7.1 Atomic Radius
PRE-HEALTH PREP
Master the Trend for Atomic Radius
General Chemistry 7.1
Lesson Overview
Atomic radius increases down a group as new electron shells are added and decreases across a period due to rising effective nuclear charge (Zeff). Cations are smaller than their parent atoms because electron loss reduces repulsion, while anions are larger because electron gain expands the electron cloud. An isoelectronic series — ions sharing the same electron count — can be ranked by nuclear charge alone, with the most positive ion being the smallest.
What Is Atomic Radius?
When chemists refer to the atomic radius of an element, they are describing the size of its electron cloud — not the nucleus. This distinction matters because it seems to contradict our intuition about mass. Most of an atom's mass is concentrated in the nucleus, where protons and neutrons reside, yet those particles contribute almost nothing to an atom's physical size. It is the electrons — which have negligible mass — that define how large an atom is.
Specifically, it is the outermost shell of electrons, called the valence shell, that sets the outer boundary of the atom. Whatever is happening in those outermost orbitals determines the atomic radius.
Why is atomic radius hard to measure precisely? The boundary of an atom is inherently "soft" — atomic orbitals don't actually stop at a defined distance from the nucleus. Instead, the probability of finding an electron asymptotically approaches zero as the distance increases (a 100% probability boundary would technically encompass the entire universe!). For this reason, the standard depiction of an atomic orbital is cut off at the 90% or 95% probability interval.
In practice, the bonding atomic radius is the most common convention: it is defined as half the bond length between two identical atoms bonded together. This is the value typically reported for periodic trend discussions.
The Atomic Radius Trend on the Periodic Table
Here is a mnemonic worth holding onto: if you spend too much time in France enjoying the wine, the cheese, and the eclairs, you are going to get fat. As it turns out, francium — named after France — is the fattest element on the periodic table. It sits in the bottom-left corner, and that is exactly where the largest atomic radii are found.
The two rules governing the general trend are:
- Atomic radius increases going down a group (top to bottom).
- Atomic radius increases going from right to left across a period.
Counter-intuitive alert: Atomic mass generally increases from left to right across a period, yet atomic radius does the opposite — it decreases left to right. Mass and size are two different things. The trend in size is governed by effective nuclear charge, not by how heavy the nucleus is.
The Horizontal Trend: 3 Elements Compared
To make the left-to-right trend concrete, compare three consecutive third-period elements: sodium (Na), magnesium (Mg), and aluminum (Al). The periodic trend predicts that sodium should have the largest atomic radius of the three, followed by magnesium, and then aluminum. Published values confirm this:
| Element | Atomic Radius (pm) | Atomic Radius (nm) | Atomic Radius (Å) |
|---|---|---|---|
| Sodium (Na) | 190 pm | 0.190 nm | 1.90 Å |
| Magnesium (Mg) | 145 pm | 0.145 nm | 1.45 Å |
| Aluminum (Al) | 118 pm | 0.118 nm | 1.18 Å |
The most common unit for atomic radii is the picometer (pm), where 1 pm = 10−12 m. You may also encounter nanometers (nm), where 1 nm = 10−9 m, or angstroms (Å), where 1 Å = 10−10 m. Converting from picometers to angstroms simply means moving the decimal two places to the left (e.g., 190 pm = 1.90 Å).
Remember: Aluminum is the heaviest of these three elements, yet it has the smallest atomic radius. Size is determined by the electron cloud, not by nuclear mass.
Effective Nuclear Charge and Atomic Radius
The key to understanding the horizontal trend is a concept called effective nuclear charge, abbreviated Zeff. To see how it works, think about what governs the position of a valence electron: it is attracted to the protons in the nucleus, but it is simultaneously repelled by all the inner (core) electrons that sit between it and the nucleus. Those inner electrons are said to shield or screen the valence electron from the full nuclear charge.
The rough formula for effective nuclear charge is:
Zeff = Z − S
where Z is the full nuclear charge (equal to the atomic number) and S is the number of shielding (core) electrons — those in shells closer to the nucleus than the valence electron in question.
