B.3 Graphs

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.—B.3.3—Slope and intercept of a linear graphDetermine the slope and intercept of a linear graph, including finding a zero-order rate constant from the gradient of a concentration–time graph.—B.3.4—Rate from a linear graphCalculate rate of change from a graph showing a linear relationship, including finding a zero-order rate constant from a concentration–time gradient.—B.3.5—Tangents and instantaneous ratesDraw and use the slope of a tangent to a curve as a measure of rate of change, including determining reaction order by the initial-rates method.

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 $[\ce{A}]$ leaves rate unchanged, doubles it or quadruples it, the order in $\ce{A}$ is 0, 1 or 2 respectively. Translate that pattern into $ext{rate}=k[\ce{A}]^m[\ce{B}]^n$ .

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.

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 $1/t\,/\,\mathrm{s^{-1}}$ or $1/T\,/\,\mathrm{K^{-1}}$ .

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.

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, $[\ce{A}]=[\ce{A}]_0-kt$ . The straight-line gradient is $-k$ and the intercept is the initial concentration. If concentration falls from $0.80$ to $0.20\,\mathrm{mol\,dm^{-3}}$ over $300\,\mathrm{s}$ , the gradient is $-2.0 imes10^{-3}\,\mathrm{mol\,dm^{-3}\,s^{-1}}$ , so $k=2.0 imes10^{-3}\,\mathrm{mol\,dm^{-3}\,s^{-1}}$ .

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.

\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 $0.360\,\mathrm{mol\,dm^{-3}}$ in $120\,\mathrm{s}$ . Its gradient is $(0.360-0.600)/120=-2.00 imes10^{-3}\,\mathrm{mol\,dm^{-3}\,s^{-1}}$ ; the disappearance rate, and zero-order $k$ , is $2.00 imes10^{-3}\,\mathrm{mol\,dm^{-3}\,s^{-1}}$ .

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.

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 $\Delta y/\Delta x$ , 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 $[\ce{A}]$ by a factor $f$ multiplies the initial rate by $f^m$ , then $m$ is the order in $\ce{A}$ . Repeat for other reactants, then combine the orders in the rate equation.

When doubling $[\ce{A}]$ at constant $[\ce{B}]$ quadruples the initial rate, $2^m=4$ , so $m=2$ . If changing $[\ce{B}]$ does not change the rate, $n=0$ and the $[\ce{B}]^0$ 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.

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—B.4 Geometry and trigonometryRead

B.3 Graphs

Learning objectives

Translate a graph into values, relationships and chemical meaning

Plot data so the relationship, not the drawing, controls the conclusion

Use the best-fit line to determine gradient and intercept

A linear gradient is a constant rate of change

A tangent turns one point on a curve into an instantaneous rate

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