Which silicate class is built of isolated SiO₄ tetrahedra joined only through other cations, giving a Si:O ratio of 1:4?

Nesosilicates, such as olivine and garnet

Phyllosilicates, such as muscovite and biotite

Nesosilicates, such as olivine and garnet

Inosilicates, such as augite and hornblende

Tectosilicates, such as quartz and feldspar

A dark, prismatic, rock-forming silicate in hand specimen shows two good cleavages that meet at nearly 90°. Which mineral group is it?

Pyroxene, whose single chains give cleavages at about 87° and 93°

Amphibole, whose double chains give cleavages at about 56° and 124°

Mica, whose weakly bonded sheets give one perfect cleavage

Pyroxene, whose single chains give cleavages at about 87° and 93°

Olivine, whose isolated tetrahedra give three perfect cleavages

Which of the following mineral and structure pairings are correct?

Olivine is a nesosilicate built of isolated SiO₄ tetrahedra

Quartz is a framework silicate in which every oxygen is shared, with no cleavage

Olivine is a nesosilicate built of isolated SiO₄ tetrahedra

Muscovite is a tectosilicate with a three-dimensional framework

Quartz is a framework silicate in which every oxygen is shared, with no cleavage

Beryl is a single-chain inosilicate with the formula unit SiO₃

In plagioclase, replacement of Na⁺ by Ca²⁺ is accompanied by replacement of Si⁴⁺ by Al³⁺. Why is this coupled substitution needed?

Because Ca²⁺ for Na⁺ adds one positive charge and Al³⁺ for Si⁴⁺ removes one, so the charge balances

Because Ca²⁺ is much larger than Na⁺ and therefore needs a larger tetrahedral site to fit into

Because Ca²⁺ for Na⁺ adds one positive charge and Al³⁺ for Si⁴⁺ removes one, so the charge balances

Because Al³⁺ and Ca²⁺ have the same ionic radius, so the two must always substitute together in a crystal

Because the number of oxygens in the formula changes from 8 to 6 as anorthite replaces albite

According to the Mohs hardness scale, which of the following statements is correct?

Quartz can scratch orthoclase but cannot scratch topaz

Corundum is exactly three times as hard as calcite, since 9 divided by 3 is 3

Calcite can scratch fluorite, since calcite has the higher hardness number

Gypsum can scratch calcite, because gypsum is softer and so cuts more readily

Quartz can scratch orthoclase but cannot scratch topaz

Which of the following statements about the physical properties of minerals are correct?

Streak, the colour of the powdered mineral, is usually more constant for a mineral than its surface colour

Cleavage is breakage along planes of weak bonding fixed by the structure, whereas fracture is not controlled by it

Streak, the colour of the powdered mineral, is usually more constant for a mineral than its surface colour

Mohs hardness is a linear scale, so corundum (9) is three times as hard as calcite (3)

Cleavage is breakage along planes of weak bonding fixed by the structure, whereas fracture is not controlled by it

Quartz has three perfect cleavages, which is why it breaks into rhombohedral pieces

Which feature most reliably distinguishes plagioclase from K-feldspar in hand specimen?

Plagioclase shows fine parallel striations on one cleavage face from polysynthetic twinning

Plagioclase has only one cleavage direction whereas K-feldspar has two cleavages near 90° to each other

Plagioclase is much softer, near 3 on the Mohs scale, while K-feldspar is near 6 on the same scale

Plagioclase shows fine parallel striations on one cleavage face from polysynthetic twinning

Plagioclase is always pink or salmon-coloured, while K-feldspar is always white or colourless

Which of the following statements about the micas are correct?

Micas have one perfect cleavage because weakly bonded potassium ions join the silicate sheets

Micas are sheet silicates with tetrahedral sheets of composition near Si₂O₅

Micas have one perfect cleavage because weakly bonded potassium ions join the silicate sheets

Muscovite is the dark, Mg–Fe-rich mica and biotite is the pale, aluminium-rich mica

Micas are sheet silicates with tetrahedral sheets of composition near Si₂O₅

Micas are framework silicates, so they show no cleavage and break with a conchoidal fracture

A student is asked which pair of minerals would not be expected to coexist in equilibrium in a rock, because they react to form a pyroxene. Which pair is it?

Forsterite-rich olivine and quartz, which react as Mg₂SiO₄ + SiO₂ → 2 MgSiO₃

Quartz and K-feldspar, which are both framework silicates of the same oxide family

Olivine and calcic plagioclase, which together make up the rock troctolite

Quartz and muscovite, which are both common in granite and in quartz-mica schist

Kyanite, andalusite and sillimanite all have the formula Al₂SiO₅. Which statement about them is correct?

