Coulombic Attraction and the Ion-Pair Potential

The single ion-pair Coulomb potential is not a historical curiosity — it is the working core of modern computational and applied chemistry wherever ionic interactions matter. Force fields used in molecular dynamics simulations of electrolytes, biomolecules, and ionic liquids represent electrostatic interactions between partial or full atomic charges using exactly this $q_1q_2/(4\pi\varepsilon_0 r)$ term (often called the "Coulomb term" in force-field energy expressions), typically combined with a Lennard-Jones or Buckingham repulsive/dispersion term that plays the same conceptual role as Born repulsion. Solid-state chemists use the ion-pair energy, scaled by Madelung constants, to rationalize and predict lattice energies, melting points, hardness, and solubility trends across families of ionic solids — from simple alkali halides to battery cathode materials like LiCoO₂ and Li-rich layered oxides, where lattice electrostatics partly govern ion mobility and structural stability. In pharmaceutical science, the same $1/r$ electrostatic law underlies models of ion-pairing between drug counter-ions and the resulting salt's crystallinity, solubility, and bioavailability. Geochemists and mineralogists apply Madelung-constant-scaled Coulomb energetics to silicate and oxide mineral stability, explaining why densely packed, highly charged mineral structures (like corundum, Al₂O₃) are exceptionally hard and refractory. Even in nuclear and astrophysical contexts, the same functional form (with appropriate charges) describes Coulomb barriers relevant to nuclear reactions — a reminder that this "simple" ion-pair law is one of the most broadly reused equations in the physical sciences.

Ask the tutor: How does adding the Madelung constant change this single ion-pair energy into the full lattice energy of NaCl?

Coulomb's law tells us the force between two point charges $q_1$ and $q_2$ separated by distance $r$ in vacuum (or a medium of permittivity $\varepsilon_0\varepsilon_r$):

This force is repulsive (positive, pushing the charges apart) when $q_1q_2 > 0$, and attractive (negative, pulling them together) when $q_1q_2 < 0$ — exactly the situation for a cation and an anion. But force is not what we want when discussing bond strength or lattice energy; we want potential energy, the work required (or released) in changing the separation. Potential energy is obtained by integrating force with respect to distance, and this integration is the single most important calculus step in the whole theory of ionic bonding — get the sign and the power of $r$ right, and the entire edifice of lattice energetics follows correctly. Physically, imagine an Na⁺ ion held fixed at the origin and a Cl⁻ ion brought in from $r = \infty$ (where the interaction energy is defined as zero, by convention) to some finite separation $r$. Because the force is attractive, the ions do positive work on each other as they approach — the system's potential energy decreases (becomes negative) as $r$ shrinks. There is no natural stopping point in the pure Coulomb picture: mathematically, $V(r) \to -\infty$ as $r \to 0$. Real ions do not collapse into each other because their electron clouds begin to overlap at short range, invoking the Pauli exclusion principle and generating a steep repulsive energy that only becomes significant once the ions are close to their "touching" distance, $r_0$ (the sum of ionic radii). The competition between these two curves — one long-ranged and attractive, one extremely short-ranged and repulsive — is what produces a genuine energy minimum at a finite bond distance.

The key qualitative features shown are: the curve lies entirely below zero (attractive), it is asymptotically flat at large $r$ (long-range tail, why Madelung sums converge slowly), and it diverges steeply and without limit as $r\to 0$ (why a repulsive term is mandatory for physical realism). The curve's shape is fixed by the exponent $-1$ on $r$; its depth at any given $r$ is fixed by the product of the ionic charges, $z_+z_-$.

The force law between two point charges $q_1$ and $q_2$ separated by $r$, in a vacuum:

where:

• $q_1, q_2$ — the two point charges (SI units: coulombs, C); for ions, $q_+ = z_+ e$ and $q_- = -z_- e$ with $z_\pm$ the (positive) integer charge numbers and $e$ the elementary charge
• $r$ — center-to-center separation (m)
• $\varepsilon_0$ — vacuum permittivity, $8.854\times10^{-12}\ \text{F m}^{-1}$
• $F(r)$ — force (N); positive = repulsive, negative = attractive, by the usual sign convention along the line joining the charges

Reading: this is an inverse-square law, structurally identical to Newton's law of gravitation, but with charge products replacing masses and (crucially) allowing either sign — gravity is always attractive, Coulomb forces can be either. Integrating the force from $r$ to infinity (with the convention $V(\infty) = 0$) gives the potential energy:

where:

• $V(r)$ — electrostatic potential energy of the two-charge system at separation $r$ (J per ion pair, or convert to J mol⁻¹ / kJ mol⁻¹ by multiplying by Avogadro's number $N_A$)
• all other symbols as above

Reading: unlike the force, the potential energy falls off as $1/r$, not $1/r^2$ — one power of $r$ is "used up" in the integration. The sign of $V$ directly tells you whether assembling the pair at that distance releases energy (bond-forming, $V<0$) or costs energy (unfavorable, $V>0$). For the specific case of a cation $+z_+e$ and anion $-z_-e$, this becomes the central ion-pair equation of ionic bonding:

where:

