Crystalline solids have a regular, long-range ordered arrangement of particles, sharp melting points, definite geometrical shapes and are anisotropic (properties differ with direction); e.g. sodium chloride and diamond. Amorphous solids have only a short-range order (irregular arrangement), no sharp melting point (they soften over a range), no definite shape and are isotropic (same properties in all directions); e.g. glass and rubber.

A unit cell is the smallest repeating three-dimensional portion of a crystal lattice which, when repeated in different directions, generates the entire crystal. In a primitive (simple) unit cell, the constituent particles are present only at the corners of the unit cell. In a centred unit cell, particles are present at positions other than the corners as well, such as body-centred (one at the centre of the body), face-centred (one at the centre of each face) or end-centred.

A corner atom is shared by 8 unit cells (contributes 1/8), a face atom by 2 (contributes 1/2) and a body-centre atom belongs entirely to one cell (contributes 1). Simple cubic: 8 corners x 1/8 = 1 atom. Body-centred cubic (bcc): (8 x 1/8) + 1 (centre) = 2 atoms. Face-centred cubic (fcc): (8 x 1/8) + (6 x 1/2) = 1 + 3 = 4 atoms.

For bcc, Z = 2. Edge a = 288 pm = 2.88 x 10^-8 cm, so a^3 = (2.88 x 10^-8)^3 = 2.39 x 10^-23 cm^3. Density d = Z M/(a^3 NA) = (2 x 56)/(2.39 x 10^-23 x 6.022 x 10^23) = 112/14.39 = 7.79 g/cm^3.

d = Z M/(a^3 NA)

Schottky defect is a vacancy defect in which an equal number of cations and anions are missing from their lattice sites, so the density of the solid decreases; it occurs in ionic solids with high coordination number and similar cation and anion sizes (e.g. NaCl, KCl). Frenkel defect is a dislocation defect in which an ion (usually the smaller cation) leaves its lattice site and occupies an interstitial site, so the density remains unchanged; it occurs in solids with a large difference in the sizes of the ions (e.g. AgCl, ZnS).

Packing efficiency is the percentage of the total space occupied by the particles. Simple cubic: packing efficiency 52.4 percent, coordination number 6. Body-centred cubic (bcc): packing efficiency 68 percent, coordination number 8. Face-centred cubic (fcc, also ccp/hcp): packing efficiency 74 percent (the highest), coordination number 12.

Molarity (M) is the number of moles of solute per litre of solution (mol/L). Molality (m) is the number of moles of solute per kilogram of solvent (mol/kg); it is independent of temperature. Mole fraction of a component is the ratio of the number of moles of that component to the total number of moles of all components in the solution; it is dimensionless and the sum of the mole fractions equals 1.

M = n/V(L); m = n/mass of solvent(kg)

Moles of glucose = mass/molar mass = 18/180 = 0.1 mol. Mass of solvent (water) = 500 g = 0.5 kg. Molality = moles of solute/mass of solvent in kg = 0.1/0.5 = 0.2 mol/kg (0.2 m).

molality = moles of solute/mass of solvent (kg)

Henry's law states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas over the liquid. Mathematically, p = KH x (mole fraction of gas), where KH is Henry's law constant. Applications: it explains why soft drinks (aerated under high pressure) fizz when opened, and why deep-sea divers can suffer from bends (nitrogen dissolving in blood under high pressure).

p = KH x (mole fraction)

Raoult's law states that for a solution of volatile liquids, the partial vapour pressure of each component is directly proportional to its mole fraction in the solution. For components A and B, pA = pA(pure) x xA and pB = pB(pure) x xB, and the total vapour pressure = pA + pB. For a solution of a non-volatile solute, the relative lowering of vapour pressure equals the mole fraction of the solute.

pA = pA(pure) x xA

Colligative properties are those properties of a solution that depend only on the number of solute particles present and not on their chemical nature. The four colligative properties are: (1) relative lowering of vapour pressure; (2) elevation of boiling point; (3) depression of freezing point; and (4) osmotic pressure. They are used to determine the molar mass of a solute.