Kyanite is the densest, high-pressure form; andalusite is the low-pressure form and sillimanite the high-temperature form

Andalusite is the densest polymorph, and it forms at the highest pressures reached in the deep continental crust of mountain belts

Sillimanite is a high-pressure, low-temperature polymorph that is characteristically found in blueschists

The three differ in chemical composition, with kyanite containing the largest proportion of aluminium

Which statement about aragonite and calcite, both CaCO₃, is correct?

Aragonite is the denser polymorph, favoured at high pressure, and inverts slowly to calcite at the surface

Aragonite is a magnesium-rich carbonate, and that magnesium content makes it denser than calcite

Calcite is the denser, high-pressure polymorph, and aragonite is the only form stable at the surface

The two form a continuous solid-solution series, with aragonite as the strontium-rich end-member of the series

Aragonite is the denser polymorph, favoured at high pressure, and inverts slowly to calcite at the surface

Take six-fold ionic radii (Å): Mg²⁺ 0.72, Fe²⁺ 0.78, Ca²⁺ 1.00, K⁺ 1.38. For which pair of ions of equal charge is simple substitution, without any coupled substitution, most favourable?

Mg²⁺ and Fe²⁺, because their radii differ by under 10 %

Mg²⁺ and Ca²⁺, because both are divalent alkaline-earth ions

Mg²⁺ and Fe²⁺, because their radii differ by under 10 %

Fe²⁺ and Ca²⁺, because both occur in pyroxene and garnet

Ca²⁺ and K⁺, because both are large cations in feldspar

A colourless, vitreous mineral of Mohs hardness 3 breaks along three cleavage planes not at right angles and effervesces vigorously in cold dilute HCl. What is it?

Calcite, which has rhombohedral cleavage and reacts vigorously with cold dilute acid

Dolomite, which effervesces only weakly unless powdered and is slightly harder than calcite

Quartz, which has hardness 7, no cleavage at all and no reaction with acid

Calcite, which has rhombohedral cleavage and reacts vigorously with cold dilute acid

Gypsum, which has hardness 2, is scratched by a fingernail and does not react with dilute HCl

Mineralogy: Silicate Structure, Solid Solution and the Properties of Rock-Forming Minerals

This chapter answers the Part A heading “Mineralogy” of the Geology and Geophysics (GG) paper: “Silicate crystal structure, physical and chemical properties of common rock-forming minerals.” It teaches the silicate tetrahedron and the classes from nesosilicates to tectosilicates with their Si:O ratios, solid solution and end-member compositions, the physical properties used to identify a mineral, the main groups (olivine, pyroxene, amphibole, mica, feldspar, quartz, garnet, aluminosilicates, carbonates) and polymorphism. The numericals are Si:O ratio and bridging oxygens, formula units per oxygen basis, mole and weight fractions of end-members, and density from the unit cell.

1. Silicate structure: the tetrahedron and its polymerisation

The building unit of every silicate is the SiO₄⁴⁻ tetrahedron: a small Si⁴⁺ ion in four-fold coordination with four O²⁻ ions. The Si–O bond is strong and partly covalent, so the tetrahedron behaves as a rigid unit, and silicate structures differ in how tetrahedra share corners. A shared oxygen is a bridging oxygen; an unshared one is non-bridging and is bonded to other cations (Mg²⁺, Fe²⁺, Ca²⁺, Na⁺, K⁺) in larger six-fold to twelve-fold sites. If each tetrahedron shares b of its corners, it owns 4 − b/2 oxygens, so O per Si = 4 − b/2. As b rises from 0 to 4 the Si:O ratio runs from 1:4 to 1:2. Al³⁺ can replace Si⁴⁺ in tetrahedra (as in feldspar), and the ratio is then (Si + Al):O.

Silicate classes by polymerisation
Class and linkage	Unit and Si:O	Examples	Consequence
Nesosilicate: isolated tetrahedra (b = 0)	SiO₄, 1:4	Olivine, garnet, zircon, kyanite, andalusite, sillimanite, topaz, staurolite	Equant, dense, hard; no strong cleavage
Sorosilicate: pairs (b = 1)	Si₂O₇, 2:7	Epidote	Common in low-grade metamorphic rocks
Cyclosilicate: rings (b = 2)	Si₆O₁₈, 1:3	Beryl, tourmaline	Prismatic crystals
Single-chain inosilicate (b = 2)	SiO₃, 1:3	Pyroxenes	Two cleavages near 90° (87° and 93°)
Double-chain inosilicate (average b = 2.5)	Si₄O₁₁, 4:11	Amphiboles	Two cleavages near 56° and 124°
Phyllosilicate: sheets (b = 3)	Si₂O₅, 2:5	Micas, chlorite, clays, talc, serpentine	One perfect cleavage, flaky
Tectosilicate: framework (b = 4)	SiO₂, 1:2	Quartz, feldspars, feldspathoids, zeolites	Hard; quartz has no cleavage, feldspars two near 90°

Worked example: the double-chain formula is Si₄O₁₁, so O per Si = 11/4 = 2.75 = 4 − b/2, giving b = 2.5: half the tetrahedra share two corners and half share three. The same ladder explains the order of crystallisation in a cooling magma, since the least polymerised mineral (olivine) forms first and the framework mineral quartz last, and it explains why cleavage in pyroxene and amphibole runs along the chains.