Calculating Zeff for Sodium, Magnesium, and Aluminum
Using the Bohr model as a rough framework, all three elements share the same core electron configuration: 2 electrons in the first shell and 8 electrons (2s² 2p⁶) in the second shell, for a total of 10 core electrons. The only difference is in the valence shell:
- Sodium (Z = 11): 1 valence electron. Zeff = 11 − 10 = +1
- Magnesium (Z = 12): 2 valence electrons. Zeff = 12 − 10 = +2
- Aluminum (Z = 13): 3 valence electrons. Zeff = 13 − 10 = +3
Why don't valence electrons shield each other? In this rough approximation, electrons within the same shell are treated as contributing zero shielding. When two electrons occupy the same s orbital, for instance, they spend most of their time on opposite sides of the orbital — they are equidistant from the nucleus and do not effectively block each other's view of it. More exact treatments (such as Slater's rules) assign a small but nonzero shielding value to same-shell electrons, but the rough approximation is sufficient to establish the trend.
As Z increases by one proton with each step across the period, the shielding constant S remains fixed at 10. The result is that Zeff climbs by +1 with each element. A higher Zeff means the valence electrons are pulled more tightly toward the nucleus — and therefore the atom is smaller. This is why atomic radius decreases from left to right across a period.
Zeff Pattern Across the 8 Main-Group Families
This approximation produces a clean, memorable pattern: the effective nuclear charge felt by valence electrons equals the number of valence electrons, which in turn equals the group number (for main-group elements).
| Group | Example Elements | Zeff on Valence Electrons |
|---|---|---|
| Group 1 (Alkali metals) | Li, Na, K, Rb, Cs | +1 |
| Group 2 (Alkaline earth metals) | Be, Mg, Ca, Sr, Ba | +2 |
| Group 13 | B, Al, Ga | +3 |
| Group 14 | C, Si, Ge | +4 |
| Group 15 | N, P, As | +5 |
| Group 16 (Chalcogens) | O, S, Se | +6 |
| Group 17 (Halogens) | F, Cl, Br, I | +7 |
| Group 18 (Noble gases) | Ne, Ar, Kr, Xe | +8 (approx.) |
The Vertical Trend: More Shells, Not More Zeff
Moving down a group, the effective nuclear charge on valence electrons remains constant — every Group 1 element has Zeff ≈ +1, for example. So why do atoms grow larger going down a group? The answer is simply that each successive period adds an entirely new electron shell. More shells mean a larger electron cloud, regardless of Zeff.
Example: Cesium vs. Rubidium
Cesium (Cs) has a larger atomic radius than rubidium (Rb). But the reason is not effective nuclear charge — both are Group 1 elements with Zeff ≈ +1 on their valence electrons. Cesium is larger because it has one additional electron shell (6 shells vs. 5 shells in Rb).
Example: Cesium vs. Barium
Cesium (Cs) has a larger atomic radius than barium (Ba). Here, both elements are in the same period (period 6), so the number of shells is identical. The difference is Zeff: Cs is in Group 1 (Zeff = +1) and Ba is in Group 2 (Zeff = +2). Barium's higher effective nuclear charge pulls its valence electrons closer to the nucleus, making it smaller. This is the horizontal trend explained by Zeff.
Atomic Radius and Bond Lengths
Atomic radius has a direct and practical consequence: it determines bond length. A bond length is defined as the nucleus-to-nucleus distance between two bonded atoms, which means it is simply the sum of the two atomic radii involved.
The type of bond determines which set of radii to use:
- Covalent bonds (nonmetal–nonmetal): use atomic radii.
- Ionic bonds (metal–nonmetal): use ionic radii for a more accurate value.
The practical takeaway is simple: larger atoms form longer bonds.
Example: Comparing C–X Bond Lengths
Consider four carbon–halogen bonds: C–F, C–Cl, C–Br, and C–I. Carbon is identical in all four, so the only variable is the halogen's atomic radius. Because atomic radius increases down a group (F < Cl < Br < I), the C–I bond is the longest of the four.
Now add a fifth option: Si–I. Both C–I and Si–I share iodine, so the question reduces to whether carbon or silicon is larger. Silicon is below carbon in Group 14, so Si > C in atomic radius — making the Si–I bond the longest of all five.
Exam strategy: When comparing a list of bond lengths, look for a common atom across the choices. The bond involving the larger of the two unique atoms will be longer. If no single atom is shared across all choices, compare in pairs to eliminate the shortest options first.
