Separate charge, distance and removal-energy claims

Two changes can compete

Imagine a fixed negative test charge near a positive source. Begin with source charge Q₀ and separation r₀. Now double the positive source charge and double the separation. A larger positive charge strengthens the attraction at fixed distance; a larger separation weakens it at fixed charge. Which change wins depends on the quantity you ask about.

Before moving the controls, predict the force ratio and the energy ratio separately. The default comparison has both inputs doubled. It gives half the original attractive-force magnitude, but the same external work to separate the charges to infinity. Explaining that difference is the central task, not memorizing an arrow on the periodic table.

Make a controlled comparison

For point charges Q and q separated by distance r, the force magnitude is k|Qq|/r², where k is the Coulomb constant. The fixed test charge q is negative; the source Q is positive, so the force on the test charge points toward the source. We compare magnitudes rather than assigning a potentially confusing signed direction to the plotted bars.

Let a = Q/Q₀ and b = r/r₀. The test charge and k are unchanged, so dividing by the baseline force cancels them: F/F₀ = a/b². Both a and b are dimensionless ratios. The input range always has positive separation; the model never evaluates division by zero or the physically inappropriate zero-distance point-charge limit.

Set a = 1, b = 1 to recover the baseline. Double only charge: a = 2, b = 1 gives force ratio 2. Return to baseline, then double only distance: a = 1, b = 2 gives 1/4. Finally combine both changes: 2/2² = 1/2. These controlled captures isolate causes before combining them.

Force is not energy

The potential energy of the attractive point-charge pair, taking infinite separation as zero, is negative. Its magnitude at separation r is k|Qq|/r. Slowly separating the fixed charges from rest to rest requires positive external work W = k|Qq|/r. This is a model of controlled separation, with no other particles or competing processes.

Its baseline ratio is W/W₀ = a/b, not a/b². Doubling both source charge and distance gives 2/2 = 1. At half the baseline separation with unchanged charge, the force is four times its baseline while separation work is twice its baseline. Check these two results in the table rather than comparing bar lengths as though force and energy had the same physical units. Each displayed ratio uses its own baseline quantity.

Optional derivation: why the exponent changes

The external work for quasistatic separation integrates the attractive-force magnitude along the outward path: W = ∫ from r to infinity of k|Qq|/x² dx = k|Qq|/r. Here x is the changing separation along the path. Integrating a force over a distance produces energy. You do not need this calculus derivation to use the ratio comparisons.

Apply the idea without fabricating atomic measurements

Real atomic electrons have spatial distributions and interact with each other. Core electrons partly shield the nuclear attraction, and shielding depends on the state. An “effective nuclear charge” is a model-dependent description, not simply the atom’s net ionic charge and not a value this widget measures. The sliders stipulate a source charge and a separation; they do not calculate an atomic radius, orbital energy or many-electron ionization energy.

Consider F−, Ne, Na+, Mg2+ and Al3+. Each has ten electrons, while proton counts increase from nine to thirteen. This is an isoelectronic comparison. Increased nuclear attraction generally contracts the electronic distribution along the sequence. The electron ledger makes the comparison useful, but it does not say the real separations remain fixed or give their sizes.

A tabulated atomic or ionic radius also needs an operational definition and environment. Do not combine a neutral atom’s van der Waals radius with a crystal’s ionic radius and call the difference a direct measurement of this sketch. Similarly, an isolated-atom removal energy cannot by itself predict the energy or rate of a bulk reaction. Use the sourced ionization packet for actual reference energies and compare their definitions before applying a trend.

Distinguish neutral trends from a change in charge

Across a representative-element period, electrons are added to the same principal outer shell while nuclear attraction increases; incomplete shielding by added valence electrons generally makes neutral atomic radii decrease. Moving down a group adds occupied shells and generally increases radius. Forming a cation removes electrons while retaining the nucleus, usually contracting the remaining distribution; losing an outer shell can make the change especially large. Adding electrons to form an anion increases electron repulsion and generally expands the distribution. These are qualitative comparisons using compatible size definitions and environments, not predictions of exact crystal radii from the sliders.

Changed comparison: Predict the neutral size ordering of Mg, Si and Ca using their periodic positions. Separately compare the size of a chlorine atom with Cl⁻, and a magnesium atom with Mg²⁺. Does either ionic change alter proton number?

Check the three comparison rules

The neutral trend predicts Si smaller than Mg, and Ca larger than Mg because Ca lies below it. Cl⁻ is generally larger than its parent Cl atom, while Mg²⁺ is smaller than its parent Mg atom. Electron gain or loss leaves proton number fixed. These parent-to-ion comparisons are not the same as the earlier isoelectronic series, which holds electron count fixed while changing nuclear charge; exact tabulated radii still require compatible definitions.

OpenStax periodic-size mechanisms, checked 31 October 2026.

Diagnose a new claim

A learner says, “If charge triples and distance halves, both force and removal energy increase sixfold.” Test the algebra under the declared two-charge model, then state which part of the claim overreaches beyond the model.

Check the changed comparison

The force ratio is 3/(0.5)² = 12. The point-charge separation-work ratio is 3/0.5 = 6. Only the second ratio is sixfold. Calling it an actual atomic removal energy would add an unsupported many-electron claim. The correct result is conditional on the stipulated point charges and controlled separation.

For a contextual return, revisit high-school periodic patterns and the charge/count audit. Electron affinity, electronegativity and chemically specific reactivity evidence require further work; this comparison does not complete all periodic properties.