2. Chemistry: solid solution, end-members and formula-unit calculations

Ions substitute for one another in a crystal if their charges can be balanced and their radii are close: Goldschmidt's rule says that a difference below about 15 % allows extensive substitution, 15 to 30 % limited substitution, and above 30 % rare substitution. Higher temperature widens solid solution; on cooling the excess can unmix (exsolution), as in perthite, sodium-rich lamellae in potassium feldspar. Simple substitution keeps the charge: Mg²⁺ ⇌ Fe²⁺ in olivine, pyroxene and garnet. Coupled substitution pairs ions of different charge: in plagioclase Na⁺ + Si⁴⁺ ⇌ Ca²⁺ + Al³⁺, so the total charge stays the same. The end-members are NaAlSi₃O₈ (albite, Ab) and CaAl₂Si₂O₈ (anorthite, An).

Olivine (Mg,Fe)₂SiO₄ runs from forsterite Mg₂SiO₄ (Fo, M = 140.7 g/mol) to fayalite Fe₂SiO₄ (Fa, M = 203.8 g/mol). Fo80 means 80 mol % Fo, formula (Mg₁.₆Fe₀.₄)SiO₄.
Plagioclase An content is the mole percentage Ca/(Ca + Na). Albite An0 to 10, oligoclase 10 to 30, andesine 30 to 50, labradorite 50 to 70, bytownite 70 to 90, anorthite 90 to 100.
Garnet A₃B₂Si₃O₁₂: pyrope Mg₃Al₂Si₃O₁₂, almandine Fe₃Al₂Si₃O₁₂, spessartine Mn₃Al₂Si₃O₁₂, grossular Ca₃Al₂Si₃O₁₂, andradite Ca₃Fe₂Si₃O₁₂.
Formula units are fixed by oxygen count: 4 O for olivine, 6 for pyroxene, 8 for feldspar, 12 for garnet. A garnet of 60 mol % pyrope and 40 % almandine has Mg = 3 × 0.6 = 1.8 and Fe = 3 × 0.4 = 1.2 per formula unit.

Mole % is not weight %. Compositions are quoted in moles, but masses differ because molar masses differ. Worked example: a mixture of 50 g forsterite and 50 g fayalite has 50/140.7 = 0.355 mol Fo and 50/203.8 = 0.245 mol Fa, so it is 0.355/0.600 = 59 mol % Fo, not 50 %. For plagioclase the two end-members have similar molar masses (albite 262.2, anorthite 278.2 g/mol), so An50 is 0.5 × 278.2 ÷ (0.5 × 278.2 + 0.5 × 262.2) = 51 wt % anorthite, close to its mole percentage.

⚠️ Oxygen basis, not cation count

Normalise a mineral formula to its fixed number of oxygens before reading cations. Counting cations to a different total, or mixing weight % with mole %, is the commonest numerical error in end-member questions.

3. Physical properties and a diagnostic table

Hand-specimen identification uses a fixed checklist. Luster is the way light reflects: metallic, vitreous (glassy), resinous, pearly, silky, adamantine or earthy. Streak is the colour of the powder, steadier than the surface colour. Hardness is resistance to scratching on the Mohs scale: talc 1, gypsum 2, calcite 3, fluorite 4, apatite 5, orthoclase 6, quartz 7, topaz 8, corundum 9, diamond 10. It is an ordinal scale, so corundum is not three times as hard as calcite. Cleavage is breakage along planes of weak bonding fixed by the structure; its quality (perfect to poor), number and angles are diagnostic. Fracture (conchoidal, uneven, splintery) has no such control. Density is quoted as specific gravity, the ratio to the density of water. Quartz shows piezoelectricity, magnetite is magnetic, and calcite effervesces in cold dilute HCl.

