Main Group Compounds and Structure

VSEPR shapes of hypervalent molecules

Hypervalent molecules are compounds of main-group elements whose central atom carries more than eight electrons in its valence shell. Their shapes are predicted by VSEPR theory, which counts the electron domains around the central atom, where each bonding pair and each lone pair is one domain, and then places the lone pairs to minimise repulsion. Five domains give a trigonal bipyramidal framework in which lone pairs occupy equatorial positions. Three bonding domains plus two lone pairs give a T-shape, and a double bond to oxygen counts as a single domain. Five bonding domains with no lone pair keep the trigonal bipyramidal shape, while a central atom with two domains and no lone pairs is linear, as in the azide ion.

The same counting explains the bonding in XeF_2. The three-centre four-electron bond along the linear F–Xe–F axis is a non-classical description that does not require d-orbital participation. The number of lone pairs on an atom equals half of its non-bonding valence electrons, so Xe carries three lone pairs and each F carries three, giving nine lone pairs in total. The centrosymmetric linear molecule belongs to the point group D_{\infty h}.

Aluminium hydride amine adducts

Aluminium hydride, AlH_3, is an electron-deficient Lewis acid that accepts electron pairs from amines and other donors to form adducts. Its coordination number can rise from four in a 1:1 adduct to five in a 1:2 adduct, and the geometry changes accordingly. A common preparative route is the protonolysis of a complex hydride such as LiAlH_4 by an amine hydrochloride, which releases hydrogen and lithium chloride and leaves the amine bound to the aluminium centre. Each successive equivalent of the ammonium salt replaces another Al–H bond, so the stepwise products can be identified from the stoichiometry. A five-coordinate aluminium centre with two neutral amine donors and three hydrides adopts a trigonal bipyramidal geometry.

Carbon dioxide phase diagram

A phase diagram plots the stable phase of a substance against temperature and pressure. The triple point is the single point at which the solid, liquid and gas phases coexist in equilibrium. A liquid phase can exist only at pressures above the triple-point pressure, so a substance whose triple point lies above atmospheric pressure, such as CO_2, cannot be liquid at one atmosphere and instead sublimes. The solid–liquid boundary is nearly vertical for most substances, and the liquid–gas boundary ends at the critical point, above which a supercritical fluid is formed. Students should read the boundaries carefully, since the liquid region lies between the solid–liquid and liquid–gas curves and not below the triple point temperature in every case.

Metal boride structures

The structure of a metal boride depends mainly on the metal-to-boron ratio, and the boron environment changes accordingly. Boron-poor, metal-rich borides such as the M_7B_3 type contain isolated boron atoms, each enclosed in a trigonal prism of metal atoms with no B–B bonds. Boron-richer compositions form B–B bonded units such as pairs, zigzag chains, or B_4 groups, and monoborides usually contain zigzag chains of bonded boron atoms. Therefore the presence of isolated boron is a diagnostic feature of the metal-rich M_7B_3 structure.

Thallium halide chemistry

Thallium forms two oxidation states, and the +1 state is the common one for its halides. Thallium(I) fluoride is the only thallium(I) halide that dissolves appreciably in water, whereas TlCl, TlBr and TlI are sparingly soluble. The 6s^2 lone pair on Tl^+ is stereochemically active and distorts the high-symmetry arrangement of ions, so TlF adopts a distorted rock-salt (NaCl) structure. TlCl and TlBr crystallise with the CsCl structure type. Polyhalide salts such as TlI_3 contain Tl^+ and the triiodide anion I_3^-, so they are isomorphic with CsI_3 and thallium stays in the +1 oxidation state.

Bragg diffraction from cubic lattice planes

Bragg's law, n\lambda = 2d\sin\theta, relates the wavelength \lambda of X-rays to the spacing d between lattice planes and the glancing angle \theta. For a cubic lattice of edge a, the spacing of planes with Miller indices (hkl) is d = a/\sqrt{h^2+k^2+l^2}. For a body-centred cubic metal, atoms touch along the body diagonal, so a = 4r/\sqrt{3}, where r is the atomic radius. The diffraction angle is conventionally reported as 2\theta. Example: for r = 0.10 nm, a = 0.231 nm; the (110) spacing is d = a/\sqrt{2} = 0.163 nm; with \lambda = 0.154 nm, \sin\theta = 0.472, so \theta \approx 28.2^\circ and 2\theta \approx 56.3^\circ. For a bcc lattice, only reflections with h + k + l even are observed, and planes of this type must be considered when indexing.

Group 1 metal properties

The alkali metals show a clear trend down the group. Lithium is unusual because it reacts directly with nitrogen at ordinary temperature to form Li_3N, a red-brown nitride, while the heavier alkali metals do not form nitrides under these conditions. Reactivity with water increases down the group as ionisation energy falls and metallic character rises. Thermal stability of the peroxides also increases down the group, because a larger cation stabilises the large peroxide anion more effectively. All Group 1 halides are colourless, predominantly ionic crystalline solids with high melting points.

