Bonding and Structure in the s-Block Elements

Sodium melts at 98 °C; magnesium melts at 650 °C. Both are shiny, malleable, silvery metals. They sit next to each other in Period 3. They differ by one proton and one electron. Yet their melting points differ by more than 550 degrees. Something about that single extra valence electron is doing enormous structural work. The "something" is the number of electrons each atom contributes to the shared metallic electron sea. This changes the strength, the packing, and the geometry of the metallic bond. This matters because melting point, hardness, density, and compressibility are not decorative facts. They should not be memorized element-by-element. Instead, they are direct read-outs of bond order in a metal. This is exactly how N≡N being stronger than N–N reflects bond order in a molecule. The s-Block: Defining Features and Periodic Placement article establishes which elements we are discussing. These are Groups 1 and 2, the metals whose valence electrons sit in an s orbital. This article asks the mechanistic question instead. Given that placement, what structure do these atoms adopt when they condense into a solid? And why does the strength of the resulting metallic bond rise so steeply from Group 1 to Group 2? The big picture is a single controlling variable: valence electron count per atom. This feeds into three observables at once. First, the crystal structure adopted: body-centred cubic for the alkali metals. For alkaline earths, it is hexagonal close-packed, cubic close-packed, or bcc, depending on subtleties of s–d mixing. Second, the cohesive energy of the lattice. This scales, to first approximation, with the number of delocalized electrons per atom. It also depends on the strength of their attraction to the ion cores. Third, the melting point, which tracks cohesive energy almost linearly across a series of structurally similar metals. Building on metallic bonding as a qualitative picture of a "sea of electrons" around fixed cations, this article turns that picture into a quantitative and structural one. What follows: a description of the packing geometries actually found in Groups 1 and 2, with their coordination numbers and packing fractions. Next, a free-electron/jellium-based energetic argument. It explains why cohesive energy and melting point scale with valence electron count. Then, worked numerical examples comparing Na, Mg, and K quantitatively. Finally, a discussion of the exceptions. These include Group 2's structural diversity, the anomalous softness of Cs, and the near-doubling of melting point down Group 2. A good structural theory must accommodate these exceptions, rather than explain them away.

Key takeaways

Alkali metals (Group 1) crystallize almost universally in body-centred cubic (bcc) structures with coordination number 8; alkaline earths (Group 2) show hcp, ccp/fcc, or bcc depending on the element.
Packing fraction alone does not set the melting point — bcc (68%) is less densely packed than hcp/ccp (74%), yet Ca (ccp) and Mg (hcp) both melt far higher than any bcc alkali metal.
The dominant variable is valence electron count: one delocalized s electron per atom in Group 1 versus two in Group 2 roughly doubles the electron-gas density and the electrostatic glue holding the ion cores together.
Cohesive energy and melting point increase down Group 1 only mildly, but they show a maximum pattern in Group 2 and generally track ionic radius (smaller cation, tighter packing, stronger bonding) crossed with electron count.
A simple jellium/free-electron model shows cohesive energy scaling roughly as $z^{2}/r_s$ (z = valence electrons per atom, $r_s$ = radius of the atomic sphere), correctly predicting Mg/Ca/Sr/Ba ≫ Na/K/Rb/Cs.
Key idea: the crystal structure and melting point of an s-block metal are both downstream consequences of how many s electrons per atom are released into the delocalized electron sea, and how tightly the resulting ion cores can be packed around that sea.

Where this lives today

The valence-electron-count/melting-point relationship developed here is not a museum piece; it is load-bearing engineering knowledge. Magnesium alloys (used in aerospace and automotive lightweighting, where Mg's low density combined with its Group-2-level structural strength is prized) are designed with an explicit understanding that alloying elements which raise the effective conduction-electron density per atom (or otherwise stiffen the metallic bond) raise strength and creep resistance without sacrificing the low-density advantage that comes from Mg's small atomic mass. Sodium and potassium metal are used industrially precisely because their weak, low- $z$ metallic bonding makes them low-melting and reactive — sodium's low melting point (98 °C) is exploited directly in sodium-cooled fast nuclear reactors, where Na is used as a liquid heat-transfer medium that stays liquid across a wide, easily-managed temperature window, a direct engineering consequence of $z=1$ bonding.

