Back to the on-screen lesson ·
Why a solid reacts only at its surface, how much surface cutting exposes, and why a powder gives the same product sooner rather than more of it.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to explain why dividing a solid reactant more finely increases the rate of a reaction involving it, calculate how the total surface area of a cube changes when it is cut into smaller cubes, and predict the effect of particle size on both the rate and the total amount of product. You will also be able to recognise which everyday effects are due to surface area rather than temperature, concentration or a catalyst.
Concentration raised the rate by putting more particles in each litre, so they met more often. That works for anything dissolved or gaseous. A solid is different: its particles are packed in a fixed lump and cannot move around to meet anything. This lesson is about how a solid reacts, and how to make it react faster without changing what it is.
A heterogeneous reaction is one between substances in different states, such as a solid in a solution or a solid in a gas; a homogeneous one has everything in one state, such as two dissolved substances. Surface area is the total area of a solid's outside, in cm² or m². Particle size is how finely the solid is divided: a lump, chips, granules or a powder.
When marble reacts with acid, an acid particle has to strike a calcium carbonate particle. The carbonate particles in the middle of a chip are surrounded by other carbonate particles and cannot be reached. Only the layer on the outside is available, and as it reacts it exposes the layer underneath. So for a solid:
$$\text{collisions per second} \propto \text{area of solid exposed}.$$
Breaking the solid into smaller pieces brings buried particles to a surface. The mass is exactly what it was — no carbonate has been added — but more of it is exposed, so more collisions happen each second and the reaction is faster.
This is the collision-frequency lever again, like concentration:
| collisions per second | share that succeed | rate | |
|---|---|---|---|
| smaller pieces, same mass | up | unchanged | up |
| larger pieces, same mass | down | unchanged | down |
It applies only to reactions involving a solid. Two solutions mixed together have no surface to speak of; every particle is already exposed.
The total amount of product is a separate question. Grinding a solid does not change how much of it there is, so if it is the limiting reactant, the powder gives the same total amount of product as the lump — sooner.
Another way: picture
A packed crowd leaving a stadium through the gates: only people at the front can get out. Split the same crowd among many small gates and many more people are at the front at once. Nobody new has arrived; more of them are at an edge.
Another way: steps
To predict a particle-size change:
Take a cube of side 4 cm. Its surface is six squares of $4 \times 4 = 16$ cm², so $96$ cm². Its volume is $4^3 = 64$ cm³.
Cut it in half along every edge. There are now $2^3 = 8$ cubes of side 2 cm, each with a surface of $6 \times 4 = 24$ cm², so $8 \times 24 = 192$ cm² in total — double. Cut it into four along every edge: $4^3 = 64$ cubes of side 1 cm, each $6$ cm², so $384$ cm² — four times.
| cuts per edge | side, cm | number of cubes | total area, cm² | total volume, cm³ |
|---|---|---|---|---|
| 1 | 4 | 1 | 96 | 64 |
| 2 | 2 | 8 | 192 | 64 |
| 4 | 1 | 64 | 384 | 64 |
The pattern is general: cutting a cube into $n$ pieces along each edge multiplies its surface area by $n$ and leaves its volume unchanged. So for the same mass, the surface area is inversely proportional to the size of the pieces — halve the size, double the area. Real powders are not perfect cubes, but the pattern holds.
Coal burns steadily in a grate. Coal dust suspended in the air of a mine burns so fast that the heat released expands the air explosively: a lump of side 5 cm ground into grains of side 0.005 cm has a thousand times the surface. Flour, custard powder, sugar, sawdust and even aluminium powder behave the same way, which is why mills and factories control dust, ban naked flames and ventilate.
The same principle is used on purpose. Kindling lights before a log; fine sugar dissolves faster than a sugar cube; chewing breaks food into small pieces so that digestive enzymes have more surface to work on; and the precious metals in a car's catalytic converter are spread as a thin coating over a honeycomb so that every gram of metal has as much surface in contact with the exhaust as possible.
Smaller pieces contain more of the solid. A 5 g powder and a 5 g lump are the same amount of substance. Grinding redistributes the particles; it does not add any.
Powder gives more product. It gives the same product sooner. Only more of the limiting reactant gives more product.
