Processing: Representing Data¶
Syllabus mapping
Working Scientifically outcome — Processing (part 2 of 2). Covers organising processed data into tables and graphs, choosing an appropriate graphical representation, and linearising a relationship so it can be read from a straight-line graph. See Precision and Uncertainty for the first half of Processing — error, uncertainty, and propagating uncertainty through a calculation.
Outcome codes: CH-11WS-04 (Year 11) and CH-12WS-04 (Year 12) — Processing data and information. Same outcome wording in both years, with a separate code for each. Wording on this page is paraphrased for teaching use — check the official syllabus before treating any line as verbatim NESA text.
You need to know: how to build a summary table from raw data, how to construct a graph that's actually readable, and how to turn a curved relationship into a straight line — because a straight line is far easier to check against a prediction than a curve is.
Assumed knowledge check¶
Before this page makes sense, you should be comfortable with:
- Graphs — maths refresher on reading and constructing line graphs and finding a gradient
Core¶
From raw data to a summary table¶
Conducting covered building a raw data table as you collect it — every individual repeat, untouched. A summary/processed table is built afterwards, and looks different: one row per independent-variable value, with a mean column (and, where relevant, an uncertainty column and any calculated quantity) rather than every individual repeat spelled out.
Worked example — raw to summary
Raw data (three repeats of the disappearing-cross time at each thiosulfate concentration):
| Concentration (mol L⁻¹) | \(t_1\) (s) | \(t_2\) (s) | \(t_3\) (s) |
|---|---|---|---|
| 0.050 | 62.1 | 60.5 | 61.8 |
| 0.100 | 30.8 | 31.6 | 30.9 |
Summary table:
| Concentration (mol L⁻¹) | Mean time (s) | \(\Delta t\) (s) | Rate, \(\frac{1}{t}\) (s⁻¹) |
|---|---|---|---|
| 0.050 | 61.5 | 0.8 | 0.0163 |
| 0.100 | 31.1 | 0.4 | 0.0322 |
The summary table is what you actually plot. Notice the extra "rate" column: a shorter time means a faster reaction, so \(\dfrac{1}{\text{time}}\) is used as a measure of rate.
Building a graph that's actually readable¶
A handful of conventions turn a scribble into something a marker (or you, a week later) can read at a glance:
- Title — what the graph shows, not just axis names restated ("Rate of reaction vs thiosulfate concentration", not "Graph 1").
- Labelled axes with units — every axis names its quantity and its unit, e.g. "Concentration (mol L⁻¹)", not just "c".
- Independent variable on the x-axis, dependent on the y-axis — this is the convention every marker expects.
- A sensible scale — spread the data across as much of the grid as reasonably possible. The scale does not have to start at zero unless you specifically need to read a y-intercept.
- A line or curve of best fit, not dot-to-dot — a best-fit line/curve represents the underlying relationship, smoothing over the scatter caused by random error in individual points.
- Error bars, where uncertainty has been calculated for each point — a vertical line through each point extending \(\pm\) the uncertainty (see Precision and Uncertainty).
Not everything is a line graph. Use a bar chart when the independent variable is a category rather than a number (e.g. mass of gas produced by different metals), and a line graph when it's a continuous number (temperature, concentration, time).
Linearising a relationship¶
Many relationships in chemistry are not straight lines when plotted directly — but a straight line is far easier to work with than a curve: it's easy to see whether data actually fits it, and its gradient and intercept can be read off directly and compared to a predicted value. Linearising means choosing what to plot on each axis so that a relationship which is naturally curved becomes a straight line.
The general approach: if a relationship is \(y = kx^n\), plotting \(y\) against \(x^n\) (instead of against \(x\)) gives a straight line through the origin with gradient \(k\).
Worked example — linearising Boyle's law
A sealed syringe of air is compressed at constant temperature (see Properties of Gases):
| Volume (mL) | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|
| Pressure (kPa) | 150 | 100 | 75 | 60 | 50 |
| \(\frac{1}{V}\) (mL⁻¹) | 0.100 | 0.067 | 0.050 | 0.040 | 0.033 |
Plotting \(P\) against \(V\) gives a curve. But \(PV = \text{constant}\) means \(P = \text{constant} \times \dfrac{1}{V}\) — the form \(y = kx\) with \(x = \dfrac{1}{V}\). Plotting \(P\) against \(\dfrac{1}{V}\) gives a straight line through the origin with gradient \(k = PV = 1500\ \text{kPa mL}\).
