Structure of Ionic Solids离子晶体的结构

Big Idea · Structure and Properties (SAP)AP Science Practice 1 · Models and RepresentationsAP link · Unit 2~40 min

Learning goals

  • Represent an ionic solid with a particulate model: alternating cations and anions, each surrounded by ions of opposite charge, with the right relative sizes and ratio
  • Compare lattice energies and melting points with Coulomb's law (ion charge and ion radius), and check the comparison against data
  • Explain from the structure why ionic solids are hard but brittle, and why they conduct when molten or dissolved but not as solids

Before you start

  • Atomic and ionic radius (2.3.3)
  • Ionic bonds and ionic compounds (2.4.1)
  • Intramolecular force and potential energy (U2.2)

Order in a grain of salt

Look at table salt under a magnifying glass and every grain is a tiny cube. That shape comes from the arrangement inside: NaX+\ce{Na+} and ClX−\ce{Cl-} alternate row after row and layer after layer in three dimensions, repeating over and over. AP topic 2.3 asks you to draw (or pick out) particulate models of ionic solids and to explain their properties with Coulomb’s law.

How the ions are arranged

In an ionic solid, cations and anions form a regular, periodically repeating three-dimensional array. The arrangement keeps oppositely charged ions as close as possible (maximum attraction) and like-charged ions as far apart as possible (minimum repulsion). In sodium chloride, each NaX+\ce{Na+} is surrounded by 6 ClX−\ce{Cl-}, and each ClX−\ce{Cl-} by 6 NaX+\ce{Na+}. There are no separate molecules; the formula NaCl gives only the ratio of ions (one formula unit, U1.1).

When you draw a particulate model of an ionic solid, check three things:

  • Alternation: every ion’s neighbors are ions of opposite charge.
  • Ratio: it must match the formula. NaCl has equal numbers of cations and anions; CaFX2\ce{CaF2} has twice as many FX−\ce{F-} as CaX2+\ce{Ca^2+}.
  • Relative size: use the ionic radii. In NaCl, ClX−\ce{Cl-} (181 pm) is much larger than NaX+\ce{Na+} (102 pm), but in KF the KX+\ce{K+} (138 pm) is actually a little larger than the FX−\ce{F-} (133 pm). Don’t assume cations are always drawn smaller.

Lattice energy and Coulomb’s law

Lattice energy晶格能 is the energy needed to separate 1 mol of an ionic solid completely into gaseous ions far apart. (This site uses positive values; some books define it as the energy released when the ions come together, with a negative sign. The magnitude is the same.) The larger the lattice energy, the more strongly the ions are held, and generally the higher the melting point.

By Coulomb’s law, the interaction between neighboring cations and anions is roughly proportional to

q+q−r++r−\frac{q_+ q_-}{r_+ + r_-}

Charge usually matters most: raising the charge product from 1 to 4 multiplies the interaction by 4. For equal charges, smaller ions interact more strongly. The table estimates each compound with this expression (NaCl = 1) and compares the estimate with measured lattice energies (CRC):

CompoundCharge productr++r−r_+ + r_- (pm)Coulomb estimateLattice energy (kJ/mol)Measured (NaCl = 1)Melting point (°C)
NaF12351.209231.17996
NaCl12831.007861.00801
NaBr12980.957470.95747
NaI13220.887040.90661
CaO42404.7234014.33—
MgO42125.3437914.82—

(Ionic radii: Shannon 1976, six-coordinate.) The Coulomb estimate doesn’t give exact values, but its ranking matches the measurements perfectly, and that ranking is exactly the reasoning AP asks for.

Look up ion sizes rather than guessing: OX2−\ce{O^2-} (140 pm) is actually larger than FX−\ce{F-} (133 pm). The ions in MgO are closer together than in NaF because MgX2+\ce{Mg^2+} (72 pm) is much smaller than NaX+\ce{Na+} (102 pm).