Common rock-forming minerals
Mineral and formula	Mohs hardness	Cleavage	Specific gravity
Quartz, SiO₂	7	None, conchoidal fracture	2.65
K-feldspar, KAlSi₃O₈	6	Two near 90°	about 2.6
Plagioclase, NaAlSi₃O₈ to CaAl₂Si₂O₈	6 to 6.5	Two near 90°, striated face	2.6 to 2.8
Muscovite, KAl₂(AlSi₃O₁₀)(OH)₂	2 to 2.5	One perfect	2.8 to 2.9
Biotite, K(Mg,Fe)₃(AlSi₃O₁₀)(OH)₂	2.5 to 3	One perfect	2.8 to 3.4
Hornblende (amphibole), Ca₂(Mg,Fe,Al)₅(Si,Al)₈O₂₂(OH)₂ (approx.)	5 to 6	Two at 56° and 124°	3.0 to 3.4
Augite (pyroxene), (Ca,Mg,Fe)₂Si₂O₆	5.5 to 6	Two at 87° and 93°	3.2 to 3.5
Olivine, (Mg,Fe)₂SiO₄	6.5 to 7	None, conchoidal fracture	3.2 to 4.4
Garnet, A₃B₂Si₃O₁₂	6.5 to 7.5	None	3.5 to 4.3
Calcite, CaCO₃	3	Three, rhombohedral (about 75°)	2.71
Dolomite, CaMg(CO₃)₂	3.5 to 4	Three, rhombohedral	2.85

Density from the formula: ρ = Z M/(N_A V), where Z is the number of formula units per unit cell, M the molar mass, N_A = 6.022 × 10²³ and V the cell volume. Worked example: forsterite is orthorhombic with a = 4.76 Å, b = 10.21 Å, c = 5.98 Å and Z = 4, so V = 290.6 Å³ = 290.6 × 10⁻²⁴ cm³ and ρ = 4 × 140.7 ÷ (6.022 × 10²³ × 290.6 × 10⁻²⁴) = 3.2 g/cm³. Replacing Mg by the heavier Fe raises the density, which is why fayalite is about 4.4.

4. The main rock-forming silicate groups

Olivine: nesosilicate with complete Mg–Fe solid solution; dense, green, no cleavage. It crystallises first from basaltic magma and is the main mineral of the upper mantle (peridotite). It alters readily to serpentine.
Pyroxene: single-chain silicate. Orthopyroxene (enstatite–ferrosilite) is orthorhombic; clinopyroxene (diopside–hedenbergite, augite) is monoclinic. Jadeite NaAlSi₂O₆ is a high-pressure pyroxene, and omphacite, a jadeite-bearing clinopyroxene, occurs with garnet in eclogite.
Amphibole: double-chain silicate containing OH. Hornblende is common in intermediate igneous and medium-grade metamorphic rocks; tremolite–actinolite forms in low-grade rocks and glaucophane in blueschists.
Mica: sheet silicate in which potassium ions weakly join the sheets, hence the single perfect cleavage. Muscovite is pale and aluminium-rich; biotite is dark and Mg–Fe-rich.
Feldspar: framework silicate, the commonest group in the crust. K-feldspar polymorphs are sanidine (high temperature), orthoclase (intermediate) and microcline (low temperature, cross-hatched twinning). Plagioclase shows fine parallel striations on one cleavage face from polysynthetic (albite) twinning; it is the usual way to tell it from K-feldspar in hand specimen.
Quartz: SiO₂ with every corner shared; no cleavage, conchoidal fracture, chemically and mechanically resistant, so it survives to make sandstone.
Garnet: nesosilicate A₃B₂(SiO₄)₃. The pyralspite series (pyrope, almandine, spessartine) has Al in the B site; the ugrandite series (uvarovite, grossular, andradite) has Ca in the A site. Almandine is the common garnet of metamorphic schists.

5. Polymorphs, carbonates and mineral stability

Polymorphs share a formula but differ in structure, and each is stable in its own pressure–temperature field; pressure favours the denser one. The aluminosilicates Al₂SiO₅ are the standard case: kyanite (triclinic, specific gravity about 3.6, highest pressure), andalusite (about 3.15, low pressure) and sillimanite (about 3.25, high temperature); they meet at a triple point near 0.4 GPa and roughly 500 to 550 °C. Kyanite has strongly anisotropic hardness, about 4.5 along the blade and 6.5 to 7 across it. For silica, α-quartz inverts to β-quartz at 573 °C (at one atmosphere), and coesite and stishovite are the high-pressure forms seen in ultra-high-pressure rocks and impact structures.

Carbonates are built on CO₃²⁻ triangles. Calcite CaCO₃ (trigonal) has three perfect rhombohedral cleavages and effervesces vigorously in cold dilute HCl. Dolomite CaMg(CO₃)₂ has an ordered Ca and Mg arrangement and effervesces weakly unless powdered. Aragonite is the orthorhombic polymorph of CaCO₃: denser (about 2.95), favoured at high pressure and secreted by many shell-forming organisms, but metastable at the surface and slowly inverting to calcite. Magnesite MgCO₃ and siderite FeCO₃ complete the calcite family.

