SCIENCE · INVESTIGATE AND EXPLAIN

Compare exposed area after fragmentation

Use ideal cube geometry to separate exposed surface from conserved volume and reaction rate.

Learning outcome: Calculate piece count and surface-area ratios without claiming a weathering-rate law.

Starting recap

The separated cubes are original geometry, not measured rock pieces.

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Try

A 2 cm cube split into eight separated 1 cm cubes changes area from 24 to 48 square centimeters while volume stays eight cubic centimeters.

Open the experiment

Predict

Does eight times as many pieces mean eight times the total area?

Your prediction and reason

Experiment

Calculate areas and volumes for n = 1, 2 and 4, then explain the requirement that faces be exposed.

Compare storm timing and initial water storage

Original sealed-store experiment · separate weathering evidence practice · Original synthetic plot with 40 mm storage capacity and maximum entry 15 mm/hour. Rain is piecewise constant by hour. Infiltration is limited by rain supply, entry limit and remaining storage; unentered rain leaves immediately as runoff. No drainage, evapotranspiration, surface ponding, routing or empirical soil forecast. Entry stops at a full store only because this store has no drainage; real saturated soil can transmit water. Exact rate segments split at filling time; stored water is not runoff.

Initial water in teaching store: 0 mm Six-hour rain profile: Uniform: 60 mm over six hours

Reset keeps your prediction, captures and practice history.

Weathering changes material; erosion moves it

Original fictional observation cards. The rain controls do not change these cards or simulate weathering. Before revealing an interpretation, identify whether the evidence supports physical fragmentation, chemical alteration, transport, or an unresolved mechanism. Real exposures may show several processes together.

Supplied qualitative evidence, not measurements from a field site
Case	Evidence to explain	Prediction task
A: broken sample	A rock breaks along cracks into smaller pieces. Mineral tests before and after show the same mineral constituents.	What changed, and what was conserved?
B: altered minerals	Mineral tests show loss of some original feldspar and formation of clay minerals; dissolved constituents occur in water leaving the sample.	Does fragmentation alone explain all observations?
C: relocated grains	Marked grains leave one patch during flowing water and accumulate downstream; their tested mineral constituents remain unchanged.	Separate removal, transport and deposition.
D: a brown surface	A photograph shows a brown coating. No mineral or chemical analysis is supplied.	What additional evidence distinguishes oxidation, staining or another coating?

Compare the weathering interpretations

A supports mechanical fragmentation; the stated evidence does not establish chemical alteration. B supports chemical weathering, including mineral transformation and dissolved export; real reactions require appropriate conditions and cannot be identified from color alone. C supplies erosion, transport and deposition evidence, not proof of new chemical weathering. D remains ambiguous: obtain compositional evidence and compare coated material with the underlying rock.

Water entering cracks, crystal growth and roots can contribute to physical breakdown under suitable conditions. Chemical weathering includes dissolution and reactions involving water, oxygen or acids. These processes can interact: new cracks expose surfaces, while alteration can weaken rock. A crack or weather event by itself does not establish an exact weathering rate.

Fragmentation exposes more surface

Original geometry: begin with a cube 2 cm on each edge. Cut every edge into equal divisions and separate all pieces so every face is accessible. Predict the total exposed area before opening the table. Cutting changes shape and exposure; it does not itself change mineral identity or create material.

Reveal area and volume calculations

All faces exposed; no material lost and no chemical reaction modeled
Divisions per original edge n	Pieces n³	Piece edge cm	Total volume cm³	Total area cm²
1	1	2	8	24
2	8	1	8	48
4	64	0.5	8	96

Each of n³ pieces has edge 2/n cm and six faces. Total area = n³ × 6 × (2/n)² = 24n cm². Total volume = n³ × (2/n)³ = 8 cm³. The area grows in proportion to n, while volume stays fixed. More available surface can change reaction opportunities, but temperature, composition, water supply, chemistry and accessible contact also matter. Twice the area does not guarantee twice the observed weathering rate.

Transfer: use a 3 cm starting cube divided into three pieces per edge. Calculate piece count, total volume and area, then identify why tightly packed pieces might not expose every face to moving water.

Background: National Park Service weathering mechanisms and USGS explanation of clay formation and weathering. These sources explain real processes; the cards and cubes here are teaching constructions.

Original storm experiment: entry and storage are different limits

Uniform: 60 mm over six hours. The receiving store holds at most 40 mm of water equivalent and admits at most 15 mm/hour while space remains. Rain that does not enter leaves the plot immediately. This is an intentionally sealed teaching store: no drainage, evapotranspiration, surface ponding or travel time is included. Real saturated soil can transmit water; these parameters are not measured soil properties.

Conservation: initial storage + rain = final storage + runoff. For this case, 0 + 60.0 = 40.0 + 20.0 mm. Newly infiltrated water equals the storage increase only because this model has no store outflow. Initial water is not counted again as new infiltration.

Read rates and accumulated water separately

The model store first fills at hour 4.0000. A full store and an exceeded entry-rate limit are distinct reasons why rain may leave as runoff.