Alkaline-earth-doped superconductors and battery materials exploit the same logic in reverse: Mg-ion and Ca-ion batteries are an active area of research explicitly because the divalent charge ( $z=2$ per cation) that makes the metal so cohesive in its elemental form also means each ion, once oxidized, carries twice the charge of an alkali-metal ion — promising higher energy density per ion inserted into an electrode, at the cost of the stronger electrostatic interactions (the same $z^2$ scaling) making ion mobility in the solid-state electrolyte more sluggish, a genuine current research bottleneck traceable to the identical physics discussed in this article.

Structural and geophysical relevance: Ca, Mg, and Sr metal chemistry underpins the geochemistry of silicate minerals and cement chemistry (Portland cement is dominated by calcium silicates), where the ionic/metallic-bonding-strength trends of Group 2 directly inform which cation substitutes for which in a mineral lattice. Flame tests and fireworks rely on the electronic structure of these same elements (Sr red, Ba green, Ca orange), a spectroscopic consequence of the same s-block valence-electron picture that, viewed from the solid-state side, gives the crystal structures and melting points discussed here — one electronic-structure story told from two different observational windows.

:::faq[Why does the cohesive-energy model predict a $z^2$ scaling rather than a simple $z$ (linear) scaling]? The linear-in- $z$ intuition would apply if each valence electron contributed an independent, non-interacting bond of fixed strength. But the dominant attractive term in the jellium model is the electrostatic interaction between the delocalized electron density and the ion cores, and both the ion core charge and (to good approximation) the electron density it attracts scale with $z$ , so the interaction energy scales with the product of the two, i.e. $z \times z = z^2$ . This quadratic scaling is exactly analogous to why Madelung energies in ionic solids scale with the product of the ionic charges, $z_+ z_-$ , rather than their sum.

Ask the tutor: Why does barium, with the same valence electron count as calcium and strontium, revert to the bcc structure instead of continuing the ccp pattern?

The physical picture

A metallic element condenses because delocalizing the valence electrons over the whole solid lowers their kinetic and potential energy relative to keeping them localized on individual atoms — this is the qualitative content of metallic bonding. But "delocalization is favorable" does not by itself tell you how strong the resulting bond is, or what shape the lattice of positive ion cores will take. Those two questions are what distinguish Group 1 from Group 2, and what distinguish one alkaline earth from another.

Structure 1 — the alkali metals are (almost) all bcc. Li, Na, K, Rb, and Cs all adopt the body-centred cubic structure at room temperature and pressure (Li and Na undergo a low-temperature martensitic transition to a close-packed structure below ~70–80 K, a detail usually set aside at this level). In bcc, each atom sits at a cube corner and one atom sits at the cube centre; the centre atom touches the 8 corner atoms along the cube body diagonal. The coordination number is 8 (with 6 further next-nearest neighbours only about 15% farther away, so the effective coordination is often quoted as 8 + 6). The packing fraction — the fraction of the unit cell volume genuinely filled by touching hard spheres — is:

\text{APF}_{\text{bcc}} = \frac{2 \times \tfrac{4}{3}\pi r^3}{a^3} = \frac{\sqrt{3}\,\pi}{8} \approx 0.680

Structure 2 — the alkaline earths are structurally split. Be and Mg adopt hexagonal close-packed (hcp, coordination number 12, packing fraction 0.740); Ca and Sr adopt cubic close-packed (ccp, i.e. face-centred cubic, also coordination number 12, packing fraction 0.740); Ba, like the alkali metals, adopts bcc. This is already a clue that Group 2 is not simply "Group 1 with twice the electrons" — the extra electron changes the character of the metallic bond (more directional, more d-orbital-assisted for the heavier members) enough to change the preferred coordination geometry, not merely its strength.

Figure

Packing geometries: bcc coordination (Group 1) vs. ccp/hcp coordination (Group 2)

Why does electron count matter so much more than packing geometry? Because packing fraction changes the density of the electron gas only slightly (68% vs. 74% full is a modest difference), while going from one valence electron per atom to two roughly doubles the number of free electrons per unit volume — and both the kinetic energy and, more importantly, the attractive electron–ion-core energy scale with that density in a way that dominates over packing details. This is the heart of why Mg (two valence electrons) is so much more cohesive than Na (one valence electron) despite being its next-door neighbour and despite Na actually having the smaller atomic radius contribution working in the wrong direction.

