and its precision
When reading a scale, read to one digit past the smallest marking (the last digit is estimated). For equipment with a single mark (a volumetric pipet or flask), the equipment itself sets the volume.
| Equipment | Record to | Used for |
|---|---|---|
| Analytical balance | 0.0001 g | Weighing samples |
| Buret (50 mL) | 0.01 mL | Titrations |
| Pipet (25 mL) | 25.00 mL | One exact portion |
| Volumetric flask (250 mL) | 250.0 mL | Making solutions |
| Graduated cylinder (100 mL) | 0.1 mL | Rough volumes |
| Beaker | Estimate only | Holding, dissolving |
Use precise equipment for quantities that go into a calculation (masses and volumes used to find the result); a quantity that isn’t used in the calculation, such as water added to dissolve a solid, only needs a rough measurement.
- Measure exactly 25.00 mL of an acid into a flask for titration
- Prepare 250.0 mL of a 0.1000 M standard solution
- Measure the volume of NaOH added during a titration (the amount needed isn’t known in advance)
- Add about 30 mL of distilled water to dissolve a weighed solid acid in a flask
- Weigh out about 0.2 g of a solid sample and record its exact mass
Describing a set of data
- Mean: the average of several trials, reported as the final result.
- Range极差: largest value − smallest value. A small range means high precision (the data agree with each other).
- Percent error百分误差: |experimental − accepted| ÷ accepted × 100%. A small percent error means high accuracy (close to the true value).
- If every result is off in the same direction, there is probably a systematic error; data scattered both ways show random error.
Using straight-line graphs
When two quantities plot as a straight line, the slope and intercept are often the quantities you want: mass vs volume gives density as the slope; absorbance vs concentration gives εb (4.4.3); ln[A] vs time gives −k (4.4.6).
- Draw a best-fit line最佳拟合直线: as close as possible to all the points, with about as many on each side; it doesn’t have to pass through every point.
- To find the slope, pick two points far apart on the line, not two data points.
- The slope has units: y-axis units ÷ x-axis units.
Example 1: concentration from buret readings
25.00 mL of hydrochloric acid is titrated with 0.1000 M NaOH. The buret reads 1.25 mL at the start and 23.70 mL at the end. Find the acid concentration and say how many significant figures it has.
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Volume of NaOH added. Final reading minus initial reading:
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Moles of NaOH.
-
Acid concentration. HCl and NaOH react 1 : 1:
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Significant figures. 22.45 mL, 0.1000 M and 25.00 mL all have four significant figures, so the result has four.
Example 2: judging a set of results
Three titrations give the concentration of an NaOH solution as 0.1012, 0.1008 and 0.1027 M; the accepted value is 0.1000 M. Find the mean, range and percent error, and comment on the data.
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Mean.
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Range. 0.1027 − 0.1008 = 0.0019 M.
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Percent error.
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Comment. All three results are above the accepted value, which points to a systematic error (for example, a problem standardizing the NaOH). The third trial is well above the other two, so the precision is only fair; another trial would show whether it’s an outlier that can be discarded.
- Accepted 0.1000 M; measured 0.1001, 0.0999, 0.1000
Close to each other and to the accepted value.
- Accepted 1.000 g/mL; measured 1.052, 1.051, 1.053
Close to each other, but all about 5% high.
- Accepted 36.0%; measured 33.1%, 38.9%, 36.4%
The mean is close, but the data are widely spread.
- Accepted 8.31; measured 8.30, 8.32, 8.31
A spread of only 0.02, and the mean equals the accepted value.
- Accepted −55.8 kJ/mol; measured −49.6, −49.9, −49.7
Close together but all too small in magnitude, probably a systematic error from heat loss.
- Accepted 0.250 M; measured 0.231, 0.262, 0.247
A range of 0.031 is too spread out.
- Accepted 63.55 g/mol; measured 63.54, 63.57, 63.55
Close and tightly grouped.
- Accepted pKa 4.76; measured 4.95, 4.96, 4.94
Tightly grouped, but all about 0.2 high.
Common mistakes
Mistake 1: “Writing more digits is always safer; more digits means more precise.” A reading can only be estimated to one digit past the smallest marking, so extra digits mean nothing. Writing 52.400 mL from a 100 mL graduated cylinder only shows a misunderstanding of significant figures.