• $z_+, z_-$ — magnitudes of cation and anion charge numbers (dimensionless positive integers, e.g. $z_+=1, z_-=1$ for Na⁺Cl⁻; $z_+=2,z_-=2$ for Mg²⁺O²⁻)
• $e = 1.602\times10^{-19}\ \text{C}$ — elementary charge
• $r$ — internuclear (center-to-center) separation, often approximated as $r_0 = r_+ + r_-$, the sum of ionic radii
• the explicit minus sign makes $V$ negative for all $r>0$, encoding "opposite charges attract"

Reading: this is the equation of ionic bonding at the single-pair level. Everything that follows in lattice energetics — the Madelung constant $A$ that sums this expression over an infinite array of neighbors, and the Born exponent $n$ that adds short-range repulsion — is built directly on top of this one term.

Example 1 (Foundational): Energy of one Na⁺–Cl⁻ pair at two different separations. Compute $V(r)$ for a single Na⁺Cl⁻ ion pair at $r_1 = 0.283\ \text{nm}$ (bonding distance) and at $r_2 = 0.566\ \text{nm}$ (twice that distance), and compare the ratio to the $1/r$ prediction. Using $V(r) = -k_e e^2/r$ with $k_ee^2 = (8.988\times10^9)(2.566\times10^{-38}) = 2.306\times10^{-28}\ \text{J m}$:

The ratio $V(r_1)/V(r_2) = 2.00$, exactly matching the prediction that doubling $r$ halves $|V|$ (since $V \propto 1/r$). This simple check — an asymmetric, non-round distance ratio still obeying the inverse relationship exactly — confirms the functional form independently of the specific numerical constants used.

Example 2 (Applied): Comparing NaCl and MgO ion-pair energies — the charge-squared effect. MgO has $z_+ = z_- = 2$ (Mg²⁺, O²⁻) and a slightly smaller ionic separation, $r_0(\text{MgO}) \approx 0.205\ \text{nm}$ (Mg²⁺ 0.072 nm + O²⁻ 0.140 nm, six-coordinate Shannon radii, giving 0.212 nm; using the experimental rock-salt Mg–O distance of 0.205 nm here for accuracy). Compute the single ion-pair Coulomb energy and compare with NaCl's $-491\ \text{kJ mol}^{-1}$.

First, the numerator inside:

Divide by $r_0$:

Multiply by $k_eN_A = (8.988\times10^9)(6.022\times10^{23}) = 5.412\times10^{33}$:

This is roughly 5.5 times more negative than NaCl's single-pair energy, even though MgO's bond distance is shorter (which alone would only give a factor of $0.283/0.205 = 1.38$). The dominant effect is the charge product: $z_+z_- = 4$ for MgO versus $1$ for NaCl, a factor of 4, combined with the modestly shorter distance (factor 1.38) to give $4 \times 1.38 \approx 5.5$ — matching the computed ratio. This factor-of-$z_+z_-$ dependence is precisely why oxides and other doubly- or triply-charged-ion compounds (MgO, Al₂O₃) have dramatically higher lattice energies and melting points than singly-charged halides (NaCl, KBr): the charge term enters as a simple product, but because both ions in a 2:2 salt carry double charge, the effect compounds multiplicatively in a way that a single-ion-charge increase would not. (Real MgO lattice energy, including Madelung summation and Born repulsion, is about $-3795\ \text{kJ mol}^{-1}$; the single-pair estimate again captures the majority of the trend and magnitude, exactly as found for NaCl.)

Example 3 (Exam / competition level): At what separation does a hypothetical +3/−1 ion pair match the Coulomb energy of Na⁺Cl⁻ at its equilibrium distance? Suppose an ion pair has $z_+ = 3, z_- = 1$ (e.g., a trivalent cation with a monovalent anion, loosely modeling something like "AlCl" as a hypothetical 1:1 contact pair, ignoring real stoichiometry for the sake of the exercise). Find the separation $r_3$ at which $V(r_3)$ for this pair equals $V(r_0) = -491\ \text{kJ mol}^{-1}$, the NaCl benchmark from Example 1. Set the two expressions equal:

The constants $k_e, N_A, e^2$ cancel from both sides, leaving a clean ratio:

With $z_+z_- = (3)(1) = 3$ and $r_0(\text{NaCl}) = 0.283\ \text{nm}$:

So a hypothetical $3+/1-$ pair would need to be separated by nearly 0.85 nm — roughly three times farther apart than Na⁺ and Cl⁻ — to have the same Coulomb energy per pair as NaCl at its normal bonding distance. This illustrates, with a genuinely asymmetric charge ratio (3:1, not the trivial 1:1 or 2:2 cases used above), how strongly the charge product $z_+z_-$ can compensate for a much larger separation, and it is exactly the reasoning chemists use informally when comparing, say, a lanthanide oxide's lattice energy to an alkali halide's despite very different ionic sizes.