The depression of freezing point is given by delta Tf = Kf x molality = Kf x (w2 x 1000)/(M2 x w1), where w2 = mass of solute, w1 = mass of solvent, M2 = molar mass of solute. So 0.93 = 1.86 x (3 x 1000)/(M2 x 100) = 55.8/M2. Therefore M2 = 55.8/0.93 = 60 g/mol.

delta Tf = Kf x (w2 x 1000)/(M2 x w1)

Osmotic pressure is given by pi = C R T, where C is the molar concentration. Here C = 0.1 mol/L, R = 0.0821 L atm K^-1 mol^-1, T = 300 K. So pi = 0.1 x 0.0821 x 300 = 2.463 atm. Osmotic pressure is a colligative property used to find the molar mass of macromolecules.

pi = C R T

The Van't Hoff factor (i) is the ratio of the observed (experimental) value of a colligative property to the calculated (theoretical) value assuming no association or dissociation; equivalently, i = (actual number of particles after association/dissociation)/(number of particles before). For a solute that dissociates (like NaCl), i is greater than 1; for a solute that associates (like acetic acid in benzene), i is less than 1; and for a solute that neither associates nor dissociates, i = 1.

i = observed value/calculated value

An ideal solution is one that obeys Raoult's law over the entire range of concentration; the enthalpy of mixing and the volume change on mixing are zero (e.g. benzene and toluene). A non-ideal solution does not obey Raoult's law; it shows either positive deviation (when A-B interactions are weaker than A-A and B-B, giving higher vapour pressure, e.g. ethanol and water) or negative deviation (when A-B interactions are stronger, giving lower vapour pressure, e.g. chloroform and acetone).

A galvanic cell is an electrochemical device that converts chemical energy into electrical energy through a spontaneous redox reaction (e.g. the Daniell cell). It has two electrodes: the anode, where oxidation (loss of electrons) takes place and which is the negative terminal; and the cathode, where reduction (gain of electrons) takes place and which is the positive terminal. A salt bridge connects the two half-cells and maintains electrical neutrality.

The standard electrode potential is the potential difference developed between an electrode and its solution of unit concentration (1 M) at 298 K and 1 bar pressure, measured with respect to a reference electrode. The standard hydrogen electrode (SHE) is taken as the reference electrode and is assigned a potential of exactly zero volt. Electrodes are compared with the SHE to obtain their standard potentials.

In the Daniell cell, zinc (lower/more negative potential) is the anode and copper is the cathode. The standard EMF = E(cathode) - E(anode) = E(Cu2+/Cu) - E(Zn2+/Zn) = (+0.34) - (-0.76) = 0.34 + 0.76 = 1.10 V. Since the EMF is positive, the cell reaction is spontaneous.

E(cell) = E(cathode) - E(anode)

The Nernst equation gives the electrode (or cell) potential under non-standard conditions. For the reaction M(n+) + n e- -> M, the electrode potential E = E(standard) - (2.303 RT/nF) log(1/[M(n+)]). At 298 K, substituting the constants, E = E(standard) - (0.0591/n) log(1/[M(n+)]), where E(standard) is the standard electrode potential, n is the number of electrons transferred, F is the Faraday constant, and the term in brackets is the reaction quotient. It shows how potential varies with concentration.

E = E(standard) - (0.0591/n) log Q

The standard Gibbs energy change of a cell reaction is related to the standard EMF by delta G(standard) = -n F E(standard), where n is the number of electrons transferred, F is the Faraday constant (96500 C/mol) and E(standard) is the standard cell EMF. A positive EMF gives a negative delta G, meaning the cell reaction is spontaneous. This relation also links electrochemistry with thermodynamics.

delta G(standard) = -n F E(standard)