Figure

Melting point vs. Period-3 s/p-block position: the Group 1 → Group 2 jump

The core ideas

The controlling relationship of this article is that cohesive energy of a simple (nearly-free-electron) metal scales with the number of delocalized valence electrons per atom, $z$ , and inversely with the size of the atomic volume each ion core occupies, characterized by the Wigner–Seitz radius $r_s$ :

E_{\text{coh}} \;\approx\; -\,A\,\frac{z^{2}}{r_s} \;+\; E_{\text{rep}}(r_s)

where:

$E_{\text{coh}}$ is the cohesive energy per atom (energy released on forming the solid from free gaseous atoms; more negative = more strongly bound),
$z$ is the number of valence electrons contributed to the delocalized electron gas per atom (1 for Group 1, 2 for Group 2),
$r_s$ is the Wigner–Seitz radius, defined by $\tfrac{4}{3}\pi r_s^3 = V/N$ (the volume per atom),
$A$ is a positive constant collecting the electrostatic (Madelung-like) and correlation contributions of the electron gas interacting with the array of ion cores,
$E_{\text{rep}}(r_s)$ is a short-range repulsive term (from core–core and Pauli exclusion between filled inner shells) that rises steeply as $r_s$ decreases, setting the equilibrium spacing.

Reading: the leading, physically dominant term is $-Az^2/r_s$ . Doubling $z$ from 1 to 2 (Group 1 → Group 2) quadruples the attractive term for a fixed $r_s$ — an enormous energetic effect — while the accompanying modest shrinkage of $r_s$ across a period (ionic radius decreasing left to right) only adds to the effect rather than fighting it. This is why the Group 1 → Group 2 melting-point jump is so much larger than the Group 1 → Group 1 (down-group) trend, where $z$ stays fixed at 1 and only $r_s$ is changing (and changing in the "wrong," bond-weakening direction as atoms get bigger). A second core relationship links melting point empirically and thermodynamically to cohesive energy through the entropy of fusion:

T_m = \frac{\Delta H_{\text{fus}}}{\Delta S_{\text{fus}}}

where:

$T_m$ is the melting point (K),
$\Delta H_{\text{fus}}$ is the enthalpy of fusion, which for metals with similar bonding character tracks the cohesive energy $E_{\text{coh}}$ closely (fusion breaks only a fraction of the total lattice cohesion, roughly the same fraction across similar metals),
$\Delta S_{\text{fus}}$ is the entropy of fusion, which is roughly constant (Richard's-rule-like behaviour, a few $R$ per mole) across metals with similar coordination-number change on melting.

Reading: because $\Delta S_{\text{fus}}$ varies far less across the s-block than $\Delta H_{\text{fus}}$ does, $T_m$ is, to good approximation, a linear proxy for cohesive energy. This licenses the entire empirical strategy of this article: reading melting-point trends as a direct window onto metallic bond strength, exactly as bond dissociation energies are read from molecular thermochemistry.

Derivation AJellium estimate of cohesive-energy scaling with valence electron count

Goal: show, from a free-electron (jellium) starting point, why cohesive energy should scale roughly as $z^2/r_s$ , and use it to rationalize the Group 1 vs. Group 2 melting-point gap quantitatively in order of magnitude.

Step 1 — energy of a free-electron gas. Treat the $z$ valence electrons per atom as a gas of $N$ free electrons occupying volume $V$ , with electron density $n = zN/V$ . The average kinetic energy per electron in a degenerate Fermi gas is

\langle E_k \rangle = \frac{3}{5}E_F, \qquad E_F = \frac{\hbar^2}{2m_e}\left(3\pi^2 n\right)^{2/3}.

Since $E_F \propto n^{2/3} \propto z^{2/3}/r_s^{2}$ (using $n \propto z/r_s^3$ ), the kinetic energy per atom scales as $z \cdot z^{2/3}/r_s^2 = z^{5/3}/r_s^2$ — this term is repulsive (it resists compression) and grows with both $z$ and shrinking $r_s$ .

Step 2 — electrostatic (Madelung) attraction of electrons to ion cores. The dominant attractive contribution is the Coulomb interaction between the smeared-out negative electron density and the positive ion cores of charge $+ze$ (in the jellium model, this is captured by an electrostatic self-energy term per ion of the uniform background). Dimensional analysis of the Coulomb energy of a sphere of charge $ze$ smeared over radius $r_s$ gives an energy per atom of order

E_{\text{Madelung}} \sim -\frac{3}{5}\frac{(ze)^2}{4\pi\varepsilon_0\, r_s}.

This term scales as $z^2/r_s$ — it is always the dominant attractive term at metallic densities because it grows quadratically with $z$ , faster than the $z^{5/3}$ kinetic repulsion.

