Appearance
10.1 — Why Atoms Bond
Two hydrogen atoms brought together release 436 kJ per mole. Two helium atoms release nothing and drift apart.
The difference is the whole subject of this Part.
Atoms bond when the resulting arrangement has lower energy than the separated atoms. That is the only rule, and everything else is working out when it applies.
Where the energy comes from
Bring two hydrogen atoms together and four electrostatic interactions appear, from Chapter 4.1:
- Electron 1 attracted to nucleus 2 — lowers energy
- Electron 2 attracted to nucleus 1 — lowers energy
- Electron 1 repelled by electron 2 — raises energy
- Nucleus 1 repelled by nucleus 2 — raises energy
At the right distance the attractions win, and there is a minimum. Push closer and the nuclear repulsion — which grows as 1/r with nothing to screen it — takes over sharply.
But that is not the whole story, and the missing part is quantum mechanical.
Chapter 7.3 established that kinetic energy is proportional to the curvature of the wavefunction: a wave squeezed into a small space must bend sharply and therefore carries a lot of kinetic energy.
When two atoms approach, the electron's wavefunction spreads over both nuclei. It now occupies twice the volume, so it can be less sharply curved, so its kinetic energy falls.
\boxed{\text{A covalent bond is electrons gaining room to spread out.}}
This is why the simple electrostatic account is incomplete and why bonding could not be explained before 1927, when Heitler and London did the first quantum calculation on H₂.
The potential energy curve
Every bond has the same shape of curve: steeply repulsive at short range, attractive at medium range, flat at long range.
Three numbers describe it:
Bond length r_0 — where the minimum sits. H₂: 74 pm.
Bond energy D_e — the depth of the well. H₂: 436 kJ/mol.
Curvature at the minimum — which sets the vibrational frequency, from Chapter 7.4's harmonic oscillator. H₂: 1.3\times10^{14} Hz.
And the zero-point energy matters. Chapter 7.9 showed the molecule cannot sit at the bottom of the well; it sits \frac{1}{2}\hbar\omega above it. For H₂ that is 26 kJ/mol, which is 6 % of the bond energy — and it is why the measured dissociation energy (432 kJ/mol) is slightly less than the well depth (436).
Why helium does not bond. Helium's 1s orbital is full. Bringing two helium atoms together would put four electrons into the shared region, and the exclusion principle (Chapter 7.7) forces two of them into a higher-energy antibonding arrangement. The antibonding penalty slightly exceeds the bonding gain, so the net is repulsive. Chapter 10.2 makes this precise.
Ionic bonding
Complete transfer of one or more electrons, between elements with a large electronegativity difference (Chapter 9.5).
\text{Na}\ ([\text{Ne}]3s^1) + \text{Cl}\ ([\text{Ne}]3s^23p^5) \to \text{Na}^+\ ([\text{Ne}]) + \text{Cl}^-\ ([\text{Ar}])
Both reach noble gas configurations.
Is the transfer favourable? Do the arithmetic
This is worth working through, because the naive answer is that it should not happen at all.
Step 1: ionise sodium. IE = +496 kJ/mol. Costs energy.
Step 2: add the electron to chlorine. EA = -349 kJ/mol. Releases energy.
Net so far: +496-349 = +147 kJ/mol. Unfavourable.
Forming isolated ion pairs from atoms costs energy. So why does sodium burn in chlorine so violently?
Step 3: the lattice. The ions do not form isolated pairs. They form a crystal in which every sodium is surrounded by six chlorides and vice versa, and every one of those interactions contributes.
The lattice energy is -787 kJ/mol, and that is what makes the whole thing work:
\Delta H = +496-349-787 = -640\ \text{kJ/mol}
Strongly favourable. (The full Born–Haber cycle also includes sublimating sodium and dissociating Cl₂, giving the standard formation enthalpy of -411 kJ/mol.)
\boxed{\text{Ionic compounds exist because of the lattice, not because the electron transfer is favourable on its own.}}
Lattice energy, derived
Consider a one-dimensional chain of alternating + and - ions spaced r apart. The energy of one ion:
U = -\frac{2ke^2}{r}+\frac{2ke^2}{2r}-\frac{2ke^2}{3r}+\dots = -\frac{2ke^2}{r}\left(1-\frac{1}{2}+\frac{1}{3}-\dots\right)
The series in brackets is \ln 2 = 0.693, so the one-dimensional Madelung constant is 2\ln 2 = 1.386.
In three dimensions the same sum over a real lattice gives the Madelung constant M, which depends only on the crystal geometry:
| Structure | M |
|---|---|
| Rock salt (NaCl) | 1.7476 |
| Caesium chloride | 1.7627 |
| Zinc blende | 1.6381 |
| Fluorite | 2.5194 |
The Born–Landé equation adds a repulsive term for the short-range overlap:
\boxed{U = -\frac{N_A M z^+z^- e^2}{4\pi\varepsilon_0 r_0}\left(1-\frac{1}{n}\right)}
where n is the Born exponent, typically 8–12, describing how steeply the repulsion rises.