Ionic Radii: Cations vs. Anions
So far, the discussion has focused on neutral atoms. When atoms gain or lose electrons to form ions, their radii change — sometimes dramatically. The resulting sizes are called ionic radii.
Cations Are Smaller Than Their Neutral Atoms
Forming a cation requires losing electrons. Fewer electrons in the cloud means less electron–electron repulsion, so the remaining electrons are drawn closer to the nucleus and the ion shrinks. The effect is substantial:
| Species | Radius (pm) | Change |
|---|---|---|
| Potassium (K) | 243 pm | — |
| Potassium ion (K+) | 138 pm | −43% (loses entire valence shell) |
| Calcium (Ca) | 194 pm | — |
| Calcium ion (Ca2+) | 99 pm | −49% (loses two electrons) |
Why is K → K+ such a dramatic size reduction? Potassium loses its single 4s electron when it ionizes — and that electron was the entire valence shell. The ion that remains has only three shells instead of four, which accounts for the sharp drop from 243 pm to 138 pm.
Anions Are Larger Than Their Neutral Atoms
Forming an anion requires gaining electrons. Adding more negatively charged electrons to the cloud increases electron–electron repulsion, causing the cloud to expand. Even adding a single electron produces a striking increase in size:
| Species | Radius (pm) | Change |
|---|---|---|
| Sulfur (S) | 88 pm | — |
| Sulfide ion (S2−) | 184 pm | +109% (more than doubles) |
| Chlorine (Cl) | 79 pm | — |
| Chloride ion (Cl−) | 181 pm | +129% (more than doubles) |
The dramatic expansion occurs because the added electrons join an already-populated valence shell. Although the rough Zeff formula treated same-shell electrons as contributing zero shielding, they do shield each other to a meaningful degree. That additional shielding reduces the effective attraction to the nucleus, allowing the electron cloud to expand significantly.
The Core Rule: Cation ionic radius < neutral atomic radius. Anion ionic radius > neutral atomic radius. Metals form cations (shrink); nonmetals form anions (grow).
Exam Questions on Ionic vs. Atomic Radius
A common exam question presents a list of elements and asks: "Which has an ionic radius smaller than its atomic radius?" Work through it in two steps:
- A smaller ionic radius means a cation was formed (losing electrons shrinks the ion).
- Cations are formed by metals. Scan the answer choices for the metal — that is your answer.
The reverse question — "Which has an ionic radius larger than its atomic radius?" — points to an anion, which is formed by a nonmetal. Identify the lone nonmetal in the answer choices.
Comparing Ionic Radii
Comparing the sizes of different ions is more nuanced than comparing neutral atoms. The simple periodic trend (francium is the fattest) applies cleanly to neutral atoms, but ions carry different charges — and charge dramatically alters size. There are two reliable situations in which ionic radii can be fairly compared.
Same Charge: Apply the Periodic Trend Directly
If a set of ions all carry the same charge, their sizes relative to one another follow the same periodic trend as their neutral atoms. The absolute sizes are all smaller (for cations) or all larger (for anions) than the neutral atoms, but the ranking is preserved.
Example: Comparing 2+ Cations
Rank Mg2+, Ca2+, Sr2+, and Ba2+ by size.
All carry a +2 charge, so apply the neutral-atom trend: atomic radius increases going down Group 2. Therefore: Mg2+ < Ca2+ < Sr2+ < Ba2+.
Mixing charges is a trap: If one ion in a comparison has a different charge from the others, you cannot reliably rank it using the periodic trend alone. For example, neutral Sr is considerably larger than Sr2+ — the ionization event itself changes the size so dramatically that simple trend-based comparisons break down. Expect exam questions to keep all compared ions at the same charge, or to use the isoelectronic approach described below.
Isoelectronic Series: Ranking 5 Species by Size
The second situation in which ions of different elements can be directly compared is when they belong to an isoelectronic series. The prefix iso means "same," so an isoelectronic series is a group of atoms or ions that all share the same number of electrons and, therefore, the same electron configuration.