Mistake 2: “If the accuracy is high, the precision must be high too.” They’re different. Widely spread data whose mean happens to land near the true value are accurate but not precise; tightly grouped data that are all off in one direction are precise but not accurate.
Mistake 3: “A best-fit line must pass through every point and through the origin.” A best-fit line shows the overall trend, with points on both sides. The intercept can mean something too: if a container wasn’t subtracted when weighing, the intercept is the container’s mass.
Quick check
A 50 mL buret has markings every 0.1 mL. A reading should be recorded to:
Read to one digit past the smallest marking, so 0.01 mL.
Which is the correct way to calculate percent error?
Divide by the accepted value; the range describes precision.
A graph of a liquid’s mass (y-axis) against its volume (x-axis) is a straight line. What does the slope represent?
Slope = Δm ÷ ΔV, which is the density, in g/mL.
A liquid is measured in a 100 mL graduated cylinder marked every 1 mL. Which recorded value is appropriate?
Markings every 1 mL, so estimate to 0.1 mL.
Which reading comes from an analytical balance?
An analytical balance reads to 0.0001 g.
When drawing a best-fit line, you should:
A best-fit line shows the trend of all the data.
AP-style practice
To find the density of ethanol, a student places a dry beaker on an electronic balance (without taring it), adds different volumes of ethanol from a buret, and records the total mass of beaker and ethanol each time:
| V (mL) | Total mass (g) |
|---|---|
| 5.00 | 55.98 |
| 10.00 | 59.83 |
| 15.00 | 63.72 |
| 20.00 | 67.58 |
(a) A graph of total mass against volume is a straight line. What does its slope represent?
(b) Estimate the slope from the first and last data points, with units.
(c) The line’s y-intercept is about 52.1 g. What does it represent?
(d) The accepted density of ethanol at this temperature is 0.789 g/mL. Calculate the percent error of the result in (b).
(e) The measured density is lower than the accepted value. The beaker sat open on the balance throughout. Suggest a reason that explains this result.
(f) If a 100 mL graduated cylinder (marked every 1 mL) were used to add the ethanol instead, how would the precision of the measured density change? Explain.
Model answer and scoring
(a) Slope = Δm ÷ ΔV, the mass per milliliter of ethanol, which is its density.
(b) Slope = (67.58 − 55.98) g ÷ (20.00 − 5.00) mL = 11.60 g ÷ 15.00 mL = 0.773 g/mL.
(c) The total mass when V = 0, which is the mass of the empty beaker (about 52.1 g). Not taring doesn’t change the slope, so the density isn’t affected.
(d) Percent error = |0.773 − 0.789| ÷ 0.789 × 100% = 2.0%.
(e) Ethanol is volatile. With the beaker open, some ethanol evaporated between adding it and weighing, so the measured masses are too small while the volumes come from the buret readings; m/V is too low. (The larger the volume and the longer the wait, the more evaporates, so the slope is too low as well.)
(f) The precision decreases. A graduated cylinder can only be estimated to 0.1 mL, while a buret reads to 0.01 mL; the volume uncertainty is about 10 times larger, so the points scatter more and the slope is less certain.
Scoring: (a) the slope is the density; (b) a correct two-point slope with units of g/mL; (c) the intercept is the beaker’s mass; (d) the percent error setup and value; (e) the chain from evaporation to masses that are too small; (f) lower precision, explained with the reading precision of the two pieces of equipment.
Summary
- Use precise equipment (analytical balance, buret, pipet, volumetric flask) for quantities used in calculations, and read to one digit past the smallest marking; other quantities only need rough measurement.
- The range describes precision and percent error describes accuracy; results that all lean one way usually mean a systematic error.
- The slope and intercept of a straight-line graph are often the quantities you want. Give the slope units, and calculate it from two points far apart on the best-fit line.
AP tip: When an FRQ asks why a buret is used rather than a graduated cylinder, answer in terms of precision and give numbers: “a buret reads to 0.01 mL, while a graduated cylinder reads only to 0.1 mL.”
Sources: density of ethanol (0.789 g/mL at 20 °C) from the Wikipedia article “Ethanol”; other data are illustrative.