Phosphine imine formation

Iminophosphoranes of the type R_3P=NR' are formed when a phosphorus(V) dihalide such as R_3PCl_2 reacts with a primary amine. The amine attacks phosphorus and displaces chloride, and a second equivalent of amine acts as a base to remove hydrogen chloride as the ammonium salt. All three groups on phosphorus are retained, so the phenyl groups on phosphorus remain intact in the product. The same P=N linkage is formed in the Staudinger reaction of a phosphine with an organic azide, which releases dinitrogen. Iminophosphoranes are useful as nitrogen-transfer reagents and as ligands.

Cubic close packing efficiency

The packing efficiency, or atomic packing fraction, is the fraction of the unit-cell volume occupied by atoms, treated as hard spheres. For a body-centred cubic cell there are two atoms per cell, and they touch along the body diagonal, so a = 4r/\sqrt{3}. The packing fraction is therefore \dfrac{2\cdot\frac{4}{3}\pi r^3}{(4r/\sqrt{3})^3} = \dfrac{\pi\sqrt{3}}{8} \approx 0.68, or about 68%. For comparison, face-centred cubic and hexagonal close packing both give about 74%, and simple cubic packing gives \pi/6 \approx 0.52. These values should be memorised as reference points for density and coordination calculations.

Silicate structures and shared oxygen corners

Silicate anions are built from SiO_4 tetrahedra, and the structural class is determined by the number of oxygen atoms shared between neighbouring tetrahedra, that is, the number of shared corners. Orthosilicates contain isolated tetrahedra with no shared corners. Pyrosilicates (for example Si_2O_7^{6-}) share one corner, cyclic silicates share two corners within each ring, and sheet silicates share three corners per tetrahedron. Chain and framework silicates have two and four shared corners respectively. The more corners that are shared, the lower the oxygen-to-silicon ratio of the anion.

Enthalpy change stoichiometry

The enthalpy change of a reaction in the gas phase can be estimated from mean bond enthalpies as \Delta H \approx \sum D(\text{bonds broken}) - \sum D(\text{bonds formed}), with each term multiplied by the number of bonds of that type in the balanced equation. Bond breaking is endothermic and bond formation is exothermic, so a negative result indicates an exothermic reaction. For example, in H_2 + Cl_2 \rightarrow 2 HCl, with D(H–H) = 436, D(Cl–Cl) = 242 and D(H–Cl) = 431 kJ mol^{-1}, the estimate is 436 + 242 - 2(431) = -184 kJ mol^{-1}. Because the method uses averaged bond energies, the result is an approximation.

Debye-Hückel mean activity coefficient

The Debye–Hückel limiting law gives the mean ionic activity coefficient as \log\gamma_\pm = -A|z_+z_-|\sqrt{I}, where A \approx 0.509 (mol kg^{-1})^{-1/2} for water at 25 ^\circC, and I is the ionic strength in mol kg^{-1}. The ionic strength is I = \tfrac{1}{2}\sum m_i z_i^2, summed over all ions present, so a 2:1 or 2:2 electrolyte contributes more than its molality alone suggests. Each ion contributes to I according to the square of its charge. The limiting law is valid only in dilute solutions, roughly up to I \approx 0.01 mol kg^{-1}. For example, a 0.001 mol kg^{-1} NaCl solution has I = 0.001 and \log\gamma_\pm = -0.016, so \gamma_\pm \approx 0.96.

Raoult's law for ideal solutions

Raoult's law states that for an ideal solution the partial vapour pressure of a component is p_A = x_A p_A^*, where x_A is its mole fraction and p_A^* is the vapour pressure of the pure component at the same temperature. When a solute is dissolved in a volatile solvent, only the solvent's mole fraction is needed, so masses must first be converted to moles using molar masses. Example: a solution containing 9 mol of solvent and 1 mol of a non-volatile solute has x_A = 0.9; if p_A^* = 20 kPa, then p_A = 18 kPa. The law holds best for solutions whose components are chemically similar, such as benzene with a related aromatic hydrocarbon.

Gibbs energy and equilibrium constant

For a cell reaction, the standard Gibbs energy change is \Delta_r G^0 = -nFE^0, where n is the number of electrons transferred per mole of reaction as written and F is the Faraday constant. Because \Delta_r G^0 = -RT\ln K, the same quantity gives the equilibrium constant. The temperature coefficient of the cell potential gives the entropy change, \Delta_r S^0 = nF\left(\partial E^0/\partial T\right)_p, and the enthalpy follows from \Delta_r H^0 = \Delta_r G^0 + T\Delta_r S^0. For example, with n = 2, E^0 = 1.00 V at 298 K and \partial E^0/\partial T = -4\times10^{-4} V K^{-1}, one obtains \Delta_r G^0 = -193 kJ mol^{-1}, \Delta_r S^0 = -77 J K^{-1} mol^{-1} and \Delta_r H^0 \approx -216 kJ mol^{-1}.