Properties of ionic solids

  • High melting and boiling points: melting means weakening the attractions between ions throughout the whole crystal, which takes a lot of energy.
  • Hard but brittle: when struck, one layer of ions shifts against another, like charges end up facing like charges, they repel, and the crystal splits along that layer.
  • Don’t conduct as solids, but conduct when molten or dissolved: conduction needs mobile charged particles. In the solid the ions are locked in the lattice; once melted or dissolved, they can move.
Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺
Before the blow: every ion is surrounded by ions of opposite charge
Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Cl⁻Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺Na⁺⇕⇕⇕⇕⇕
After the top layer shifts by one ion: like charges face each other (⇕) and repel

Choose different compounds to compare ion sizes and Coulomb estimates, then press “Shift a layer” to see why ionic solids shatter:

Try it: look at one layer of an ionic crystal and compare ionic compounds

Ionic radii: Shannon 1976 (six-coordinate, trends.ts); lattice energies and melting points: CRC (bonds.ts). All of these salts have the rock-salt (NaCl) structure and are drawn to the same scale.

Example 1: Ranking lattice energies

Rank MgO, CaO, NaF, and NaCl from largest to smallest lattice energy, justify the ranking, and check it against data.

Step-by-step solution

  1. Identify: what sets lattice energy. By Coulomb’s law, the attraction between ions is proportional to the product of their charges and inversely proportional to their separation:

    E∝q1q2rE \propto \frac{q_1 q_2}{r}

    Charge usually matters more than radius, so compare charges first, then radii.

  2. Compare charge products.

    • MgO, CaO: 2 × 2 = 4
    • NaF, NaCl: 1 × 1 = 1

    MgO and CaO have far larger lattice energies than NaF and NaCl.

  3. Same charges: compare ion sizes.

    • MgX2+\ce{Mg^2+} (72 pm) < CaX2+\ce{Ca^2+} (100 pm), so MgO > CaO
    • FX−\ce{F-} (133 pm) < ClX−\ce{Cl-} (181 pm), so NaF > NaCl

  4. The ranking.

    MgO>CaO>NaF>NaCl\text{MgO} > \text{CaO} > \text{NaF} > \text{NaCl}

  5. Check with data. 3791 > 3401 > 923 > 786 (kJ/mol), in the same order; the biggest gap is between the two pairs whose charge products differ by a factor of 4.

Example 2: Why is the lattice energy of MgO more than 4 times that of NaF?

The two compounds have similar ion separations (212 pm and 235 pm), yet the lattice energy of MgO (3791 kJ/mol) is more than 4 times that of NaF (923 kJ/mol). Explain with Coulomb’s law.

Step-by-step solution

  1. Identify: write Coulomb’s law as a proportion. Lattice energy E ∝ q₁q₂/r, so compare the two compounds through the ratio of q₁q₂/r.

  2. Compare charge products.

    (q1q2)MgO(q1q2)NaF=2×21×1=4\frac{(q_1 q_2)_{\ce{MgO}}}{(q_1 q_2)_{\ce{NaF}}} = \frac{2 \times 2}{1 \times 1} = 4

  3. Compare separations. MgO’s 212 pm is a bit smaller than NaF’s 235 pm, so its attraction is a little stronger still:

    rNaFrMgO=235212=1.11\frac{r_{\ce{NaF}}}{r_{\ce{MgO}}} = \frac{235}{212} = 1.11

  4. Combine them.

    EMgOENaF≈4×1.11≈4.4\begin{aligned} \frac{E_{\ce{MgO}}}{E_{\ce{NaF}}} &\approx 4 \times 1.11 \\ &\approx 4.4 \end{aligned}

  5. Compare with the measured ratio.

    3791923=4.1\frac{3791}{923} = 4.1

    The estimate of 4.4 is very close to the measured 4.1.

  6. Connect to the conclusion. The factor of 4 comes from charge and only about 1.1 from distance: the main reason is the larger ion charges; the smaller separation is secondary.