That straight line is the evidence that pressure and volume really are inversely proportional — far more convincing than "the curve looks about right".
Other chemistry relationships that need (or don't need) the same treatment:
- Rate vs concentration (Rates of Reactions): time gets shorter as concentration goes up, so time vs concentration is a curve. Plotting \(\dfrac{1}{t}\) (rate) against concentration gives a straight line if rate is proportional to concentration.
- Volume vs temperature (Properties of Gases): already a straight line — through the origin if temperature is in kelvin. In °C, the line crosses zero volume at about −273 °C (absolute zero).
- Calibration curves (Measures of Concentration): the light absorbed by a coloured solution is directly proportional to its concentration, so absorbance vs concentration is a straight line — which is what lets you read off an unknown concentration.
Advanced
Choosing the transformation when it isn't obvious. Not every relationship arrives pre-labelled as \(y = kx^n\). Given an unfamiliar formula, isolate the two quantities you can actually measure, then rearrange until one is a constant multiple of some power of the other — as done above for Boyle's law. If a relationship involves two variables you're changing, hold one constant while linearising against the other, one variable at a time.
Using the gradient. In the Boyle's law example, the gradient \(PV = 1500\ \text{kPa mL} = 1.5\ \text{kPa L}\) equals \(nRT\). At 25 °C (298 K), \(n = \dfrac{1.5}{8.314 \times 298} = 6.1 \times 10^{-4}\ \text{mol}\) of air was in the syringe. A gradient isn't just a number — it usually means something.
Extension
Beyond the syllabus for this course — won't appear in the HSC, included for interest / depth study inspiration.
📎 Depth study idea
Not every relationship linearises with a simple power — exponential relationships (radioactive decay, see Atomic Structure; and how reaction rate changes with temperature) need a logarithmic axis instead. For \(N = N_0e^{-\lambda t}\), taking the natural log of both sides gives \(\ln N = \ln N_0 - \lambda t\) — a straight line if you plot \(\ln N\) against \(t\), with gradient \(-\lambda\). For rates, plotting \(\ln(\text{rate})\) against \(\dfrac{1}{T}\) (the Arrhenius plot) gives a straight line whose gradient lets you calculate the activation energy. Log-linearising real rate data is a strong, technically substantial depth study angle.
Video/visual resources¶
- 🎥 Khan Academy — TODO: source a graphing/line-of-best-fit explainer that covers linearising a relationship specifically, not just basic best-fit lines.
Check yourself¶
-
Explain why a graph of \(P\) against \(V\) for a gas at constant temperature is curved, but a graph of \(P\) against \(\dfrac{1}{V}\) is a straight line.
Answer
\(PV = \text{constant}\), so \(P = \dfrac{\text{constant}}{V}\) — pressure is inversely proportional to volume, which is a curve when plotted against \(V\). Writing it as \(P = \text{constant} \times \dfrac{1}{V}\) shows it's the linear form \(y = kx\) with \(x = \dfrac{1}{V}\), so plotting \(P\) against \(\dfrac{1}{V}\) gives a straight line through the origin.
-
A student plots the volume of a gas against its temperature in °C and gets a straight line. When they extend the line back, it reaches zero volume at \(-268\ ^\circ\text{C}\). What does this point represent, and how good is their result?
Answer
It's an estimate of absolute zero — the temperature at which an ideal gas would have zero volume. The accepted value is \(-273.15\ ^\circ\text{C}\), so the result is within about 2% — a good result, especially since it relies on extrapolating well beyond the measured data.
-
A student has three repeated disappearing-cross times at each of five thiosulfate concentrations. Describe what their summary table should contain, and what they should plot to test whether rate is proportional to concentration.
Answer
One row per concentration (five rows), with columns for concentration, mean time, the uncertainty in the mean (e.g. half-range), and rate as \(\dfrac{1}{\text{mean time}}\) — not all fifteen raw times. They should plot rate (\(\dfrac{1}{t}\)) on the y-axis against concentration on the x-axis. A straight line through the origin would show rate is proportional to concentration.