Rate from the gradient of a tangent

Rate and extent of chemical change

GCSE Chemistry (8462) · 4.6.1.1 · Higher

ORIGINAL

The rate at one instant is the gradient of the tangent there: rise ÷ run between two points on the tangent, with units.

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Question

A student measures the volume of hydrogen given off when magnesium reacts with hydrochloric acid, and plots volume in cm³ against time in s. A tangent drawn to the curve at 30 s passes through the points (0 s, 18 cm³) and (60 s, 66 cm³). Calculate the rate of reaction at 30 s, with its unit.

Every step worked, with the reasoning.

1
Rate at 30 s = gradient of the tangent at 30 s
The steepness of the curve at an instant is the rate then; the tangent touches the curve at that one point and has the same gradient.
2
$\text{change in volume} = 66 - 18 = 48\ \text{cm}^3$
Read the rise between two well-separated points on the tangent, not on the curve.
3
$\text{change in time} = 60 - 0 = 60\ \text{s}$
The run between the same two points.
4
$\text{rate} = \dfrac{48}{60} = 0.80\ \text{cm}^3\text{/s}$
Rate = rise ÷ run, in cm³/s. Common error: dividing one reading by its time (the volume at 30 s ÷ 30) — that is the mean rate since the start, not the rate at 30 s.

Answer: $0.80\ \text{cm}^3\text{/s}$ (2 s.f., matching the graph readings)

Question

Stuck on rate from a graph?