Why a bond has a length
In Topic 2.1 you classified bonds by how electrons are shared. This topic asks a different question: why do two bonded atoms sit at one particular distance apart, and what makes one bond stronger than another?
Picture two hydrogen atoms approaching each other. Each has one proton and one electron. As they get closer, four kinds of Coulomb interaction act:
- Attractions: each nucleus attracts the other atom’s electron (as well as its own).
- Repulsions: the two nuclei repel each other, and the two electrons repel each other.
What happens depends on the internuclear distance, the distance between the two nuclei:
- Far apart. The atoms do not interact. We set the potential energy at zero here.
- Closer. The attractions grow faster than the repulsions. The potential energy falls below zero: the pair is becoming more stable, and energy is released.
- At one particular distance, attraction and repulsion are balanced. The potential energy is at its minimum. This is the bond.
- Even closer. The nuclei are now so close that their repulsion dominates. The potential energy rises steeply.
Reading a potential energy graph
Plotting potential energy against internuclear distance gives a curve with a deep, lopsided well.
Two quantities come straight off the graph:
- Bond length (equilibrium bond length): the internuclear distance at the lowest point of the curve. For H₂ it is 74 pm.
- Bond energy: the energy needed to separate the bonded atoms completely, from the minimum back up to zero. It equals the depth of the well, written as a positive number: 436 kJ mol⁻¹ for H–H.
Two points about signs:
- The curve’s minimum is at −436 kJ mol⁻¹, but the bond energy is +436 kJ mol⁻¹. Bond energy is the energy you must put in to break the bond.
- Forming one mole of H–H bonds from separate atoms releases 436 kJ. Breaking it absorbs 436 kJ. Breaking a bond never releases energy.
(Background: bonded atoms are always vibrating, so the distance oscillates a little about the bond length. You do not need this for calculations.)
What sets bond length and bond energy in covalent bonds
Two factors matter most for covalent bonds.
Factor 1: the size of the atoms’ cores
A bond joins two atoms whose bonding electrons sit outside their cores (the nucleus plus inner electrons). Larger atoms, with more occupied shells, have bigger cores, so the nuclei cannot get as close. The shared pair is further from each nucleus and is attracted less strongly. Longer bonds are therefore usually weaker.
| Bond | Bond length (pm) | Bond energy (kJ mol⁻¹) |
|---|---|---|
| C–F | 134 | 485 |
| C–Cl | 176 | 339 |
| C–Br | 193 | 276 |
| C–I | 213 | 240 |
Going down group 17, each halogen adds a shell. The bond gets longer and weaker every step. (Bond energies in tables are averages over many compounds, so treat them as typical values.)
Factor 2: bond order
Bond order is the number of shared electron pairs between two atoms: 1 for a single bond, 2 for a double bond, 3 for a triple bond. More shared pairs between the same two nuclei pull them closer together and hold them more strongly.
| Bond | Bond order | Bond length (pm) | Bond energy (kJ mol⁻¹) |
|---|---|---|---|
| C–C | 1 | 154 | 347 |
| C=C | 2 | 134 | 614 |
| C≡C | 3 | 120 | 839 |
Higher bond order: shorter bond, larger bond energy. Notice that the double bond is not twice as strong as the single bond (614 ÷ 347 = 1.77), and the triple bond is not three times as strong (839 ÷ 347 = 2.42).
Interactions between ions: Coulomb’s law
For a cation and an anion, the attraction follows Coulomb’s law, which you met in Topic 1.5:
F ∝ q₁q₂ / r²
Here q₁ and q₂ are the sizes of the ion charges and r is the distance between the centres of the ions (their nuclei). For ions in contact, r is roughly the sum of the two ionic radii. Two consequences:
- Larger charges, stronger attraction. The strength is proportional to the charge on each ion.
- Smaller ions, stronger attraction. Small ions can get closer, so r is smaller.
A potential energy curve for an ion pair has the same shape as Figure 1: attraction pulls the ions together until their electron clouds overlap and repulsion takes over. Stronger attraction means a deeper well. (The potential energy itself is proportional to q₁q₂ / r. For the ion pairs compared on this page, both forms give the same ranking.)
Worked example 1: reading the H₂ graph
Question. Use Figure 1. (a) State the bond length of H₂. (b) State the energy needed to break one mole of H–H bonds. (c) Calculate the energy needed to break the bond in one H₂ molecule. (d) Describe what happens to the potential energy as two H atoms approach from 300 pm to 50 pm, and explain the shape.
(a) The minimum is at 74 pm.
(b) The well is 436 kJ mol⁻¹ deep, so 436 kJ is needed per mole of bonds.
(c) Divide by Avogadro’s number, converting kJ to J first: 436 000 J mol⁻¹ ÷ 6.022 × 10²³ mol⁻¹ = 7.24 × 10⁻¹⁹ J per molecule.
