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Lesson 2 · 25 min

Covalent bonding: sharing a pair

Why does the energy of two approaching hydrogen atoms first fall and then rise? Read bond length, bond energy and zero-point energy off a curve computed from spectroscopic data, then see how bond order changes length and strength.

In this lesson

  • Explain how two neutral atoms can be held together by a shared pair of electrons
  • Read bond length, well depth and dissociation energy off a potential-energy curve, and explain the zero-point energy
  • Relate bond order to bond length and bond energy
  • Estimate reaction enthalpies from average bond enthalpies and know where such estimates go wrong

Hydrogen is the simplest molecule there is: two protons and two electrons. There are no ions here; the two atoms are identical and neither “wants” the electrons more. Yet pulling H₂ apart into atoms takes 436 kJ per mole — stronger than many bonds in ionic crystals. Run it backwards and hydrogen atoms recombine so violently that an atomic-hydrogen torch burns above 3000 °C. What pulls two neutral atoms together?

Predict first

Two hydrogen atoms approach slowly from far apart. At what internuclear distance is the energy lowest?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Why a shared pair makes a bond

The essence of a covalent bond is two nuclei attracting the same pair of electrons. An electron between the nuclei feels both of them, which is more stable than belonging to one atom alone; for two electrons to crowd into the same region their spins must be opposite — hence the “pair”.

The minimum of the curve gives the equilibrium bond length rer_\mathrm{e} (74.1 pm for H₂) and the well depth DeD_\mathrm{e} (458 kJ/mol). But a molecule never rests at the bottom of the well: quantum mechanics insists it keeps some vibrational energy even at 0 K, the zero-point energy (about 26 kJ/mol for H₂). Breaking the bond therefore takes D0=De−D_0 = D_\mathrm{e} - zero-point energy =432= 432 kJ/mol; the bond enthalpy measured at 298 K is a little higher again, 436 kJ/mol.

V(r)=De[(1−e−a (r−re))2−1],a=2πc ν~eμ2DeV(r) = D_\mathrm{e}\left[\left(1 - e^{-a\,(r - r_\mathrm{e})}\right)^2 - 1\right],\qquad a = 2\pi c\,\tilde{\nu}_\mathrm{e}\sqrt{\frac{\mu}{2D_\mathrm{e}}}
The Morse potential: an empirical function that draws the whole curve from three measured numbers — bond length, well depth and vibrational frequency.
rer_\mathrm{e}
equilibrium bond length
DeD_\mathrm{e}
well depth = measured dissociation energy D₀ + zero-point energy
ν~e\tilde{\nu}_\mathrm{e}
vibrational wavenumber (4401 cm⁻¹ for H₂); sets how sharply the well curves
μ\mu
reduced mass of the two atoms
Morse potential · computed from spectroscopic constants

Potential-energy curve of a bond

separated atoms H₂ HCl F₂ Cl₂ O₂ N₂ 50100150200250300350400450Internuclear distance r / pm−1000−5000Energy V / (kJ/mol)
Distance
159 pm
Energy
−441 kJ/mol
Force between the atoms
stretched by 49 pm: pulled back together
Bond length rₑ
109.8 pm
Well depth Dₑ
956 kJ/mol
Dissociation energy D₀
942 kJ/mol
Zero-point energy
14.1 kJ/mol
Vibrational wavenumber ωₑ
2359 cm⁻¹
Bond order
3
Bound levels in the Morse well
68

A molecule never rests at the bottom of the well: even at 0 K it vibrates in the v = 0 level. Breaking the bond therefore takes D₀ = Dₑ − zero-point energy.