Stability: some pairs cannot coexist at equilibrium because they react. Magnesian olivine and quartz make orthopyroxene: Mg₂SiO₄ + SiO₂ → 2 MgSiO₃. Olivine therefore occurs in silica-poor rocks (basalt, gabbro, peridotite) and is absent from granite. Iron-rich olivine (fayalite) can persist with quartz in some iron-rich rocks, so the rule is stated for the magnesian end.

Chemical stability at the surface runs opposite to the order of crystallisation. In Goldich's weathering series olivine and calcic plagioclase weather fastest, then pyroxene, amphibole and biotite, then sodic plagioclase and K-feldspar, with muscovite and quartz most resistant. Feldspar breaks down by hydrolysis to clay: 2 KAlSi₃O₈ + 2 H₂CO₃ + 9 H₂O → Al₂Si₂O₅(OH)₄ + 4 H₄SiO₄ + 2 K⁺ + 2 HCO₃⁻, giving kaolinite, silica and potassium in solution. Quartz and muscovite therefore survive to sand and silt, while olivine is rare in sediments.

⚠️ Class from the ratio, not the name

Read the class from the Si:O ratio and the cleavage: 1:4 means isolated tetrahedra, 1:3 single chains (pyroxene, cleavage near 90°) or rings, 4:11 double chains (amphibole, cleavage near 120°), 2:5 sheets, 1:2 framework. Beryl is a ring silicate, not a chain silicate, and kyanite, andalusite and sillimanite are all nesosilicates despite their different looks.

Key takeaways

The SiO₄ tetrahedron links by sharing corners: O per Si = 4 − b/2. Nesosilicates (1:4), sorosilicates (2:7), cyclosilicates and single chains (1:3), double chains (4:11), sheets (2:5) and frameworks (1:2) show rising polymerisation. Cleavage follows the structure: pyroxene near 90°, amphibole 56° and 124°, mica one plane.
Substitution needs similar radii (within about 15 %) and balanced charge: Mg ⇌ Fe is simple, Na Si ⇌ Ca Al in plagioclase is coupled. Exsolution gives perthite on cooling.
Compositions are mole fractions: Fo80 = (Mg₁.₆Fe₀.₄)SiO₄ and An content = Ca/(Ca + Na). Mole % differs from weight % by the molar masses (50 g each of Fo and Fa is 59 mol % Fo). Formula units are normalised to oxygens: 4 for olivine, 6 pyroxene, 8 feldspar, 12 garnet. Density = Z M/(N_A V).
Mohs hardness is ordinal (talc 1 to diamond 10). Cleavage reflects structure, fracture does not; streak is steadier than colour. Quartz 7 with no cleavage, feldspar 6 with two cleavages near 90°, calcite 3 with three rhombohedral cleavages that effervesce, mica 2 to 3 with one.
Olivine (Mg–Fe, first to crystallise), pyroxene (jadeite at high pressure), amphibole (contains OH), mica (muscovite pale, biotite dark), feldspar (plagioclase striations; K-feldspar sanidine to microcline), quartz and garnet (pyralspite and ugrandite) are the main groups.
Polymorphs share a formula: kyanite (high pressure), andalusite (low pressure), sillimanite (high temperature); calcite and denser aragonite; quartz, coesite and stishovite. Magnesian olivine and quartz react to give pyroxene, so they do not coexist.