(d) From 300 pm to 74 pm the potential energy decreases from almost zero to −436 kJ mol⁻¹, because the attraction of each nucleus for both electrons grows as the atoms approach. Below 74 pm it increases steeply: the nuclei are now so close that nucleus–nucleus (and electron–electron) repulsion dominates. By 50 pm the energy is rising quickly, and at still shorter distances it becomes positive (Figure 1).
Check. A very small energy per molecule is expected: molar quantities are about 10²³ times larger.
Worked example 2: sketching two curves on one graph
Question. The Cl–Cl bond has a length of 199 pm and a bond energy of 239 kJ mol⁻¹. The I–I bond has a length of 267 pm and a bond energy of 149 kJ mol⁻¹. (a) Describe how to sketch both potential energy curves on the same axes. (b) Explain the differences using atomic structure.
(a) A good sketch has:
- Axes with quantities and units: potential energy (kJ mol⁻¹) vertically, with zero marked; internuclear distance (pm) horizontally, starting at 0.
- A sensible scale: the distance axis must reach beyond 267 pm (say 0 to 500 pm); the energy axis must reach below −239 (say +200 to −300 kJ mol⁻¹).
- Cl₂: minimum at 199 pm and −239 kJ mol⁻¹.
- I₂: minimum further right (267 pm) and shallower (−149 kJ mol⁻¹).
- Shape: each curve rises steeply to the left of its minimum and levels off towards zero at large distance.
(b) Iodine is two periods below chlorine, so its valence electrons are in the n = 5 shell, not n = 3. Its larger core keeps the nuclei 68 pm further apart. The shared pair is further from each nucleus and more shielded, so it is attracted less strongly. The bond is therefore longer and weaker, by 90 kJ mol⁻¹.
Worked example 3: ranking ionic attractions
Question. Ionic radii: Na⁺ 102 pm, Mg²⁺ 72 pm, F⁻ 133 pm, Cl⁻ 181 pm, O²⁻ 140 pm. Rank the attraction between the ions in each pair: Na⁺ with F⁻, Na⁺ with Cl⁻, Mg²⁺ with O²⁻. Use Coulomb’s law to compare them.
| Ion pair | q₁ × q₂ | r = sum of radii (pm) | q₁q₂ / r² (pm⁻²) |
|---|---|---|---|
| Na⁺ F⁻ | 1 × 1 = 1 | 102 + 133 = 235 | 1.81 × 10⁻⁵ |
| Na⁺ Cl⁻ | 1 × 1 = 1 | 102 + 181 = 283 | 1.25 × 10⁻⁵ |
| Mg²⁺ O²⁻ | 2 × 2 = 4 | 72 + 140 = 212 | 8.90 × 10⁻⁵ |
Ranking (strongest first): Mg²⁺–O²⁻ > Na⁺–F⁻ > Na⁺–Cl⁻.
Interpretation.
- Na⁺F⁻ vs Na⁺Cl⁻ (same charges): only size differs. F⁻ is smaller, so the ions are closer and the attraction is (283 ÷ 235)² = 1.45 times stronger.
- Mg²⁺O²⁻ vs Na⁺F⁻: the charge product is 4 times larger, and the ions are slightly closer, so the attraction is 4.92 times stronger. Charge has the bigger effect here.
This is a model comparison, not a measured energy. In Topic 2.3 you will use the same reasoning to compare the melting points of ionic solids.
Common misconceptions
- “Breaking bonds releases energy.” Breaking a bond always requires energy (it lifts the pair from the bottom of the well to zero). Energy is released when bonds form.
- “Bonded atoms sit as close together as possible.” At distances shorter than the bond length, repulsion between the nuclei makes the energy rise steeply.
- “The bottom of the curve is at zero energy.” Zero is the energy of separated atoms; the bonded pair is below zero.
- “Bond energy is negative.” The minimum of the curve is negative, but bond energy is quoted as the positive energy needed to break the bond.
- “A double bond is twice as strong as a single bond.” C=C is 1.77 times as strong as C–C, not 2.
- “Bigger atoms form stronger bonds because they have more electrons.” Larger atoms usually form longer, weaker bonds: the shared pair is further from the nuclei.
- Using only charge (or only size) for ions. Both matter; check both, then decide which effect is larger.
Where this leads
The energy ideas here run through the rest of the course: bond energies return in Topic 6.7 for estimating enthalpy changes, and Coulomb’s law for ions leads straight into Topic 2.3, Structure of Ionic Solids. Bond order will make more sense once you can draw Lewis diagrams (Topic 2.5). Try the practice questions now, then use the revision notes and the checklist to consolidate.
Quick questions
Why is the potential energy negative when atoms are bonded?
Zero is chosen to be the energy of the two atoms when they are far apart and not interacting. A bonded pair is more stable than that, so its energy is below zero.
Is a double bond twice as strong as a single bond?
No. A C=C bond (614 kJ mol⁻¹) is stronger than a C–C bond (347 kJ mol⁻¹) but less than twice as strong. The second pair of electrons is shared less effectively than the first.