The curve is the Morse function V = Dₑ[(1 − e^(−a(r−rₑ)))² − 1]. rₑ, ωₑ, ωₑxₑ and D₀ come from Huber & Herzberg’s Constants of Diatomic Molecules (NIST WebBook); Dₑ = D₀ + zero-point energy and a follows from ωₑ and the reduced mass — no parameter is tuned by hand. The Morse form is an approximation: for H₂ it supports 17 bound vibrational levels, while real H₂ has 15.
Overlay F₂, O₂ and N₂: from single to double to triple bond the well gets deeper and narrower and the bond shorter. Compare F₂ with Cl₂ too: both single bonds, yet F–F is surprisingly weak.

Bond order, length and energy

Two atoms can share one, two or three pairs of electrons — a single, double or triple bond; the number of shared pairs is the bond order. The higher the bond order, the more electron density sits between the nuclei, pulling them closer and binding them harder. Carbon–carbon bonds show it most cleanly in the table below: 154 pm, 134 pm and 120 pm, with bond enthalpies rising in step. But a double bond is not twice a single bond: the second and third pairs occupy different regions of space (π bonds, see the hybridisation lesson) and overlap less well than the first.

Bond order, length and enthalpy (N≡N and O=O are values for the diatomic molecules themselves; the rest are averages over many molecules)
BondOrderLength / pmEnthalpy / (kJ/mol)
C–C1154348
C=C2134614
C≡C3120839
N–N1145163
N=N2125418
N≡N3110945
O–O1148146
O=O2121498
Source: Average bond enthalpies and typical lengths: T. L. Brown et al., Chemistry: The Central Science, Table 8.4; N≡N and O=O: D₂₉₈ and rₑ of N₂ and O₂ from the CRC Handbook of Chemistry and Physics and NIST WebBook.

Average bond enthalpies are handy for estimating reaction enthalpies: breaking bonds absorbs energy, forming bonds releases it, and the difference is the reaction enthalpy. It is an estimate because the same kind of bond has different strengths in different molecules (C–H dissociation energies range from about 400 to 440 kJ/mol).

Worked example

Estimate ΔH for H₂ + Cl₂ → 2HCl from bond enthalpies

Bond enthalpies (kJ/mol): H–H 436, Cl–Cl 242, H–Cl 431. Estimate ΔH for HX2+ClX2→2 HCl\ce{H2} + \ce{Cl2} \mathrel{\ce{->}} \ce{2HCl} and compare with the measured ΔfH∘(HCl)=−92.3\Delta_\mathrm{f}H^\circ(\ce{HCl}) = -92.3 kJ/mol.

Concept check

Why is H₂’s dissociation energy D₀ (432 kJ/mol) smaller than its well depth Dₑ (458 kJ/mol)?

Try the calculation

Estimate ΔH for HX2+FX2→2 HF\ce{H2} + \ce{F2} \mathrel{\ce{->}} \ce{2HF} from bond enthalpies: H–H 436, F–F 159, H–F 570 kJ/mol.

kJ/mol

Key takeaways

  • A covalent bond is two nuclei attracting the same pair of electrons, whose spins are opposite.
  • The minimum of the curve gives the bond length; the dissociation energy is measured from the v = 0 level: D₀ = Dₑ − zero-point energy.
  • Higher bond order means shorter, stronger bonds — though a double bond is not twice a single one.
  • Breaking bonds absorbs energy and forming them releases it; average bond enthalpies estimate reaction enthalpies, with errors because bond strengths vary between molecules.
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Data and references

  1. K. P. Huber, G. Herzberg, Constants of Diatomic Molecules (Van Nostrand Reinhold, 1979), via the NIST Chemistry WebBook: rₑ, ωₑ, ωₑxₑ, D₀ for H₂, HCl, F₂, Cl₂, O₂, N₂.
  2. P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels”, Phys. Rev. 34, 57 (1929).
  3. W. M. Haynes (ed.), CRC Handbook of Chemistry and Physics, 97th ed., “Bond Dissociation Energies” (Y.-R. Luo) and standard enthalpies of formation.
  4. T. L. Brown, H. E. LeMay, B. E. Bursten et al., Chemistry: The Central Science, Table 8.4 (average bond enthalpies).