B.3 Graphs
- Syllabus
- 2017
- Topic
- —
- Level
- AS
Learning objectives
—B.3.1—Translating between data formsTranslate information between graphical, numerical and algebraic forms, including interpreting spectra. A2 applications include determining reaction order and deriving rate expressions from graphs.—B.3.2—Plotting two variablesPlot two variables from experimental or other data, including concentration–time graphs with an appropriate best-fit curve.
Translate a graph into values, relationships and chemical meaning
| Form | Evidence to extract | Translation |
|---|---|---|
| table | paired values, units, repeats, anomaly | plot or calculate a relationship |
| graph | coordinates, trend, intercept, gradient | numerical value or algebraic model |
| equation | variables, powers, constants | predicted graph shape and changes |
| spectrum | axis quantity, peak position and stated intensity measure | chemical feature supported by that spectrum |
To read a calibration graph, locate the measured response on its axis, draw to the best-fit line, then project to the concentration axis. Show both construction lines. If the sample was diluted, the graph gives the diluted concentration; apply the dilution factor afterwards to recover the original concentration.
At A2, compare initial rates at controlled concentrations. If doubling [A] leaves rate unchanged, doubles it or quadruples it, the order in A is 0, 1 or 2 respectively. Translate that pattern into extrate=k[A]m[B]n.
Interpret each spectrum using its own axes and conventions. Peak position and signal size may carry different meanings in mass, infrared or NMR spectra, so identify what the supplied spectrum measures before assigning a chemical feature.
Interpolation within calibrated data is supported more strongly than extrapolation beyond it. A plotted correlation supplies a model or estimate; it does not by itself prove the proposed chemical cause.
Plot data so the relationship, not the drawing, controls the conclusion
| Feature | Requirement |
|---|---|
| axes | independent variable on x, dependent on y; label quantity and unit |
| scale | linear unless specified, easy intervals, data covering at least half the grid in both directions |
| points | small accurate crosses at the supplied coordinates |
| fit | one straight line or smooth curve representing the overall trend |
Decide between a straight line and a smooth curve from the pattern and the stated model. A best-fit line should balance scatter rather than pass through every point. Retain a suspected anomaly unless there is evidence to exclude it; do not bend the fit solely to capture that point.
Include the origin only when it is a supplied point or the chemical relationship justifies it. An axis may use a clearly marked break, but the numerical scale must remain uniform on each section. Preserve transformed labels such as 1/t/s−1 or 1/T/K−1.
Joining points dot-to-dot is not a best-fit curve, and a non-linear scale can create a false shape. Reversing axes changes the gradient and may invalidate the intended chemical interpretation.
Use the best-fit line to determine gradient and intercept
m=\frac{\Delta y}{\Delta x}\qquad y=mx+c
| Quantity | Graph method | Unit |
|---|---|---|
| gradient m | choose two far-apart points on the best-fit line and calculate rise/run | y-unit divided by x-unit |
| intercept c | read y where the fitted line reaches x=0 | same as y |
For an A2 zero-order concentration-time graph, [A]=[A]0−kt. The straight-line gradient is −k and the intercept is the initial concentration. If concentration falls from 0.80 to 0.20moldm−3 over 300s, the gradient is −2.0imes10−3moldm−3s−1, so k=2.0imes10−3moldm−3s−1.
Use points on the best-fit line, not automatically raw data points. A small triangle magnifies reading error, while omitting units or the negative sign loses physical information even when the arithmetic is correct.
A linear gradient is a constant rate of change
\text{rate of disappearance of A}=-\frac{\Delta[\ce{A}]}{\Delta t}
A straight concentration-time line has the same gradient throughout, so the concentration changes by the same amount per unit time. A falling reactant concentration gives a negative graph gradient; the rate of disappearance is reported as its positive magnitude.
| Graph feature | Chemical interpretation |
|---|---|
| horizontal line | zero change in the plotted quantity per unit time |
| steeper positive line | faster increase |
| steeper negative line | faster decrease |
| constant negative concentration gradient | zero-order disappearance; k is the gradient magnitude |
A best-fit line changes from 0.600 to 0.360moldm−3 in 120s. Its gradient is (0.360−0.600)/120=−2.00imes10−3moldm−3s−1; the disappearance rate, and zero-order k, is 2.00imes10−3moldm−3s−1.
The sign describes direction, while the rate magnitude describes speed. A curved graph does not have one constant rate and must be handled with a tangent at the required time.
A tangent turns one point on a curve into an instantaneous rate
| Step | Action |
|---|---|
| locate | mark the required time, using t=0 for an initial rate |
| draw | place a straight tangent touching the curve locally without cutting across it nearby |
| measure | choose two far-apart points on the tangent, not on the curve |
| calculate | use Δy/Δx, attach units and interpret the sign |
\text{rate}=k[\ce{A}]^m[\ce{B}]^n
For the initial-rates method, compare experiments in which only one reactant concentration changes. If multiplying [A] by a factor f multiplies the initial rate by fm, then m is the order in A. Repeat for other reactants, then combine the orders in the rate equation.
When doubling [A] at constant [B] quadruples the initial rate, 2m=4, so m=2. If changing [B] does not change the rate, n=0 and the [B]0 factor may be omitted.
A chord between two curve points gives an average rate, not the instantaneous rate. Rate comparisons reveal an order only when other relevant concentrations and conditions, especially temperature, are controlled.