Practice: which has the larger lattice energy?
  1. NaCl or KCl
  2. MgO or NaF
  3. CaO or MgO
  4. NaF or NaI
  5. LiF or NaF
  6. NaBr or NaCl
  7. CaO or KF
  8. KCl or NaBr

Common mistakes

Mistake 1: “A NaCl crystal is made of NaCl molecules.” An ionic solid has no molecules: each NaX+\ce{Na+} is surrounded by 6 ClX−\ce{Cl-} at once, and no ClX−\ce{Cl-} belongs to any particular NaX+\ce{Na+}. NaCl only states the 1 : 1 ratio.

Mistake 2: “Cations are always smaller than anions.” Comparing ions from the same period, cations are indeed much smaller, but KX+\ce{K+} (138 pm) is larger than FX−\ce{F-} (133 pm), and CsX+\ce{Cs+} (167 pm) larger still. Draw particulate models using actual radii.

Mistake 3: “Ionic compounds contain ions, so the solid conducts electricity.” Conduction needs charged particles that can move. In the solid the ions are locked in place, so it doesn’t conduct; once melted or dissolved, the ions move freely and it conducts.

Quick check

Which diagram best represents one layer of ions in solid KCl?

MgO has a much higher melting point than NaF. Which is the best explanation?

Solid NaCl does not conduct electricity, but molten NaCl does. Why?

Which ranks the lattice energies correctly, from largest to smallest?

Which diagram best represents one layer of ions in solid CaFX2\ce{CaF2}? (CaX2+\ce{Ca^2+} 100 pm, FX−\ce{F-} 133 pm)

AP-style free response

AP-style free response

(a) Describe three features that a particulate model of solid NaCl should show.

(b) The lattice energy of NaF is 923 kJ/mol and that of NaI is 704 kJ/mol. Explain the difference using Coulomb’s law.

(c) Predict whether KF or NaF has the larger lattice energy, and justify your prediction.

(d) Using the particulate model of NaCl, explain why a NaCl crystal shatters rather than bends when struck with a hammer.

Model answer and scoring

(a) (1) NaX+\ce{Na+} and ClX−\ce{Cl-} alternate, so every ion’s neighbors have the opposite charge; (2) there are equal numbers of the two ions (1 : 1); (3) ClX−\ce{Cl-} is drawn larger than NaX+\ce{Na+}.

(b) Both have the same ion charges (+1 and −1). FX−\ce{F-} is much smaller than IX−\ce{I-}, so the cation and anion centers are closer in NaF. By Coulomb’s law, a smaller distance means stronger attraction, so more energy is needed to separate the crystal, and NaF has the larger lattice energy.

(c) NaF. The charges are the same and so is the anion; NaX+\ce{Na+} is smaller than KX+\ce{K+} (one fewer electron shell), so the ions in NaF are closer and attract more strongly.

(d) When struck, one layer of ions shifts relative to another, so ions that were next to opposite charges end up facing like charges. The strong repulsion between them splits the crystal along that layer, which is why ionic solids are brittle.

Scoring: (a) alternation, ratio, and relative size; (b) notes the equal charges and compares ion size and distance; (c) both the prediction and the reason (smaller NaX+\ce{Na+}); (d) “like charges face each other and repel”.

Summary

  • In an ionic solid, cations and anions form a periodic three-dimensional array, with opposite charges adjacent and like charges kept apart; there are no molecules, and the formula gives the ratio.
  • Draw particulate models correctly: alternating ions, the right ratio, and correct relative sizes (KX+\ce{K+} can be larger than FX−\ce{F-}).
  • Lattice energy ∝ q+q−/(r++r−)q_+ q_- / (r_+ + r_-): larger charges and smaller ions mean a larger lattice energy and usually a higher melting point; charge usually matters most.
  • Ionic solids have high melting points and are hard but brittle; they don’t conduct as solids but do when molten or dissolved.

AP tip: Two question types are common here: rank several ionic compounds by lattice energy or melting point and justify the order with Coulomb’s law (charge first, then ion size), and pick the particle diagram that correctly represents an ionic solid (check alternation, ratio, and relative size). Ionic solids are compared with other types of solids in U3.2.

Practice this chapter: Unit 2 · Compound Structure and Properties · 12 questionsPractice this topic: U2.3 · Structure of Ionic Solids · 7 questions

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