A graph is not a picture of your results. It is a calculating device: the gradient is a rate or a constant, the intercepts are quantities you could not measure directly, and the area under the curve is a number in its own right.
📚 What you need to know
A sketch graph has labelled axes but no scale and no data points; it shows the shape of a relationship.
Directly proportional means a straight line through the origin. A straight line that misses the origin is proportional, but not directly.
Inversely proportional (y = k / x) gives a curve that never touches either axis.
The independent variable goes on the x-axis, the dependent variable on the y-axis.
A line of best fit may be straight or curved; it shows the trend and need not pass through every point.
Interpolation reads within the plotted points; extrapolation extends the line beyond them.
Find a gradient with a large triangle on a straight line, or by drawing a tangent to a curve.
The three sketch shapes
The red dot on the first graph is the whole point of the distinction. Without it the line is still linear, and it may still be a perfectly good relationship — it just is not direct proportionality.
Two related terms describe the direction rather than the form. A positive correlation means y rises as x rises; a negative correlation means y falls as x rises. They apply to curves as well as straight lines, and they say nothing about whether the line passes through the origin.
Drawing a graph that earns its marks
🧩 The checklist
Put the independent variable on x and the dependent variable on y.
Give a clear title, and label both axes with quantity and unit.
Choose linear scales with no jumps, and make the plotted data occupy at least half the grid.
Mark data points with small crosses, in pencil.
Draw a single line of best fit, straight or curved, with points scattered evenly on both sides.
Add uncertainty bars where the data warrants them.
A quick test for the scale: if you could double it and the points would still fit on the grid, the scale is too coarse and you are throwing away precision. Cramming the data into one corner is one of the easiest marks to lose in a practical.
A line of best fit does not have to be straight. Students hear “best fit” and reach for a ruler, then force a straight line through data that is obviously curving. The line follows the trend of the data — if the trend curves, so does the line.
Gradients
For a straight line, draw a triangle using two points on the line (not two data points) and divide the change in y by the change in x. Make the triangle as large as the graph allows: reading two points that are close together doubles the effect of every small misreading.
For a curve the gradient changes from moment to moment, so you find it at one particular point by drawing a tangent there.
Drawing several tangents along the same curve shows how the rate falls as the reaction proceeds — which is how a rate is shown to depend on concentration.
Intercepts
An intercept is where the line crosses an axis, and in chemistry it usually is the answer. The clearest example is a plot of Gibbs energy change against temperature, because rearranging ΔG = ΔH – TΔS into the form y = mx + c puts a real quantity at each end.
Reading the y-intercept means extrapolating back to T = 0, which is far outside the plotted range — a rare case where extrapolation is exactly what the analysis calls for.
Interpolation, extrapolation and the rest
Interpolation means reading the line between plotted points, and is safe: you have evidence on both sides. Extrapolation means extending the line past them, and is only justified when you have a physical reason to believe the relationship continues.
The classic legitimate use is the temperature correction in calorimetry. Heat escapes while you are taking readings, so the maximum temperature you actually observe is lower than the true one. You draw the cooling section of the graph, extrapolate that line back to the moment of mixing, and read off the temperature the mixture would have reached with no heat loss.
Graph feature
What it tells you
Typical example
Gradient of a straight line
the constant linking the two variables
absorbance against concentration
Gradient of a tangent
the rate at one instant
rate at a given concentration
y-intercept
the value when x is zero
ΔH on a ΔG against T plot
x-intercept
the value of x when y is zero
the temperature where ΔG = 0
Maximum
gradient goes positive, zero, negative
the transition state on an energy profile
Area under a curve
a total, not a rate
molecules above Ea on a Maxwell–Boltzmann curve
Uncertainty bars are drawn above and below a point to show its absolute uncertainty, and can be drawn horizontally too. Their real use is as a test: if a straight line can be drawn touching every bar, the data supports a linear relationship. If no such line exists, either the relationship is not linear or the uncertainty has been underestimated.
WORKED EXAMPLE
A graph of reaction rate against concentration is a straight line with a positive gradient that crosses the y-axis at a rate of 0.02 mol dm–3 s–1. State whether rate is directly proportional to concentration, and justify your answer.
What direct proportionality requiresThe relationship must be of the form y = kx, which means the line has to pass through the origin.What this graph doesIt is a straight line, so the two variables are linearly related, but it crosses the y-axis above zero.at c = 0, rate = 0.02, not 0not directly proportionalYou can still say the rate increases linearly with concentration, and that there is a positive correlation. Just not “directly proportional”.
WORKED EXAMPLE
A tangent drawn to a volume-against-time curve passes through the points (10 s, 44.8 cm3) and (30 s, 81.6 cm3). Calculate the rate of reaction at 20 s, and state the units.
Step 1 — the change in y81.6 − 44.8 = 36.8 cm³Step 2 — the change in x30 − 10 = 20 sStep 3 — divide36.8 ÷ 20 = 1.841.84 cm³ s⁻¹The units come straight from the axes: y-units divided by x-units. If the axes were labelled in minutes, dividing by 60 would convert the rate to per second.
WORKED EXAMPLE
A plot of ΔG against T for a reaction is a straight line of gradient +0.199 kJ K–1 mol–1 with a y-intercept of –92.2 kJ mol–1. Deduce ΔH and ΔS, and find the temperature above which the reaction is no longer feasible.
Step 1 — rearrange into y = mx + c∆G = (−∆S)T + ∆HSo the gradient is −∆S and the y-intercept is ∆H.Step 2 — read them off∆H = −92.2 kJ mol⁻¹∆S = −0.199 kJ K⁻¹ mol⁻¹ = −199 J K⁻¹ mol⁻¹Step 3 — the x-interceptThe reaction stops being feasible when ∆G reaches zero.0 = 0.199T − 92.2, so T = 92.2 ÷ 0.199T = 463 KWatch the sign and the units. A positive gradient here means a negative entropy change, and ∆S must be converted from kJ to J to be quoted conventionally.
💡 Exam tip
Say through the origin whenever you claim direct proportionality; without it the claim is wrong.
Label axes with quantity and unit, and take gradient units from the axes.
Use a large triangle and take the points from the line, not from the data.
For a curve, draw a tangent and say you have done so.
Distinguish interpolation from extrapolation, and justify any extrapolation you rely on.
For ΔG against T, remember the gradient is −ΔS, not ΔS.
⚠️ Common mix-up
Calling any rising straight line directly proportional. It must pass through the origin.
Joining the points dot-to-dot instead of drawing one line of best fit.
Forcing a ruler through curved data because “best fit” was assumed to mean straight.
Taking gradient points from the data rather than from the line itself.
Reading ΔS straight off the gradient of a ΔG against T graph without the minus sign.
That completes Tool 3, and with it the Tools. Between the three of them you now have the safety and technique (Tool 1), the instruments and software (Tool 2), and the mathematics that turns readings into a defensible result (Tool 3).
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