Bond Energies and Potential Curves

01 / UNDERSTAND & APPLY

1. Choose the energy zero before reading a bond curve

A diatomic potential-energy curve plots energy against internuclear separation r for a specified electronic state. A common convention sets U(∞)=0 for separated atoms. A bound state then lies in a negative-energy well. The negative potential energy means stabilization relative to that reference, not a negative amount of energy required to break the bond.

At large separation the interaction approaches the chosen zero. At intermediate separation attraction lowers the energy. At very small separation strong repulsive effects involving overlapping electron distributions and nuclear repulsion raise it steeply. A stable equilibrium separation r_e occurs at a local minimum.

The well depth D_e is the positive difference between the minimum and the separated-atom limit: D_e=U(∞)−U(r_e). If the minimum is −4.0 eV and the limit is zero, the well depth is +4.0 eV. Changing the plotted energy zero shifts the entire curve but does not change forces or energy differences.

A curve is not a literal orbit followed by atoms and is not a graph of energy versus time. The nuclei can vibrate on the potential, and the allowed quantum states depend on both the curve and their masses. Different electronic states can have different curves, so one drawing is not every possible state of a molecule.

02 / UNDERSTAND & APPLY

2. Force is minus the slope, not the energy itself

For a one-dimensional separation coordinate, the generalized radial force is F_r=−dU/dr, with positive direction chosen toward increasing separation. On the compressed side of a usual bonding minimum, U decreases as r increases, so its slope is negative and the force is positive: the atoms are pushed apart.

On the stretched side, U increases toward its asymptote as r increases, so the slope is positive and the force is negative: attraction tends to reduce separation. At the minimum dU/dr=0, so the net force is zero even though the attractive and repulsive contributions individually need not vanish.

A stationary point is stable when small displacements lead to restoring forces; mathematically d²U/dr²>0 at a smooth minimum. A maximum has negative curvature and is unstable along that coordinate. Zero first derivative alone is therefore not enough to establish stability.

Energy and force have different units. If U is energy per pair in joules and r is metres, the derivative gives newtons. If U is in kJ mol⁻¹ and r in picometres, the numerical slope must be converted from molar energy and length before it can be interpreted as force on one pair.

03 / UNDERSTAND & APPLY

3. Curvature controls small-amplitude vibration

Near a minimum, Taylor expansion gives U(r)≈U(r_e)+½k(r−r_e)², where k=(d²U/dr²) at r_e is the force constant. The linear term vanishes because the slope at equilibrium is zero. A larger k means a steeper, narrower local well and greater resistance to a small displacement.

Differentiation gives F_r≈−k(r−r_e), the restoring-force law. With k in N m⁻¹ and displacement in m, the force is in N. This approximation is local; extending a parabola to arbitrary separation would predict infinite dissociation energy, which is not physically suitable for a real breakable bond.

For a diatomic harmonic oscillator, angular frequency ω=√(k/μ), with reduced mass μ=m₁m₂/(m₁+m₂). Ordinary frequency is ν=ω/(2π). Increasing reduced mass at fixed force constant lowers vibrational frequency; isotope substitution can therefore shift spectra without a large change in electronic bonding.

Well depth and curvature are related only through a specified potential model. Two different curves can have the same depth and different stiffness, or the same stiffness and different depths. A bond dissociation energy alone does not uniquely determine its vibrational frequency.

04 / UNDERSTAND & APPLY

4. An anharmonic model can approach a dissociation limit

A useful model is U(r)=D_e[(1−e^(−a(r−r_e)))²−1], with D_e>0 and a>0. This version sets separated atoms at zero and the minimum at −D_e. The parameter a has inverse-length units, so the exponent is dimensionless. Other books add D_e to the whole expression and put the minimum at zero; that is a reference change, not a different force.

The repulsive side rises steeply while the stretched side approaches zero, unlike a symmetric harmonic parabola. Differentiation gives F_r=2D_e a[e^(−2a(r−r_e))−e^(−a(r−r_e))]. Near the minimum k=2D_ea² when D_e is energy per pair and consistent length units are used.

Anharmonicity means the potential is not exactly quadratic. In a quantum treatment, high vibrational levels become more closely spaced as dissociation is approached, rather than remaining equally spaced forever. The Morse model is useful but not exact, and its far-distance tail does not reproduce every real interaction law.

The plotted curve in this lesson uses explicitly hypothetical parameters. It illustrates minima, force directions, zero choice and finite dissociation; it is not a measured potential for a named molecule. Real potential-energy surfaces for polyatomic molecules involve multiple bond lengths and angles, not just one coordinate.