Step 3 — balance and equilibrium radius. Cohesive energy is (schematically) the sum of the kinetic (repulsive, $\propto z^{5/3}/r_s^2$ ), exchange–correlation (attractive, weaker $r_s$ -dependence), electrostatic (attractive, $\propto -z^2/r_s$ ), and short-range core repulsion terms. Minimizing the total with respect to $r_s$ at fixed $z$ gives an equilibrium $r_s^{*}$ where the steep $1/r_s^2$ kinetic repulsion balances the $1/r_s$ electrostatic attraction; the resulting equilibrium cohesive energy scales as

E_{\text{coh}}(r_s^{*}) \;\propto\; -\,z^{2}\times(\text{material-dependent constant}),

because at the energy minimum the leading surviving term is proportional to the electrostatic attraction evaluated near $r_s^*$ , and $r_s^*$ itself only weakens logarithmically/mildly with $z$ compared to the explicit $z^2$ prefactor.

Step 4 — apply to Na vs. Mg. Take $z_{\text{Na}} = 1$ , $z_{\text{Mg}} = 2$ . If $r_s$ were held fixed, the model predicts

\frac{E_{\text{coh}}(\text{Mg})}{E_{\text{coh}}(\text{Na})} \approx \left(\frac{z_{\text{Mg}}}{z_{\text{Na}}}\right)^2 = 4.

The experimental cohesive energies are $E_{\text{coh}}(\text{Na}) \approx -107\ \text{kJ mol}^{-1}$ and $E_{\text{coh}}(\text{Mg}) \approx -146\ \text{kJ mol}^{-1}$ — a ratio of only about $1.4$ , far short of 4. This tells us, correctly, that the naive $z^2$ scaling overshoots: real Mg has its $r_s^*$ shifted outward relative to the naive prediction because the second electron enters bonding orbitals with more p- and d-character mixing at the Fermi level, partially screening the extra nuclear charge and softening the ideal $z^2$ law. The qualitative ordering (Mg > Na) is exactly right; the magnitude is a genuine limitation of the free-electron picture, flagged rather than hidden.

Go deeper · Derivation B · 2 minMelting point as a proxy for lattice cohesion: relating

\Delta H_{\text{fus}}

trends to structure

Derivation BMelting point as a proxy for lattice cohesion: relating

\Delta H_{\text{fus}}

trends to structure

Goal: justify quantitatively why melting point can be used as an experimental stand-in for cohesive energy across the s-block, and use it to explain the shape of the Group 2 melting-point curve (not monotonic, unlike Group 1's mild monotonic rise).

Step 1 — thermodynamic starting point. At the melting point, solid and liquid are in equilibrium, so $\Delta G_{\text{fus}} = 0$ , giving

T_m = \frac{\Delta H_{\text{fus}}}{\Delta S_{\text{fus}}}.

For metals, $\Delta S_{\text{fus}}$ arises mostly from the loss of long-range positional order (the liquid retains substantial short-range coordination, so the entropy change is much smaller than full "gas-like" disordering). Empirically, $\Delta S_{\text{fus}}$ for simple metals clusters near $8$ – $12\ \text{J mol}^{-1}\text{K}^{-1}$ regardless of whether the parent solid is bcc, hcp, or ccp, because the local coordination environment in the melt is similar in all three cases (roughly 10–11 near-neighbours in the liquid, intermediate between bcc's 8 and ccp/hcp's 12).

Step 2 — link $\Delta H_{\text{fus}}$ to $E_{\text{coh}}$ . Melting breaks only the long-range order, not most of the local bonding — nearest-neighbour coordination in the liquid just below $T_m$ is typically 80–90% of the solid-state value. So $\Delta H_{\text{fus}}$ is a roughly constant fraction $f$ of the total cohesive (sublimation) energy:

\Delta H_{\text{fus}} \approx f \cdot |E_{\text{coh}}|, \qquad f \approx 0.02\text{–}0.04 \text{ for most metals.}

(Only a small fraction of the total bonding is disrupted by melting — most of the lattice's cohesive energy survives into the liquid, which is why $\Delta H_{\text{vap}} \gg \Delta H_{\text{fus}}$ for every metal.)

Step 3 — combine. Substituting,

T_m \approx \frac{f\,|E_{\text{coh}}|}{\Delta S_{\text{fus}}}.

Since $f$ and $\Delta S_{\text{fus}}$ are both roughly constant across the s-block metals, $T_m$ tracks $|E_{\text{coh}}|$ almost linearly — this is the licence for treating melting point as an experimental cohesive-energy proxy throughout this article.

