Read bond length and separation energy from a curve

Find two different features of the curve

A bond has an equilibrium separation and a cost of pulling its atoms apart. Those are different quantities. In the lab, curve A has its minimum at 1.25 Å and depth 2 eV; curve B has the same minimum position and depth 4 eV. The invented comparison changes depth while holding distance fixed. Curve C keeps B’s 4 eV depth but moves the minimum to 1 Å.

An ångström, Å, is 10⁻¹⁰ metre. An electron volt, eV, is an energy unit here assigned per interacting pair. The three curves are original teaching parameterizations, not measurements of named molecules. Their independent controls isolate two features; they do not imply real chemistry allows arbitrary bond length and strength combinations.

Select each curve at its own minimum. Before capturing, read the horizontal position and the vertical depth separately. A more negative minimum means more energy must be supplied to reach the declared zero at infinite separation. It does not, by itself, give a shorter equilibrium distance.

Why there is a minimum

At larger separations than the minimum, net attraction tends to reduce the distance. At sufficiently short distances, repulsive effects dominate and tend to increase it. The minimum marks balance of the net radial force in this model. The curve summarizes electronic and nuclear interactions; its short-range repulsion is not simply another attractive point-charge term.

Use curve B at 0.9 and 1.5 Å. The first point lies on the repulsive side, the second on the attractive side. Now inspect positions with negative energy on both sides of a minimum: the sign of potential alone cannot tell you the direction of force. Direction depends on the slope. A horizontal tangent at the minimum means zero net force, even though potential is −D, not zero.

The earlier Coulomb comparison separated inverse-square force from inverse-distance work for unlike point charges. Its attractive term alone cannot supply a finite equilibrium bond length. This curve adds a different empirical representation rather than silently replacing one formula with another.

Declare a reference before adding work

We set U(infinity) = 0 for separated atoms. U(r) is the potential at separation r, and rₑ names the minimum position. The well depth D is 0 − U(rₑ), a positive energy. External quasistatic work means controlled slow movement with negligible change in kinetic energy, so work on the pair equals final potential minus initial potential.

Suppose a new curve has a minimum of −5 eV and a point at a larger finite distance R with U(R) = −1 eV. Moving from the minimum to R costs 4 eV; moving from R to infinity costs another 1 eV. The total is the 5 eV depth. A finite graph endpoint is not automatically complete dissociation.

Check curve B at 3 Å. The attraction and negative potential have become smaller, but neither is exactly zero. The lab reports the remaining work explicitly. At a compressed point with potential above zero, the signed remaining work can be negative: ideal external control could extract energy as separation increases. That is not negative required energy input and does not describe the uncontrolled trajectory or a barrier along a different path.

Inspect the model without treating it as measured chemistry

The teaching function is U(r) = D[1 − exp(−a(r − rₑ))]² − D. Here D is in eV, r and rₑ are in Å, and the positive width parameter a is in Å⁻¹, making the exponential’s argument dimensionless. At r = rₑ the squared bracket vanishes; at very large r it approaches one. Thus the function has the declared minimum and zero limit.

The optional derivative gives radial force F = −dU/dr. Writing x = exp(−a(r − rₑ)), the calculation is F = 2aDx(x − 1), in eV/Å. Positive F increases separation. The numerical direction can be understood from the curve before using calculus.

This Morse-style function follows a historical empirical model, not an exact many-electron calculation or a fit to supplied experimental data. Morse’s 1929 article supplies the model provenance. Its quantum vibrational problem is not solved here. Well depth from the potential minimum differs from dissociation energy starting at a molecule’s vibrational ground state because zero-point energy is omitted. No temperature, spectrum, rate or reaction enthalpy is predicted.

Higher bond order often corresponds to shorter, stronger bonds in comparable covalent systems, but the curve must supply the actual comparison and its context. Return to high-school chemistry for the existing bond-energy reaction ledger; a single pair’s potential does not replace that complete reaction accounting.