Practice questions (18)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Which silicate class is built of isolated SiO₄ tetrahedra joined only through other cations, giving a Si:O ratio of 1:4?
1. Phyllosilicates, such as muscovite and biotite
2. Nesosilicates, such as olivine and garnet
3. Inosilicates, such as augite and hornblende
4. Tectosilicates, such as quartz and feldspar
Show answer
Answer: B — Nesosilicates, such as olivine and garnet
Isolated tetrahedra with no shared oxygens give Si:O = 1:4 and are the nesosilicates (olivine, garnet, zircon, the Al₂SiO₅ minerals). Inosilicates share two or more corners and have ratios of 1:3 and 4:11, sheets give 2:5 and frameworks 1:2, so the other classes are all more polymerised.
A dark, prismatic, rock-forming silicate in hand specimen shows two good cleavages that meet at nearly 90°. Which mineral group is it?
1. Amphibole, whose double chains give cleavages at about 56° and 124°
2. Mica, whose weakly bonded sheets give one perfect cleavage
3. Pyroxene, whose single chains give cleavages at about 87° and 93°
4. Olivine, whose isolated tetrahedra give three perfect cleavages
Show answer
Answer: C — Pyroxene, whose single chains give cleavages at about 87° and 93°
Two cleavages near 90° follow the single silicate chains of a pyroxene. Amphibole is the tempting alternative because it is also dark and prismatic, but its double chains give cleavage angles of about 56° and 124°. Mica has one cleavage, and olivine has no strong cleavage at all.
An amphibole has the double-chain unit Si₄O₁₁. Using O per Si = 4 − b/2, where b is the average number of bridging oxygens per tetrahedron, find b to one decimal place. Give the number only.
Numerical answer — type the value.
Show answer
Answer: 2.5
O per Si = 11/4 = 2.75, so 2.75 = 4 − b/2 and b = 2 × (4 − 2.75) = 2.5. Single chains (SiO₃, b = 2) give 3 oxygens per Si and sheets (Si₂O₅, b = 3) give 2.5 oxygens per Si; taking the oxygen count 2.75 itself as b is the usual slip.
Which of the following mineral and structure pairings are correct?
1. Olivine is a nesosilicate built of isolated SiO₄ tetrahedra
2. Muscovite is a tectosilicate with a three-dimensional framework
3. Quartz is a framework silicate in which every oxygen is shared, with no cleavage
4. Beryl is a single-chain inosilicate with the formula unit SiO₃
Show answer
Answer: A — Olivine is a nesosilicate built of isolated SiO₄ tetrahedra; C — Quartz is a framework silicate in which every oxygen is shared, with no cleavage
Olivine has isolated tetrahedra, and quartz is SiO₂ with all corners shared and no cleavage. Muscovite is a sheet silicate (phyllosilicate), not a framework one, and beryl contains Si₆O₁₈ rings, so it is a cyclosilicate; its ratio of 1:3 matches that of a single chain, which is why the pairing is tempting.
In plagioclase, replacement of Na⁺ by Ca²⁺ is accompanied by replacement of Si⁴⁺ by Al³⁺. Why is this coupled substitution needed?
1. Because Ca²⁺ is much larger than Na⁺ and therefore needs a larger tetrahedral site to fit into
2. Because Ca²⁺ for Na⁺ adds one positive charge and Al³⁺ for Si⁴⁺ removes one, so the charge balances
3. Because Al³⁺ and Ca²⁺ have the same ionic radius, so the two must always substitute together in a crystal
4. Because the number of oxygens in the formula changes from 8 to 6 as anorthite replaces albite
Show answer
Answer: B — Because Ca²⁺ for Na⁺ adds one positive charge and Al³⁺ for Si⁴⁺ removes one, so the charge balances
Ca²⁺ in place of Na⁺ adds +1 charge, and Al³⁺ in place of Si⁴⁺ gives −1, so the crystal stays neutral with 8 oxygens throughout. Na⁺ and Ca²⁺ have nearly the same radius, so size is not the issue, which is the tempting but wrong explanation. The oxygen count stays at 8 from albite to anorthite.
A garnet is a solid solution of 60 mol % pyrope Mg₃Al₂Si₃O₁₂ and 40 mol % almandine Fe₃Al₂Si₃O₁₂. How many Fe atoms does it have per formula unit of 12 oxygens? Give the number to one decimal place.
Numerical answer — type the value.
Show answer
Answer: 1.2
Each almandine formula unit carries 3 Fe atoms, and the almandine mole fraction is 0.40, so Fe = 3 × 0.40 = 1.2 per formula unit (and Mg = 3 × 0.60 = 1.8). Using 0.40 itself, or 2 × 0.40, mistakes the number of A-site cations.
A mixture contains 50 g of forsterite (M = 140.7 g/mol) and 50 g of fayalite (M = 203.8 g/mol). What is the mole percentage of forsterite, to the nearest whole number? Give the number only.
Numerical answer — type the value.
Show answer
Answer: 59
n(Fo) = 50 ÷ 140.7 = 0.3554 mol and n(Fa) = 50 ÷ 203.8 = 0.2453 mol. Mole % Fo = 0.3554 ÷ (0.3554 + 0.2453) × 100 = 59.2, which rounds to 59. Equal masses do not mean equal moles, and answering 50 ignores the difference in molar mass.
A plagioclase has the composition An50, meaning 50 mol % CaAl₂Si₂O₈ (M = 278.2 g/mol) and 50 mol % NaAlSi₃O₈ (M = 262.2 g/mol). What is its weight percentage of anorthite, to the nearest whole number? Give the number only.