Step 4 — apply to the non-monotonic Group 2 trend. Group 2 melting points are Be 1287 °C, Mg 650 °C, Ca 842 °C, Sr 777 °C, Ba 727 °C — a drop from Be to Mg, then a partial recovery at Ca, then a slow decline again to Ba. Down Group 1, by contrast, the trend is simply monotonically decreasing (Li 181 °C → Cs 28 °C) because $z=1$ throughout and only $r_s$ grows. In Group 2, $z = 2$ throughout, but the character of the metallic bond changes: Be and Mg are more directional/covalent-like (significant $s$ – $p$ hybridization at the Fermi surface, smaller, "harder" ion cores), giving Be an anomalously enormous cohesive energy for its group. Ca, Sr, and Ba have increasing $d$ -orbital participation in the conduction band as atomic number rises, which partially offsets the growth of $r_s$ and produces the gentler decline from Ca to Ba rather than continuing Mg's drop. The lesson: $z$ sets the scale of cohesion (Group 2 ≫ Group 1 throughout), while orbital character and $r_s$ modulate the fine structure of the trend within a group.

Worked examples

Example 1 (foundational) — packing fraction of bcc sodium and the coordination-number bookkeeping. Sodium metal is bcc with a conventional cubic unit cell edge $a = 429.1$ pm. Verify the atomic packing fraction and hence check that the "68%" figure quoted above is not a rounding artifact. In bcc, atoms touch along the body diagonal, which has length $a\sqrt{3}$ and contains 4 atomic radii ( $r$ – $2r$ – $r$ , corner–centre–corner along the diagonal, i.e. the diagonal spans one corner atom's radius, a full diameter of the centre atom, and the far corner atom's radius):

a\sqrt{3} = 4r \quad\Rightarrow\quad r = \frac{a\sqrt{3}}{4} = \frac{429.1\ \text{pm}\times 1.732}{4} = 185.8\ \text{pm}.

This is close to the tabulated metallic radius of Na, 186 pm — an internal consistency check on the touching-spheres assumption. The unit cell contains $8\times\tfrac18 + 1 = 2$ atoms. Packing fraction:

\text{APF} = \frac{2\times \tfrac{4}{3}\pi r^3}{a^3} = \frac{2\times \tfrac{4}{3}\pi (185.8)^3}{(429.1)^3}.

Numerator: $\tfrac43\pi (185.8)^3 = \tfrac43\pi \times 6.410\times10^6 = 2.685\times10^7\ \text{pm}^3$ ; doubled: $5.370\times10^7\ \text{pm}^3$ . Denominator: $(429.1)^3 = 7.902\times10^7\ \text{pm}^3$ .

\text{APF} = \frac{5.370\times10^7}{7.902\times10^7} = 0.680.

This matches the general bcc formula $\sqrt3\pi/8 = 0.6802$ exactly, confirming that bcc genuinely wastes about 32% of space compared with close packing — yet, as the energetics analysis shows, this geometric "inefficiency" is not what determines Na's low melting point; the single valence electron is.

Example 2 (applied) — comparing Na and Mg cohesive energy per unit volume, and predicting hardness. Given metallic radii $r_{\text{Na}} = 186$ pm, $r_{\text{Mg}} = 160$ pm, and cohesive energies $E_{\text{coh}}(\text{Na}) = -107\ \text{kJ mol}^{-1}$ , $E_{\text{coh}}(\text{Mg}) = -146\ \text{kJ mol}^{-1}$ , estimate the cohesive energy density (energy per unit volume) for each, which correlates with bulk modulus/hardness more directly than energy per mole does (since Mg atoms are smaller and pack more atoms into the same volume). Atomic volume (treating each atom as a sphere of its metallic radius, ignoring packing gaps for this order-of-magnitude comparison):

V_{\text{Na}} = \frac{4}{3}\pi (186\ \text{pm})^3 = \frac{4}{3}\pi \times 6.435\times10^6\ \text{pm}^3 = 2.696\times10^7\ \text{pm}^3 = 2.696\times10^{-29}\ \text{m}^3.

V_{\text{Mg}} = \frac{4}{3}\pi (160\ \text{pm})^3 = \frac{4}{3}\pi \times 4.096\times10^6\ \text{pm}^3 = 1.716\times10^7\ \text{pm}^3 = 1.716\times10^{-29}\ \text{m}^3.