Building an Isoelectronic Series
Consider the species that are isoelectronic with argon (Ar), which has 18 electrons and the configuration 1s² 2s² 2p⁶ 3s² 3p⁶:
| Species | Atomic Number (Z) | Electrons | Nuclear Charge |
|---|---|---|---|
| S2− | 16 | 18 | +16 |
| Cl− | 17 | 18 | +17 |
| Ar | 18 | 18 | +18 |
| K+ | 19 | 18 | +19 |
| Ca2+ | 20 | 18 | +20 |
Ranking Size in an Isoelectronic Series
Because every species in the series has the same electron configuration, the electron cloud is essentially identical in structure. The only variable is the nuclear charge — how many protons are pulling those 18 electrons inward.
More protons = stronger attraction = smaller radius. The ranking follows directly from the atomic number:
- Largest: S2− (Z = 16 — fewest protons, weakest pull, most diffuse cloud)
- Cl− (Z = 17)
- Ar (Z = 18)
- K+ (Z = 19)
- Smallest: Ca2+ (Z = 20 — most protons, strongest pull, most compact cloud)
The isoelectronic rule: Within an isoelectronic series, the most negative ion is the largest (fewest protons for the same number of electrons) and the most positive ion is the smallest. The ranking is determined by nuclear charge Z alone — effective nuclear charge is not needed here because the shielding is identical for every member of the series.
A note on argon in this series: In some published illustrations, argon appears to be smaller than Ca2+. Keep in mind that argon is the only neutral atom in this series — its atomic radius is determined differently from the ionic radii of the others, and the noble gas convention is difficult to compare directly with ionic measurements. Regardless of what the data appear to show in such illustrations, the correct ranking for exam purposes remains: S2− > Cl− > Ar > K+ > Ca2+.
Slater's Rules
Slater's rules provide a more accurate approximation of the screening constant S and therefore the effective nuclear charge Zeff. Rather than treating all core electrons as equally shielding (as the simplified formula does), Slater's rules assign specific screening values depending on which shell and subshell the shielding electrons occupy. The sum of all these individual screening values gives the overall screening constant S used in:
Zeff = Z − S
An important starting point: electrons with a higher principal quantum number than the electron of interest contribute zero screening — only electrons in the same shell or inner shells matter. With that established, Slater's rules are organized into three groups based on the orbital type of the electron being screened:
1. [1s] electrons
The screening value experienced by a 1s electron from the other 1s electron is 0.30.
2. [ns, np] electrons
- Screening from any other electron in the same [ns, np] group: 0.35
- Screening from an electron with principal quantum number (n − 1): 0.85
- Screening from an electron with principal quantum number ≤(n − 2): 1.00
3. [nd] or [nf] electrons
- Screening from any other electron in the same [nd] or [nf] group: 0.35
- Screening from all other electrons with principal quantum number ≤n: 1.00
| SLATER'S RULES | ||||
|---|---|---|---|---|
| Electron Group | Same Group | Same n (lower l) | (n − 1) | ≤(n − 2) |
| [1s] | 0.30 | — | — | — |
| [ns, np] | 0.35 | — | 0.85 | 1 |
| [nd] or [nf] | 0.35 | 1 | 1 | 1 |
8 Key Takeaways
- Atomic radius measures the electron cloud, not the nucleus. The valence shell defines the outer boundary of an atom; mass is irrelevant to size.
- Francium is the largest element. Atomic radius increases going down a group and increases going from right to left across a period.
- Effective nuclear charge (Zeff) drives the horizontal trend. Across a period, the nuclear charge Z increases by one with each element while the shielding constant S stays fixed, so Zeff rises and atoms shrink left to right.
- The vertical trend is explained by shell count, not Zeff. Going down a group, Zeff is unchanged; atoms grow larger simply because each period adds a new electron shell.
- Bond length equals the sum of the two atomic radii. Larger atoms form longer bonds. Use atomic radii for covalent bonds and ionic radii for ionic bonds.
- Cations are smaller than their neutral atoms; anions are larger. Losing electrons reduces repulsion (smaller ion); gaining electrons increases repulsion (larger ion).
- Ions of the same charge can be ranked using the periodic trend. If all compared ions have identical charges, their relative sizes follow the same order as the neutral atoms.
- In an isoelectronic series, size is governed by nuclear charge alone. The most negative ion is the largest; the most positive ion is the smallest.