Numerical answer — type the value.
Show answer
Answer: 51
Take 1 mol of mineral: 0.5 × 278.2 = 139.1 g anorthite and 0.5 × 262.2 = 131.1 g albite, total 270.2 g. Weight % An = 139.1 ÷ 270.2 × 100 = 51.48, which rounds to 51. The answer is close to 50 only because the two molar masses are similar.
Forsterite is orthorhombic with unit-cell edges a = 4.76 Å, b = 10.21 Å, c = 5.98 Å and Z = 4 formula units per cell (M = 140.7 g/mol, N_A = 6.022 × 10²³, 1 Å³ = 10⁻²⁴ cm³). What is its density in g/cm³, to one decimal place? Give the number only.
Numerical answer — type the value.
Show answer
Answer: 3.2
V = 4.76 × 10.21 × 5.98 = 290.6 Å³ = 290.6 × 10⁻²⁴ cm³. Mass per cell = 4 × 140.7 ÷ (6.022 × 10²³) = 9.35 × 10⁻²² g. Density = 9.35 × 10⁻²² ÷ (2.906 × 10⁻²²) = 3.2 g/cm³. Leaving out Z gives 0.8, and forgetting to convert Å³ to cm³ gives an absurd power of ten.
According to the Mohs hardness scale, which of the following statements is correct?
1. Corundum is exactly three times as hard as calcite, since 9 divided by 3 is 3
2. Calcite can scratch fluorite, since calcite has the higher hardness number
3. Gypsum can scratch calcite, because gypsum is softer and so cuts more readily
4. Quartz can scratch orthoclase but cannot scratch topaz
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Answer: D — Quartz can scratch orthoclase but cannot scratch topaz
A mineral scratches every mineral of lower Mohs number and is scratched by those of higher number: quartz (7) scratches orthoclase (6) but is scratched by topaz (8). The scale is ordinal, so the ratio of hardness numbers has no meaning, which defeats the corundum statement. Calcite (3) is softer than fluorite (4), and gypsum (2) is softer than calcite.
Which of the following statements about the physical properties of minerals are correct?
1. Streak, the colour of the powdered mineral, is usually more constant for a mineral than its surface colour
2. Mohs hardness is a linear scale, so corundum (9) is three times as hard as calcite (3)
3. Cleavage is breakage along planes of weak bonding fixed by the structure, whereas fracture is not controlled by it
4. Quartz has three perfect cleavages, which is why it breaks into rhombohedral pieces
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Answer: A — Streak, the colour of the powdered mineral, is usually more constant for a mineral than its surface colour; C — Cleavage is breakage along planes of weak bonding fixed by the structure, whereas fracture is not controlled by it
Streak is steadier than surface colour, which varies with impurities, and cleavage is controlled by crystal structure while fracture is not. The Mohs scale is ordinal, not linear, and quartz has no cleavage and breaks with a conchoidal fracture; the rhombohedral cleavage belongs to calcite.
Which feature most reliably distinguishes plagioclase from K-feldspar in hand specimen?
1. Plagioclase has only one cleavage direction whereas K-feldspar has two cleavages near 90° to each other
2. Plagioclase is much softer, near 3 on the Mohs scale, while K-feldspar is near 6 on the same scale
3. Plagioclase shows fine parallel striations on one cleavage face from polysynthetic twinning
4. Plagioclase is always pink or salmon-coloured, while K-feldspar is always white or colourless
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Answer: C — Plagioclase shows fine parallel striations on one cleavage face from polysynthetic twinning
The striations of polysynthetic (albite) twinning on one cleavage face are the diagnostic hand-specimen feature of plagioclase. Both feldspars have two cleavages near 90° and similar hardness (6 to 6.5), so those cannot separate them. Colour is unreliable, and K-feldspar is the one that is often pink or salmon, not plagioclase.
Which of the following statements about the micas are correct?
1. Micas have one perfect cleavage because weakly bonded potassium ions join the silicate sheets
2. Muscovite is the dark, Mg–Fe-rich mica and biotite is the pale, aluminium-rich mica
3. Micas are sheet silicates with tetrahedral sheets of composition near Si₂O₅
4. Micas are framework silicates, so they show no cleavage and break with a conchoidal fracture
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Answer: A — Micas have one perfect cleavage because weakly bonded potassium ions join the silicate sheets; C — Micas are sheet silicates with tetrahedral sheets of composition near Si₂O₅
The sheets are weakly bonded by potassium, which gives the single perfect cleavage, and each tetrahedral sheet has the Si₂O₅ composition (with some Al replacing Si). The statement that muscovite is the dark Mg–Fe mica has the two micas the wrong way round: biotite is the dark Mg–Fe mica. Micas are phyllosilicates, not frameworks.
A student is asked which pair of minerals would not be expected to coexist in equilibrium in a rock, because they react to form a pyroxene. Which pair is it?
