Why not build bridges from pure iron?
Pure iron is fairly soft and bends easily under a load. Add a small amount of carbon (usually less than 2%) and it becomes hard steel. Pure copper is fairly soft too; add zinc and it becomes brass, hard enough for gears and musical instruments. Why does adding so few other atoms change the properties so much? AP topic 2.4 asks you to answer with a particulate model.
The particulate model of a metal
Lesson 2.4.3 introduced the sea-of-electrons model: a metal’s valence electrons don’t belong to any one atom but are delocalized离域 through the whole piece of metal and shared by all the metal cations, which sit in an orderly lattice immersed in that sea of electrons. The model explains the two most important properties:
- Electrical conductivity: the delocalized electrons can move freely; connected to a power source, they drift in one direction and carry a current.
- Malleability and ductility: the sea of electrons surrounds every cation, so metallic bonding is nondirectional. Under a force, layers of atoms can slide past each other. After sliding, every cation is still surrounded by the sea of electrons, so the metallic bonding isn’t broken, and the metal can be hammered into sheets and drawn into wire. Ionic solids are different (U2.3): when one layer of ions shifts, like charges face each other and the crystal shatters.
A quick review: run a current through a metal, hammer it, and compare with an ionic solid:
Two kinds of alloy
An alloy合金 is a material made by melting other elements (metals or nonmetals) into a metal that still has metallic properties. It is a mixture (a solid solution) whose composition can vary within a range; it has no fixed formula. AP sorts alloys into two kinds by whether the added atoms can take the place of host metal atoms:
- Substitutional alloy替代型合金: the added atoms are similar in size to the host atoms and simply replace some of them in the lattice.
- Interstitial alloy间隙型合金: the added atoms are much smaller. They don’t take lattice sites but fit into the spaces (interstices) between host atoms. Atoms small enough to do this are usually H, B, C, and N.
To decide which kind, compare atomic radii (Slater empirical radii):
| Alloy | Host metal | Added element | Radius ratio | Type |
|---|---|---|---|---|
| Brass | Cu, 135 pm | Zn, 135 pm | 1.00 | substitutional |
| Bronze | Cu, 135 pm | Sn, 145 pm | 1.07 | substitutional |
| Cupronickel | Cu, 135 pm | Ni, 135 pm | 1.00 | substitutional |
| Stainless steel (chromium) | Fe, 140 pm | Cr, 140 pm | 1.00 | substitutional |
| Carbon steel | Fe, 140 pm | C, 70 pm | 0.50 | interstitial |
| Nitrided steel | Fe, 140 pm | N, 65 pm | 0.46 | interstitial |
| Nickel–boron alloy | Ni, 135 pm | B, 85 pm | 0.63 | interstitial |
One alloy can contain both kinds: in stainless steel, chromium replaces iron atoms, while its small amount of carbon still sits in the interstices.
Enrichment (beyond the AP core): Metallurgists use the Hume-Rothery rules of thumb: two kinds of atom can replace each other over a wide range only if their radii differ by no more than about 15% and their crystal structures, electronegativities, and valences are also similar. For example, Cu and Ni can replace each other in any proportion, while the zinc in brass and the tin in bronze have upper limits, beyond which other crystal structures appear. AP only asks you to compare radii and decide the type.
Properties of alloys
- Greater hardness and strength: in either kind, the foreign atoms disrupt the orderly layers and make them harder to slide. In a substitutional alloy, the substituted atoms are not quite the same as the host atoms (slightly different in size and number of electrons), so the layers are no longer perfectly regular; in an interstitial alloy, small atoms wedged between the layers have an even bigger effect, which is why steel is much harder than pure iron. In a substitutional alloy, the more the added atoms differ in size from the host atoms, the more they disrupt the layers, and the harder the alloy tends to be.
- Malleability and ductility: substitutional alloys largely keep the metal’s malleability and ductility (brass can still be drawn into wire and pressed into sheets); interstitial alloys are more rigid, with clearly reduced malleability and ductility.
- Electrical conductivity: alloys still have a sea of electrons, so they still conduct; the foreign atoms disturb the lattice, so they generally conduct somewhat less well than the pure metal.
- Density: a substitutional alloy’s density is usually between those of the two metals. An interstitial alloy adds atoms to the existing lattice with almost no change in volume, so the model predicts a slightly higher density (the real change is small).
Alloying can change a metal’s chemistry too: the chromium in stainless steel forms a thin, tight layer of chromium oxide on the surface that seals the iron off from air and water, so it resists rusting.
Choose the pure metal or an alloy, change the number of added atoms, then press “Push” to compare how the layers slide:
Atomic radii: Slater 1964 empirical radii (alloys.ts), all drawn to one scale. The arrangement is schematic: iron atoms in steel are not really in a square grid.