Surface area matters for every reaction. It matters only when a solid is involved. For two solutions, concentration is the frequency lever instead.
Grinding lowers the activation energy. The reaction is the same at every surface; the barrier has not moved. The number of collisions per second has.
A cube of side 6 cm is cut into small cubes of side 2 cm. Find the total surface area before and after.
Find the number of cuts per edge first.
Before: $6 \times 6^2 = 216$ cm². Cuts per edge: $6 \div 2 = 3$, so $3^3 = 27$ small cubes.
The number of small cubes is the cuts per edge cubed.
Each small cube has $6 \times 2^2 = 24$ cm², so the total is $27 \times 24 = 648$ cm², which is $3 \times 216$.
Three cuts per edge, three times the area; the volume stays $216$ cm³.
4 g of magnesium ribbon and 4 g of magnesium powder are each added to excess acid.
Same mass, different particle size; the magnesium is limiting.
The powder exposes far more magnesium atoms, so acid particles strike them more often and it reacts faster.
Frequency lever: more surface, more collisions per second.
Both are 4 g of the limiting reactant, so both give the same volume of hydrogen in the end — the powder's curve is steeper and levels off at the same height.
Rate follows surface; total follows amount.
Cuts per edge: $3 \div 1 = 3$.
Divide the big side by the small side.
Cutting into $n$ along each edge multiplies the area by $n$, so the factor is…
…$3$: from $54$ cm² to $162$ cm², with the volume still $27$ cm³.
A cube of calcium carbonate with sides of $3$ cm is cut into small cubes, each with sides of $0.6$ cm. What is the total surface area of all the small cubes? Give the area with its unit.
Answer: unit: m2 / cm2
A cube of marble with sides of $3$ cm is cut into 2, 4 and 5 pieces along each edge. For each way of cutting, fill in the side of each small cube, the number of small cubes, the total surface area and the total volume.
| side of each cube, in cm | number of cubes | total surface area, in cm² | total volume, in cm³ | |
|---|---|---|---|---|
| not cut | 3 | 1 | 54 | 27 |
| cut 2 ways along each edge | ||||
| cut 4 ways along each edge | ||||
| cut 5 ways along each edge |
$6$ g of marble chips and $6$ g of powdered marble are each added to the same volume of the same hydrochloric acid. The powder fizzes much faster. Why?
Four $20$ g samples of zinc are dropped into identical beakers of the same dilute sulfuric acid. Put them in order of initial rate, fastest first.
Number the steps in order (write the number in the box):
$5$ g of marble chips react with hydrochloric acid, which is in excess, so the marble is the limiting reactant. For each change, say what happens to the initial rate and to the total volume of carbon dioxide.
| initial rate | total carbon dioxide | |
|---|---|---|
| the same mass, ground to powder | ||
| the same mass, as a few large lumps | ||
| twice the mass of the same chips |
Each everyday observation is explained by one of the ways a rate can be changed. Match each observation to the change responsible.
| surface area | temperature | concentration | a catalyst | |
|---|---|---|---|---|
| flour dust in a mill's air can explode, while a sack of flour only smoulders | ||||
| milk keeps for days longer in a fridge | ||||
| vinegar at half strength fizzes more slowly with baking soda | ||||
| a pinch of manganese(IV) oxide makes hydrogen peroxide froth and is still there afterwards |
A mine-safety engineer compares a cube-shaped lump of coal with sides of $2$ cm with the same coal ground into dust, each grain a tiny cube with sides of $0.02$ cm. By what factor is the total surface area of the coal increased?
Answer: times
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Two $10$ g samples of the same metal are made of cube-shaped pieces: sample P has cubes with sides of $24$ mm, sample Q has cubes with sides of $4$ mm. Both go into the same excess acid. If the initial rate is proportional to the total surface area, how many times faster does Q start than P?
Answer: times
You can explain surface area as a collision-frequency lever for solids and calculate how cutting changes it. Say out loud why 5 g of powder and a 5 g lump give the same volume of gas at different rates. Next: catalysts, which change the share of collisions that succeed without being used up.
10. Your turn: a cube of side 3 cm is cut into cubes of side 1 cm. By what factor does the surface area increase?, step 3