Worked example: NaCl. r_0 = 282 pm, M = 1.7476, n = 8.
U = -\frac{(6.022\times10^{23})(1.7476)(1)(1)(1.602\times10^{-19})^2}{4\pi(8.854\times10^{-12})(2.82\times10^{-10})}\left(1-\frac{1}{8}\right)
Numerator: (6.022\times10^{23})(1.7476)(2.566\times10^{-38}) = 2.700\times10^{-14}
Denominator: (1.113\times10^{-10})(2.82\times10^{-10}) = 3.139\times10^{-20}
U = -\frac{2.700\times10^{-14}}{3.139\times10^{-20}}\times0.875 = -(8.601\times10^{5})(0.875) = -753\ \text{kJ/mol}
Experimental value: -787 kJ/mol. Within 4 %, from pure electrostatics plus one empirical exponent.
What the formula predicts
Lattice energy \propto z^+z^-/r_0, and both factors matter enormously.
Charge dominates. MgO has doubly charged ions, so z^+z^- = 4 instead of 1:
U_{\text{MgO}} = -3795\ \text{kJ/mol} \quad\text{against}\quad U_{\text{NaCl}} = -787
And the melting points follow: MgO melts at 2852 °C, NaCl at 801 °C. That is why magnesium oxide is used as a refractory lining in furnaces.
Size matters inversely. Down group 1 the ions get bigger and the lattice energy falls: LiF -1036, NaF -923, KF -821, RbF -785 kJ/mol.
Properties, explained
High melting points — you must overcome the whole lattice.
Hard but brittle. Push a layer sideways by one ion spacing and like charges suddenly face each other, and the crystal cleaves along that plane. This is why salt crystals shatter cleanly rather than deforming, and it is the opposite of metals.
Conduct only when molten or dissolved. The ions are the charge carriers and they are locked in place in the solid.
Soluble in water and not in petrol. Water's high dielectric constant of 80 (Chapter 4.3) cuts the attraction between ions eightyfold, and its polar molecules surround each ion — hydration — releasing energy that pays for breaking the lattice.
Whether something dissolves is a close-run competition between lattice energy and hydration energy, which is why NaCl dissolves freely and AgCl does not, despite both being ionic.
Covalent bonding
Sharing rather than transfer, between atoms of similar electronegativity.
Lewis's insight, 1916. Gilbert Lewis proposed that atoms share electron pairs to reach eight valence electrons — the octet rule — and this was eight years before quantum mechanics explained why eight.
The reason is now obvious: eight is 2 + 6, filling one s and three p orbitals. The octet rule is the ns^2np^6 configuration in disguise.
Drawing Lewis structures:
- Count total valence electrons.
- Connect atoms with single bonds.
- Complete octets on the outer atoms.
- Put remaining electrons on the central atom.
- If the central atom is short, form multiple bonds.
Worked example: CO₂. Carbon has 4 valence electrons, each oxygen 6, total 16.
Single bonds use 4. Completing octets on both oxygens uses 12 more, total 16 — but carbon then has only 4 electrons around it.
Form two double bonds: O=C=O. Now carbon has 8 and each oxygen has 8. ✔
Formal charge identifies the best structure when several are possible:
FC = (\text{valence electrons}) - (\text{lone pair electrons}) - \tfrac{1}{2}(\text{bonding electrons})
The best structure minimises formal charges and puts any negative charge on the most electronegative atom.
Where the octet rule fails
Incomplete octets. BF₃ has only 6 electrons around boron, and boron is content with it. This is why BF₃ is a powerful Lewis acid — it actively wants a lone pair from something else.
Expanded octets. SF₆ has 12 electrons around sulphur, PCl₅ has 10. Only elements in period 3 and beyond can do this. The traditional explanation invokes empty d orbitals; modern calculations say the d contribution is small and the real reason is simply that a larger atom can physically accommodate more neighbours, with the bonding described better by delocalised multi-centre orbitals.
Odd electrons. NO has 11 valence electrons and one must be unpaired. Such radicals are usually very reactive, and nitric oxide is a biological signalling molecule precisely because it is reactive and short-lived — a discovery that won the 1998 Nobel Prize in Medicine.
Bond order, length and strength
| Bond | Order | Length (pm) | Energy (kJ/mol) |
|---|---|---|---|
| C–C | 1 | 154 | 348 |
| C=C | 2 | 134 | 614 |
| C≡C | 3 | 120 | 839 |
| N–N | 1 | 145 | 163 |
| N≡N | 3 | 110 | 945 |
| O–O | 1 | 148 | 146 |
| O=O | 2 | 121 | 498 |
Higher order means shorter and stronger. More shared electrons pull the nuclei closer.
But not proportionally. A C=C double bond is 614, not 2\times348 = 696. The second bond is a sideways (pi) overlap and is inherently weaker than the head-on (sigma) one, which Chapter 10.2 explains.