05 / UNDERSTAND & APPLY

5. D_e and D_0 refer to different starting levels

Even the lowest quantum vibrational state lies above the classical minimum. In the harmonic approximation its zero-point energy is E_ZPE=½hν. Therefore the dissociation energy from the lowest vibrational level, D₀, is smaller than the well depth D_e: D₀=D_e−E_ZPE in the simple one-channel picture.

The difference matters when comparing a potential minimum with a spectroscopic dissociation measurement. A molecule in an excited vibrational level needs still less additional energy to reach the same dissociation limit, although selection rules and kinetics affect how it gets there.

A tabulated bond dissociation enthalpy at a stated temperature is not automatically identical to either D_e or D₀. Thermal contributions, zero-point energies, pressure conventions and the specified fragments matter. Keep the symbol and physical process attached to the number.

Useful conversions include 1 eV per particle=96.485 kJ mol⁻¹ and 1 pm=10⁻¹² m. Multiplying by Avogadro’s constant converts per-particle energy to molar energy; it does not change whether the process is bond breaking or bond formation.

06 / UNDERSTAND & APPLY

6. Bond parameters describe length, angle, order, polarity and strength

Bond length is an equilibrium or experimentally averaged internuclear distance, depending on the measurement. Bond angle describes the angle between bonds sharing an atom. A double bond between the same pair of elements is usually shorter and stronger than a comparable single bond, and a triple bond usually shorter still, but different chemical environments can complicate simple rankings.

Bond order describes bonding multiplicity in a model. Lewis structures use single, double and triple bonds; delocalized systems can have fractional average bond orders. Equivalent bonds in carbonate do not alternate permanently between one long single and one short double bond. Resonance structures are contributors to one delocalized state, not molecules rapidly switching drawings.

Bond enthalpy or dissociation energy quantifies a specified breaking process, while a force constant describes local stiffness. Bond dipole describes charge separation and contributes vectorially to molecular dipole. A short bond, a strong bond and a highly polar bond are related concepts only in context; none is a universal synonym for the others.

Angles depend on electron-domain geometry, lone pairs, multiple bonds and molecular environment. Tetrahedral, trigonal-planar and linear reference angles are about 109.5°, 120° and 180°, but real molecules can deviate. A measured molecular dipole combines geometry and charge distribution, so polar bonds can cancel in a symmetric molecule.

07 / UNDERSTAND & APPLY

7. Bond-energy estimates use broken minus formed

Breaking a bond into the specified gas-phase fragments requires energy; forming the same bond from those fragments releases it. A common estimate is ΔH_rxn≈ΣD(bonds broken)−ΣD(bonds formed). A negative result indicates that forming product bonds releases more energy than breaking reactant bonds requires.

Mean bond enthalpies are averages over different molecular environments. They are useful for estimates, but a C–H bond in every molecule does not have precisely the same dissociation enthalpy. Successive bond removals from one molecule can also differ because the fragments change after each step.

The usual tabulated calculation assumes gas-phase molecules. If a reactant or product is liquid or solid in the stated reaction, phase-change terms may be needed. Ionic lattice energy is a collective solid-state quantity and should not be inserted as if it were one ordinary covalent bond in a molecule.

Atom balance comes first. Count only the bonds whose net changes are represented, or count all bonds consistently on both sides. A stoichiometric coefficient multiplies the number of molecules and their bond counts. The method is a Hess-law energy cycle, not a literal claim that the reaction mechanism first breaks every bond at once.

08 / UNDERSTAND & APPLY

8. Avoid turning a trend into a universal identity

Higher bond order often shortens bonds for a comparable atom pair, but comparing different elements also changes atomic size, orbital overlap and electron repulsion. A short bond across one pair cannot automatically be ranked against a longer bond across a different pair using length alone.

A deeper well means greater classical separation energy from its minimum, while a steeper curvature means greater small-displacement stiffness. These features can vary independently in a general family of curves. In the Morse model they are linked by the additional parameter a, demonstrating why D_e alone is insufficient.

The minimum marks zero net force, not zero potential energy, zero kinetic energy or absence of quantum motion. At finite temperature, vibration samples a range of separations; anharmonicity can make average separation differ from r_e.

For multiple dissociation channels or changing electronic states, a one-curve diagram may not capture the full process. The school-level curve is powerful when its coordinate, state and energy reference are explicitly stated.

09 / UNDERSTAND & APPLY

9. Read the graph before calculating

Mark the separated-fragment reference, minimum location, well depth and any vibrational level. Determine force from slope and stability from curvature. Then decide whether the requested quantity is D_e, D₀, a finite-temperature enthalpy, stiffness or molecular geometry.

For numerical work, distinguish energy per pair from energy per mole and picometres from metres. For reaction estimates, balance atoms, count broken and formed bonds, preserve signs and account for phases.

All plotted parameters and numerical exercises here are supplied original models. They illustrate how to reason from curves and bond parameters rather than pretending to provide measured data for an unspecified molecule.