Example 1: Identifying the type of alloy
Use atomic radii to identify the type of each alloy, and describe its particulate model: (1) brass (Cu 135 pm, Zn 135 pm); (2) carbon steel (Fe 140 pm, C 70 pm); (3) bronze (Cu 135 pm, Sn 145 pm).
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Identify: what decides the type. Compare the relative size of the added atom and the host metal atom: similar radii means it replaces host atoms on lattice sites (substitutional alloy); much smaller means it squeezes into the spaces between them (interstitial alloy).
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Compute the radius ratios.
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Compare and classify.
- Brass (1.00): same size, substitutional
- Carbon steel (0.50): C is half the size of Fe, interstitial
- Bronze (1.07): similar size, substitutional
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Describe the particulate models.
- Brass: some Cu atoms are replaced by Zn atoms of the same size; every atom still sits on the same set of lattice sites.
- Carbon steel: Fe atoms form the lattice, and the much smaller C atoms are drawn in the spaces between them.
- Bronze: Sn atoms replace some Cu atoms and are drawn slightly larger.
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Note. The deciding factor is relative size, not the amount added: brass is often 30% Zn or more and is still substitutional.
Example 2: Why is steel harder than iron but harder to bend?
Use a particulate model to explain why carbon steel is harder than pure iron and less malleable and ductile, yet still conducts electricity.
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Identify: the structure of pure iron. The iron atoms are all the same size and packed neatly in layers. Under a force, the layers slide past each other along a slip plane without breaking the metallic bonding (the attraction between the atoms and the delocalized electrons), so pure iron is fairly soft and quite malleable and ductile.
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Compare: after adding carbon. The small C atoms squeeze into the spaces between Fe atoms and sit between the layers, wedging them so they can’t slide easily.
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Connect to hardness and malleability. With the layers hard to slide:
- it resists deformation better → steel is harder and stronger
- it is harder to bend or stretch → less malleable and ductile
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Connect to conductivity. The valence electrons of Fe are still delocalized and free to move through the solid, and a little carbon doesn’t change that, so steel still conducts electricity.
- Zn (135 pm) added to Cu (135 pm)
Ratio 1.00, the same size, so Zn replaces Cu. This is brass.
- C (70 pm) added to Fe (140 pm)
Ratio 0.50; the C atoms are much smaller and go into the interstices. This is carbon steel.
- Sn (145 pm) added to Cu (135 pm)
Ratio about 1.07, similar sizes. This is bronze.
- Cr (140 pm) added to Fe (140 pm)
The same radius, so Cr replaces Fe, as the chromium in stainless steel does.
- N (65 pm) added to Fe (140 pm)
Ratio about 0.46; N atoms go into the interstices. Nitriding hardens the surface of steel.
- B (85 pm) added to Ni (135 pm)
Ratio about 0.63; B atoms are much smaller than Ni atoms and, like C and N, go into the interstices.
- H (25 pm) added to Ti (140 pm)
H atoms are tiny, with a ratio of about 0.18, so they can only go into the interstices.
- Cu (135 pm) added to Ni (135 pm)
The same radius; Cu and Ni can replace each other in any proportion.
Common mistakes
Mistake 1: “An alloy is a compound formed from two elements.” An alloy is a mixture (a solid solution) whose composition can vary within a range; it has no fixed formula, and its atoms are still held by metallic bonding. Brasses with different proportions of zinc are simply different grades of brass.
Mistake 2: “Steel contains carbon, a nonmetal, so it is no longer a metal and doesn’t conduct.” Carbon is only a small part of steel. The valence electrons of iron still form a sea of electrons, so steel still has a metallic luster and conducts electricity. Every alloy keeps metallic properties; that is part of what makes it an alloy.
Mistake 3: “A small amount of added element makes an interstitial alloy, and a large amount makes a substitutional one.” The type depends on the relative size of the atoms, not the amount added. However little zinc is added, it replaces Cu atoms; however much carbon steel contains, the carbon is in the interstices.
Quick check
Which diagram best represents the structure of brass (Cu 135 pm, Zn 135 pm)?
Cu and Zn have the same radius, so Zn atoms occupy some Cu lattice sites; they are drawn the same size and mixed in among the Cu atoms. A draws Zn too large; B draws Zn as small atoms in the interstices, which is an interstitial alloy; D puts Zn in a separate layer instead of mixing it with Cu.
Nickel forms an alloy with a small amount of boron (Ni 135 pm, B 85 pm). Which diagram best represents its structure?
B atoms are much smaller than Ni atoms (ratio about 0.63), so like C and N they fit into the interstices between Ni atoms instead of taking lattice sites. B puts the small atoms on lattice sites; C and D draw B as large as Ni.