1. Forsterite-rich olivine and quartz, which react as Mg₂SiO₄ + SiO₂ → 2 MgSiO₃
2. Quartz and K-feldspar, which are both framework silicates of the same oxide family
3. Olivine and calcic plagioclase, which together make up the rock troctolite
4. Quartz and muscovite, which are both common in granite and in quartz-mica schist
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Answer: A — Forsterite-rich olivine and quartz, which react as Mg₂SiO₄ + SiO₂ → 2 MgSiO₃
Magnesian olivine is silica-undersaturated and reacts with free silica to give orthopyroxene, so forsterite and quartz do not coexist at equilibrium. Olivine and calcic plagioclase coexist in troctolite and gabbro, quartz with K-feldspar or muscovite is normal in granite, and sharing a structural class (framework silicates) says nothing about reactivity.
Kyanite, andalusite and sillimanite all have the formula Al₂SiO₅. Which statement about them is correct?
1. Kyanite is the densest, high-pressure form; andalusite is the low-pressure form and sillimanite the high-temperature form
2. Andalusite is the densest polymorph, and it forms at the highest pressures reached in the deep continental crust of mountain belts
3. Sillimanite is a high-pressure, low-temperature polymorph that is characteristically found in blueschists
4. The three differ in chemical composition, with kyanite containing the largest proportion of aluminium
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Answer: A — Kyanite is the densest, high-pressure form; andalusite is the low-pressure form and sillimanite the high-temperature form
The three are polymorphs with identical composition; pressure favours the densest, kyanite (about 3.6), the least dense andalusite (about 3.15) occurs at low pressure, and sillimanite at high temperature. Swapping the roles of kyanite and andalusite is the tempting error: pressure favours the denser structure, so the low-pressure form is the least dense. They do not differ in composition.
Which statement about aragonite and calcite, both CaCO₃, is correct?
1. Aragonite is a magnesium-rich carbonate, and that magnesium content makes it denser than calcite
2. Calcite is the denser, high-pressure polymorph, and aragonite is the only form stable at the surface
3. The two form a continuous solid-solution series, with aragonite as the strontium-rich end-member of the series
4. Aragonite is the denser polymorph, favoured at high pressure, and inverts slowly to calcite at the surface
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Answer: D — Aragonite is the denser polymorph, favoured at high pressure, and inverts slowly to calcite at the surface
Aragonite and calcite are polymorphs of CaCO₃: aragonite is orthorhombic and denser (about 2.95 against 2.71), is favoured by pressure and secreted by many organisms, and is metastable at the surface so it inverts to calcite. The statement that calcite is the denser, high-pressure form reverses the facts. Polymorphs have the same composition, so neither a magnesium nor a strontium difference defines them.
Take six-fold ionic radii (Å): Mg²⁺ 0.72, Fe²⁺ 0.78, Ca²⁺ 1.00, K⁺ 1.38. For which pair of ions of equal charge is simple substitution, without any coupled substitution, most favourable?
1. Mg²⁺ and Ca²⁺, because both are divalent alkaline-earth ions
2. Mg²⁺ and Fe²⁺, because their radii differ by under 10 %
3. Fe²⁺ and Ca²⁺, because both occur in pyroxene and garnet
4. Ca²⁺ and K⁺, because both are large cations in feldspar
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Answer: B — Mg²⁺ and Fe²⁺, because their radii differ by under 10 %
Goldschmidt's rule needs the same charge and radii within about 15 %; Mg²⁺ (0.72) and Fe²⁺ (0.78) differ by about 8 %, which is why olivine shows complete Mg–Fe solid solution. Mg²⁺ and Ca²⁺ differ by about 39 %, beyond the 30 % limit, so substitution between them is minimal despite both being divalent. Ca²⁺ and K⁺ differ in charge as well as in size.
A colourless, vitreous mineral of Mohs hardness 3 breaks along three cleavage planes not at right angles and effervesces vigorously in cold dilute HCl. What is it?
1. Dolomite, which effervesces only weakly unless powdered and is slightly harder than calcite
2. Quartz, which has hardness 7, no cleavage at all and no reaction with acid
3. Calcite, which has rhombohedral cleavage and reacts vigorously with cold dilute acid
4. Gypsum, which has hardness 2, is scratched by a fingernail and does not react with dilute HCl
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Answer: C — Calcite, which has rhombohedral cleavage and reacts vigorously with cold dilute acid
Hardness 3, three rhombohedral cleavages and vigorous effervescence in cold dilute HCl identify calcite. Dolomite is the tempting neighbour, but it is harder (3.5 to 4) and reacts weakly with cold acid unless it is powdered. Quartz has no cleavage and no acid reaction, and gypsum is softer and does not effervesce.