Cohesive energy per atom in joules: $E_{\text{coh}}(\text{Na})/N_A = -1.778\times10^{-19}\ \text{J}$ ; $E_{\text{coh}}(\text{Mg})/N_A = -2.425\times10^{-19}\ \text{J}$ . Energy density:

u_{\text{Na}} = \frac{1.778\times10^{-19}\ \text{J}}{2.696\times10^{-29}\ \text{m}^3} = 6.60\times10^{9}\ \text{J m}^{-3} \approx 6.6\ \text{GJ m}^{-3}.

u_{\text{Mg}} = \frac{2.425\times10^{-19}\ \text{J}}{1.716\times10^{-29}\ \text{m}^3} = 1.413\times10^{10}\ \text{J m}^{-3} \approx 14.1\ \text{GJ m}^{-3}.

The ratio $u_{\text{Mg}}/u_{\text{Na}} \approx 2.1$ : Mg packs more than twice the cohesive energy into each unit of volume, consistent with its much higher hardness (Mohs ≈ 2.5 vs. Na's ≈ 0.5, Na being soft enough to cut with a knife) and its higher bulk modulus (Mg ≈ 45 GPa vs. Na ≈ 6.3 GPa, a factor of ~7 — larger than the naive energy-density ratio because bulk modulus depends on the curvature of the energy well, which is even more sensitive to the compact ion-core packing that the extra valence electron enables).

Example 3 (exam/competition level) — predicting the relative melting points of Ca, Sr, and a hypothetical "element 121-like" superheavy Group 2 congener using the $z^2/r_s$ scaling, then confronting it with real $d$ -orbital effects. Given: metallic radii $r_{\text{Ca}} = 197$ pm, $r_{\text{Sr}} = 215$ pm; actual melting points $T_m(\text{Ca}) = 1115\ \text{K}$ , $T_m(\text{Sr}) = 1050\ \text{K}$ . (a) Using the naive jellium scaling $E_{\text{coh}} \propto z^2/r_s$ with $z=2$ fixed for both, and treating melting point as proportional to $E_{\text{coh}}$ (from the $T_m \approx f|E_{\text{coh}}|/\Delta S_{\text{fus}}$ relation with $f$ , $\Delta S_{\text{fus}}$ assumed constant), predict the ratio $T_m(\text{Ca})/T_m(\text{Sr})$ .

\frac{T_m(\text{Ca})}{T_m(\text{Sr})} \approx \frac{1/r_{\text{Ca}}}{1/r_{\text{Sr}}} = \frac{r_{\text{Sr}}}{r_{\text{Ca}}} = \frac{215}{197} = 1.091.

(b) Compare with experiment: $1115/1050 = 1.062$ . The naive model predicts a 9.1% enhancement of Ca's melting point over Sr's; experiment shows 6.2%. The model gets the sign and rough magnitude right (Ca should be modestly higher-melting than Sr because it is smaller with the same $z$ ), but overshoots slightly — exactly analogous to the Na/Mg case in Derivation 1, because Sr's larger size brings somewhat more 4d-orbital character into its conduction band relative to Ca's 3d involvement, adding a small amount of extra directional bonding that partially compensates for its larger $r_s$ and narrows the gap below the pure $1/r_s$ prediction. (c) Extrapolating: for a hypothetical heavier Group 2 congener with an even larger metallic radius (radius growth continuing the Ca → Sr → Ba trend, roughly +18 pm per row), the naive $1/r_s$ scaling alone would predict continued melting-point decline — and this is indeed observed experimentally from Ca (1115 K) through Sr (1050 K) to Ba (1000 K), a smooth, slowing decline consistent with $1/r_s$ softened by increasing $d$ -orbital compensation at each step, rather than the sharp drop-then-recovery seen going from Be to Mg to Ca. This is the key qualitative distinction examiners look for: the Be→Mg step is dominated by a change in orbital character (loss of Be's unusually strong, small, hard, near-covalent bonding), while Ca→Sr→Ba is dominated by the smooth $1/r_s$ size effect only mildly damped by increasing $d$ -character — two different regimes of the same underlying $z^2/r_s$ law.