High-carbon steel is harder and more difficult to bend than pure iron. Which is the best explanation?
Hardness depends on whether the layers can slide. A: the carbon in steel is spread out as single atoms in the interstices of the iron, not present as lumps of carbon; C: the density changes very little and isn't the cause of hardness; D: thermal conductivity has nothing to do with hardness.
A metal can be hammered into a thin sheet without shattering. Which is the best explanation?
The sea of electrons has no direction: after the layers shift, every cation is still surrounded by the sea of electrons, and like charges never end up facing each other as they do in an ionic solid. A: metallic bonds have no fixed direction; C: the cations do repel each other, and the sea of electrons holds them together; D describes localized covalent bonds.
What is the main reason copper and zinc can form a substitutional alloy?
Whether atoms can substitute depends mainly on size (135 pm and 135 pm). A describes forming an ionic bond; B is a true fact but not the reason; C is a wrong fact, since Cu is in group 11 and Zn in group 12.
Metal M has an atomic radius of 140 pm. Equal proportions of metal X (139 pm) and of metal Z (124 pm) are separately added to M, and both form substitutional alloys. Which alloy is likely to be harder?
Both are substitutional (radius ratios about 0.99 and 0.89). The more the added atoms differ in size from the host, the less regular the layers and the harder it is for them to slide, so the alloy tends to be harder. B: Z is only somewhat smaller than M and takes lattice sites, not interstices; D: added atoms do change the hardness.
Stainless steel consists of Fe (140 pm), Cr (140 pm), and a little C (70 pm), with a thin film of chromium oxide on its surface. Which diagram best represents stainless steel near its surface?
Cr is the same size as Fe and replaces some Fe atoms; C is much smaller and sits in the interstices; the surface is a tight film of chromium oxide that seals the iron inside off from air and water. A is missing the oxide layer; B draws Cr as an interstitial atom; C swaps the places of Cr and C.
AP-style free response
Slater atomic radii: Fe 140 pm, Cr 140 pm, C 70 pm. Stainless steel consists mainly of Fe and Cr, with a small amount of C.
(a) Identify the type of alloy that Cr forms with iron and the type that C forms with iron, and justify each.
(b) Describe the features that a particulate diagram of stainless steel containing Cr and C should show.
(c) Carbon steel is harder to bend than pure iron. Explain using a particulate model.
(d) A student says, “Carbon, a nonmetal, has been added to stainless steel, so it doesn’t conduct electricity.” Evaluate this claim.
Model answer and scoring
(a) Cr: substitutional. Cr has the same radius as Fe (140 pm), so Cr atoms can occupy Fe lattice sites. C: interstitial. The radius of C (70 pm) is only half that of Fe, so C atoms fit into the spaces between Fe atoms.
(b) (1) Fe atoms in a regular lattice; (2) some lattice sites occupied by Cr atoms drawn the same size as Fe atoms; (3) a few much smaller C atoms in the spaces between atoms, not on lattice sites; (4) Fe atoms the most numerous, then Cr, with C the fewest.
(c) In pure iron the atoms are all the same size, so under a force the layers can slide along a slip plane. In carbon steel, small C atoms wedge between the layers of Fe atoms and keep them from sliding past each other, so the steel is harder and more difficult to bend.
(d) The claim is incorrect. Stainless steel is still an alloy: the valence electrons of Fe and Cr are still delocalized in a sea of electrons and can move freely, so stainless steel conducts electricity.
Scoring: (a) both types correct, each justified by comparing radii; (b) the lattice, substituted atoms of the same size, and small interstitial atoms in the spaces; (c) states that the layers can’t slide easily; (d) identifies the claim as wrong and explains with delocalized electrons (the sea of electrons).
Summary
- Particulate model of a metal: metal cations in a lattice, with valence electrons delocalized in a sea of electrons. The mobile electrons make metals conduct; metallic bonding is nondirectional and survives when layers slide, so metals are malleable and ductile.
- An alloy is a mixture that keeps metallic properties. If the added atoms are similar in size to the host atoms, they form a substitutional alloy (brass, bronze); if they are much smaller, they fit into the interstices and form an interstitial alloy (carbon steel).
- Both kinds are harder than the pure metal (the more a substituted atom differs in size from the host, the harder the alloy tends to be); substitutional alloys largely keep malleability and ductility, while interstitial alloys are more rigid and less malleable and ductile. Alloys still conduct.
AP tip: Typical questions give atomic radii and ask you to identify the type of alloy, pick the correct particle diagram, or explain how an alloy’s properties differ from the pure metal’s. Take each explanation down to the particle level: atomic size decides the type, whether the layers can slide decides hardness and malleability, and delocalized electrons decide conductivity. Metallic solids are compared with other types of solids in U3.2.