Note the N–N single bond at 163 kJ/mol — remarkably weak, because the lone pairs on adjacent nitrogens repel. Combined with the 945 kJ/mol triple bond, this is what makes nitrogen compounds explosive: breaking weak N–N single bonds and forming N≡N releases an enormous amount.
Polar covalent bonds
Between the extremes. When \Delta\chi is nonzero but not large, electrons are shared unequally.
\text{H}^{\delta+}\!\!-\!\!\text{Cl}^{\delta-}
The bond has a dipole moment:
\mu = q\,d
measured in debyes (1 D = 3.336\times10^{-30} C·m).
Worked example: HCl. The measured dipole is 1.08 D and the bond length is 127 pm. What fraction of an electron has moved?
\mu_{\text{full}} = ed = (1.602\times10^{-19})(1.27\times10^{-10}) = 2.035\times10^{-29}\ \text{C m} = 6.10\ \text{D}
\text{Fractional charge} = \frac{1.08}{6.10} = 0.177
About 18 % of an electron transferred, which is close to the 21 % ionic character computed by Pauling's formula in Chapter 9.P. Two completely different methods agreeing.
Molecular polarity is a vector sum. CO₂ has two polar bonds pointing in opposite directions, so they cancel and the molecule is non-polar. Water has two polar bonds at 104.5°, so they add to a substantial 1.85 D. This one difference is why water is a liquid at room temperature and CO₂ is a gas.
Metallic bonding
The problem. Sodium has one valence electron and eight nearest neighbours. It cannot form eight covalent bonds with one electron.
The solution: delocalisation. The valence electrons are shared over the entire crystal rather than between specific pairs. A lattice of positive ions in a sea of mobile electrons.
Why this is favourable is the kinetic energy argument again, now at maximum scale. An electron delocalised over 10^{23} atoms has an enormously long wavelength and therefore very low kinetic energy. Chapter 7.4's box formula, E \propto 1/L^2, with L the size of the crystal instead of one atom.
Every metallic property follows:
Electrical conductivity — electrons are free to move (Chapter 4.4).
Thermal conductivity — the same electrons carry heat, which is the Wiedemann–Franz relation of Chapter 3.7.
Malleability. Push a layer of atoms sideways and the bonding is unchanged, because it was never directional. Contrast with ionic crystals, which shatter. This one difference — non-directional versus directional bonding — is why you can beat gold into leaf and cannot beat salt into anything.
Opacity and lustre. The free electrons absorb and re-emit light at all visible frequencies.
Strength varies with valence electron count. Sodium contributes one electron and melts at 98 °C; magnesium contributes two and melts at 650 °C; aluminium three, at 660 °C; tungsten has partly filled d orbitals contributing to directional bonding as well and melts at 3422 °C.
Band theory replaces the electron sea with the proper quantum treatment (Volume III, Chapter 2): the atomic levels broaden into bands, and whether a material conducts depends on whether the highest occupied band is full.
Comparing the three
| Ionic | Covalent | Metallic | |
|---|---|---|---|
| Electrons | Transferred | Shared locally | Shared globally |
| Between | Metal + non-metal | Non-metals | Metals |
| Directional | No | Yes | No |
| Melting point | High | Low (molecular) | Variable |
| Conducts solid | No | No | Yes |
| Mechanical | Brittle | Soft or hard | Malleable |
The three are corners of a triangle, not separate categories. Silicon carbide is covalent with 12 % ionic character; brass is metallic with covalent contributions; and every real bond sits somewhere inside.
The van Arkel–Ketelaar triangle plots average electronegativity against difference, and every compound lands somewhere in it, with the three bonding types at the corners and continuous gradation between.
Where this shows up in your life
Salt dissolves and sugar dissolves, for completely different reasons. Salt is ionic and water pulls the ions apart; sugar is covalent and polar, and water hydrogen-bonds to its hydroxyl groups.
Why you can hammer a copper pipe and not a ceramic one. Non-directional metallic bonding versus directional ionic and covalent.
Why diamond and salt are both hard but only one conducts heat well. Both have strong three-dimensional bonding; diamond's is covalent with a very stiff light lattice, giving exceptional phonon conduction.
Why frying pans have metal bases and plastic handles. Delocalised electrons carry heat; covalent molecular solids do not.
Why electrical wire is copper and insulation is polymer. One has a partly filled band, the other a large gap.
And why table salt is safe to eat while sodium metal explodes in water and chlorine gas is a chemical weapon. The ions are not the atoms, and a filled shell behaves nothing like an open one.
What the next chapter fixes
Lewis structures say which atoms are connected and nothing about the shape. Shape decides everything in biology — whether a drug fits a receptor, whether an enzyme works, whether water is polar. Chapter 10.2 derives molecular geometry from electron pair repulsion, explains what hybridisation actually means, and then shows where the simple picture fails badly with the case of oxygen's magnetism, which needs molecular orbital theory to get right.