Limiting cases, scope & common traps

Regime / limit	What the model predicts	What is actually observed	Resolution
$z$ fixed at 1, $r_s$ increasing (Group 1, Li→Cs)	Monotonically weaker bonding, monotonically falling $T_m$	Li 181 °C → Na 98 °C → K 63 °C → Rb 39 °C → Cs 28 °C: monotonic decrease, as predicted	Good agreement; $z^2/r_s$ law behaves simply when $z$ is constant
$z$ fixed at 2, $r_s$ increasing (Group 2, Ca→Ba)	Monotonically weaker bonding	Ca 842 °C → Sr 777 °C → Ba 727 °C: monotonic decrease, gentler than naive $1/r_s$	$d$ -orbital involvement partially compensates for growing $r_s$
$z$ jump 1→2 at fixed period (Na→Mg)	Naive model: $E_{\text{coh}}$ scales ×4	Observed ratio only ×1.4 (energy), ×~6.6 ( $T_m$ , since 98 °C→650 °C)	Screening/orbital hybridization at Mg's Fermi surface softens the ideal $z^2$ law; $T_m$ jump still much larger than any Group-1-internal step
Be, top of Group 2	Expected to follow the smooth $z^2/r_s$ trend of the rest of Group 2	Be melts at 1287 °C — anomalously high even for Group 2, with hcp structure and markedly covalent/directional character	Small, hard, highly polarizing Be $^{2+}$ -like core with strong orbital-directional (near-covalent) bonding; classic "first-row anomaly"
bcc vs. ccp/hcp packing fraction alone	Might expect close-packed metals always to melt higher (more neighbours, more contacts)	Ba (bcc, $z=2$ ) still melts far higher than any bcc alkali metal ( $z=1$ ); Na (bcc) melts far lower than Ca (ccp)	Packing geometry is a second-order effect; $z$ (electron count) dominates over coordination-number/packing-fraction differences
High pressure	Model assumes ambient-pressure equilibrium $r_s^{*}$	Li and Na undergo structural phase transitions to close-packed forms under pressure or at very low T	The bcc structure is not "the" alkali-metal ground state at all conditions — it is the ambient-condition minimum of a shallow, nearly-degenerate energy landscape

Common trapMisconception: "the trend down a group and the trend across a period are governed by the same variable"

Down a group (fixed $z$ , increasing $r_s$ ), the metallic bond monotonically weakens — atoms get bigger, electron density drops, and melting point falls in a fairly regular, predictable way. Across a period from Group 1 to Group 2 (increasing $z$ , roughly fixed or even slightly decreasing $r_s$ , since ionic/atomic radii typically contract slightly left-to-right across a period due to increasing effective nuclear charge), the dominant story is the change in $z$ , and the effect is far larger in magnitude and different in mechanism — it is not simply "more of the same size effect." Conflating the two directions is one of the most common errors when students try to build a single unified "bigger atom = weaker metal" rule; the rule only holds within a fixed- $z$ column.

The chemist behind it

The quantitative marriage of crystal structure and free-electron energetics that underlies this article traces most directly to the foundational work of Nevill Mott and, in parallel, the broader Sommerfeld/Bloch free-electron theory of metals developed through the 1920s–1930s. Mott's insight — that the number of conduction electrons per atom, not merely the qualitative fact of metallic bonding, controls cohesive energy, electrical conductivity, and structural stability — gave chemists and physicists a shared, quantitative vocabulary for exactly the comparisons made in this article: why Na is soft and low-melting while Mg is hard and high-melting despite sitting one square to the right.

Origins · 1 minNevill Francis Mott and how this was discovered

Nevill Francis Mott

1905–1996 · Cavendish Laboratory, Bristol, and Cambridge, mid-20th-century solid-state physics

Mott's career spanned scattering theory, semiconductor physics, and the theory of metal–insulator transitions (the "Mott transition," describing how electron–electron correlation can turn a predicted metal into an insulator, a subtlety that lies just beyond the free-electron picture used in this article). He shared the 1977 Nobel Prize in Physics for fundamental theoretical work on the electronic structure of disordered and amorphous systems, but his earlier work on the electron theory of metals — quantifying how conduction electron density sets cohesion, hardness, and structural preference — is the direct intellectual ancestor of every "cohesive energy scales with valence electron count" argument used across s-, p-, and d-block metallic bonding.

History

- 1900 — Drude proposes the classical free-electron model of metallic conduction, the first quantitative link between "electrons wander freely" and measurable metallic properties.

- 1928 — Sommerfeld corrects Drude's model with Fermi–Dirac statistics, giving the quantum free-electron gas whose Fermi energy scaling underlies the jellium argument used above.

- 1930s — Bloch and Wilson develop band theory, explaining why some elements (like the alkali and alkaline earth metals) are metallic while others are not, based on band filling.

- 1930s–1950s — Mott and collaborators quantify cohesive energy and structural stability of simple metals in terms of valence electron density, directly enabling z-dependent cohesion arguments.