One-hour totals; quantities are mm of water equivalent
Hour interval	Rain	Initial store	Infiltration	Runoff	Final store
0–1	10.0	0.0	10.0	0.0	10.0
1–2	10.0	10.0	10.0	0.0	20.0
2–3	10.0	20.0	10.0	0.0	30.0
3–4	10.0	30.0	10.0	0.0	40.0
4–5	10.0	40.0	0.0	10.0	40.0
5–6	10.0	40.0	0.0	10.0	40.0

Inspect exact rate intervals and the filling event

Within-hour rate changes occur at calculated filling times
Start–end hours	Rain mm/hour	Infiltration mm/hour	Runoff mm/hour	Store start–end mm
0.0000–1.0000	10	10	0	0.00–10.00
1.0000–2.0000	10	10	0	10.00–20.00
2.0000–3.0000	10	10	0	20.00–30.00
3.0000–4.0000	10	10	0	30.00–40.00
4.0000–5.0000	10	0	10	40.00–40.00
5.0000–6.0000	10	0	10	40.00–40.00

For a one-hour bin, entry depth is the minimum of rain depth, 15 mm and remaining storage. Before filling, entry rate is the smaller of rain rate and 15 mm/hour. Time to fill is remaining storage divided by that rate. If the rate is zero, no filling time is calculated from division by zero. At filling, entry stops and the rest of the rain leaves; the plot is not drawn as though the average hourly entry occurred uniformly.

Choose a fair comparison and explain its limits

Compare uniform and concentrated storms from the same initial storage: both supply 60 mm, isolating timing in this model. Light rain supplies 20 mm and the dry case supplies zero, so those comparisons also change total input. With initial storage zero, uniform rain admits 40 mm and exports 20 mm; the concentrated storm admits 30 mm and exports 30 mm. With 20 mm initially stored, both export 40 mm, but the time patterns differ.

Real runoff depends on drainage, surface storage, soil structure, vegetation, slope and spatial connections omitted here. This plot's immediate export is not a catchment outlet hydrograph. Runoff depth alone does not determine sediment mass; that requires concentration or sediment-process evidence. Background: USGS surface runoff and USACE infiltration and runoff-volume concepts.

Rain 60.0 mm = runoff 20.0 mm + storage increase 40.0 mm.

Initial store 0 mm; final store 40.0 mm. Store first full at hour 4.000. Full means the sealed teaching store is full, not a diagnosis of real soil saturation; real saturated soil can transmit water. Synthetic bounded accounting, not a catchment forecast.

Observation to attach to a capture

Captures preserve initial water storage and the synthetic storm profile. Replay recalculates the sealed-store comparison; it does not measure real runoff, simulate weathering or establish an independent field result.

Understand

Expose more surface without creating more material

Imagine one cube cut into smaller cubes that are then separated. The material occupies the same total volume, but previously internal faces can now contact the surroundings. Predict whether doubling the number of divisions along each edge doubles the total exposed area, the total volume, or both.

This is original ideal geometry. Real rock pieces are irregular, can remain touching and may have pores or coatings. The calculation is a controlled way to reason about area; it is not a calibrated weathering experiment or a claim that every newly cut surface reacts equally.

Count faces, pieces and dimensions

For a cube with edge length L, one face has area L² and all six faces total 6L². Volume is L³. If L = 2 cm, the surface area is 24 cm² and volume is 8 cm³. Keep the squared and cubed units visible: area and volume describe different quantities.

Now divide each original edge into n equal lengths. There are n³ small cubes, each with edge L/n. For n = 2 there are eight cubes with 1 cm edges. Each has six square centimeters of surface, so their combined exposed area is 48 cm² after separation. Their combined volume is still eight cubic centimeters.

For n = 4 there are 64 cubes with 0.5 cm edges. Each face is 0.25 cm²; each cube has 1.5 cm² of surface. Multiplying 64 by 1.5 gives 96 cm². Each small volume is 0.125 cm³; multiplying by 64 again gives 8 cm³.

Derive a ratio you can check

The total exposed area is n³ × 6(L/n)² = 6L²n. Read that as number of pieces times surface area per piece. The area multiplier is n, while the volume multiplier is one. Cutting is not chemical conversion and creates no new total material volume in this idealization.

Draw the n = 1, 2 and 4 cases on paper. Label their piece counts 1, 8 and 64, areas 24, 48 and 96 cm², and equal total volumes. Explain why eight times as many pieces does not mean eight times the exposed area: each piece is smaller.

The area calculation assumes every small cube is separated so all faces are exposed. If pieces remain tightly packed, touching faces may not be accessible to water or gas. Distinguish total geometric area from the portion accessible to a particular reacting fluid.

Connect area with weathering evidence

Physical fragmentation can create additional contact surfaces. Chemical weathering changes minerals, and additional accessible surface can influence opportunities for reactions. The National Park Service description distinguishes those processes. It does not supply a constant that turns our cube areas into actual reaction times.

Revisit fictional cards A and B. Fragmentation alone is not proof that a chemical reaction occurred. Clay and dissolved products supply a different kind of evidence from the count of fragments. A mechanism explanation can link the processes while preserving the evidence needed for each.

Do not announce that doubling area halves the time required to weather a rock. Reaction conditions, mineral type, temperature, fluid access, solution chemistry and protective layers can matter. The browser has no rate law for these factors. The storm controls also do not alter the cube or calculate chemical release.

Transfer: a different original cube

On paper divide a 3 cm cube into three equal sections along each edge and separate every resulting cube. Calculate piece count, original and final total surface areas, and total volume. Explain whether the calculation alone establishes a threefold chemical-weathering rate.

Check the weathering-surface transfer

There are 3³ = 27 cubes, each 1 cm on an edge. Original area is 6 × 3² = 54 cm²; separated total area is 27 × 6 = 162 cm², three times larger. Total volume stays 27 cm³. The area increase does not establish a reaction-rate multiplier without a defensible chemical and transport model.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Identify the missing chemical and transport conditions before inferring a rate multiplier.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Distinguish weathering from movement Next: Compare rainfall timing with water entry

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