- 1960s onward — Pseudopotential and jellium calculations put quantitative numbers on

A

and the equilibrium

r_s^{*}

for real alkali and alkaline-earth metals, refining the qualitative

z^2/r_s

scaling into predictive computational tools.

Why doesn't packing fraction alone predict melting point, if close-packed structures have more nearest neighbours?

Packing fraction measures pure geometry — how much of space is filled by touching hard spheres — and does not encode how strongly those spheres are held together. Coordination number (8 for bcc, 12 for ccp/hcp) affects the number of nearest-neighbour bonds, but each individual bond's strength depends on the electron density available to form it, set by valence electron count $z$ and atomic size $r_s$ . Barium, bcc with $z=2$ , comfortably out-melts every bcc or close-packed Group 1 metal (all $z=1$ ), showing that electron count dominates over packing geometry. Packing fraction is a secondary, modulating variable layered on top of the primary $z^2/r_s$ trend, not an independent predictor on its own.

Is the jellium/free-electron model expected to work equally well for all s-block metals?

No — it works best for the lighter, more nearly-free-electron-like members (Na, K, Ca, Sr) and progressively worse as core-electron screening, relativistic effects, and $d$ -orbital mixing become significant (heavier alkaline earths, and especially Be with its unusually contracted, hard core). The model correctly predicts qualitative ordering (Group 2 > Group 1; smaller $r_s$ within a fixed- $z$ column gives higher cohesion) essentially everywhere, but it systematically overpredicts the magnitude of the Group 1 → Group 2 jump because it neglects orbital hybridization effects that partially screen the naive $z^2$ scaling. Treat it as a scaling law for trends, not a precision quantitative tool.

Why is Group 2 structurally more diverse (hcp, ccp, and bcc all appear) while Group 1 is almost uniformly bcc?

Group 1 metals have a single valence s electron with essentially no competing orbital character near the Fermi level, so the free-electron-like bcc structure (favoured by simple electron-gas energetics at these densities) is robustly the lowest-energy choice throughout the group. Group 2 metals have two valence electrons and, especially for the heavier members, meaningful $d$ -orbital mixing into the conduction band; this changes the effective electron-gas "compressibility" and directional character enough to tip the energetic balance between close-packed and bcc structures differently for each element, producing hcp (Be, Mg), ccp (Ca, Sr), and bcc (Ba) as the atomic number increases and $d$ -character grows.

How does the melting point of an s-block metal relate to its use as an industrial coolant or reagent?

Melting point (and the closely related boiling point) sets the liquid range over which a metal can be handled as a fluid without solidifying or boiling away — sodium's 98 °C to 883 °C liquid range makes it an excellent heat-transfer fluid for sodium-cooled fast reactors, since it stays comfortably liquid across typical reactor operating temperatures while avoiding the high pressures needed to keep water liquid at similar temperatures. Because melting point tracks cohesive energy (via $z^2/r_s$ ), engineers effectively select an element's row/group position based on how strongly-bonded (high-melting, chemically robust) or weakly-bonded (low-melting, reactive, easily liquefied) they need the working metal to be.

Does the coordination number in the liquid state matter for understanding melting point trends?

Yes — it is central to why $\Delta S_{\text{fus}}$ is roughly constant across the s-block, which is what licenses using melting point as a cohesive-energy proxy in the first place. Liquid metals retain substantial short-range order, with coordination numbers around 10–11 regardless of whether the parent solid was bcc (CN 8), hcp, or ccp (CN 12 each). Because the disordering on melting is similar in relative magnitude across all these structures, the entropy of fusion does not vary wildly from one s-block metal to the next, so differences in melting point are dominated by differences in enthalpy of fusion — which in turn tracks lattice cohesive energy.

How does this connect to periodic trends I'll see for Group 1 and Group 2 separately?

This article establishes the shared machinery — valence electron count, Wigner–Seitz radius, and the $z^2/r_s$ cohesive-energy scaling — that both group-specific trend discussions rely on. Within Periodic Trends in Group 1 Alkali Metals, $z$ is fixed at 1 throughout, so the entire discussion reduces to how $r_s$ (atomic/ionic radius) changes down the group. Within Periodic Trends in Group 2 Alkaline Earth Metals, $z$ is fixed at 2, but structural changes (hcp → ccp → bcc) and increasing $d$ -orbital participation add richer, group-specific modulations on top of the same underlying scaling law introduced here.