Six of these share a face with the central cube's six faces 12 of these share an edge with the central cube's 12 edges and eight of these share a corner with the central cube's eight corners. The central cube is in contact with 26 neighbouring cubes. However, complications arise if we try to continue this trend for distances much further out.įigure 2 - Two adjacent sodium chloride unit cells, sharing a faceĬonsider, for example, a large cube of three cubes by three cubes by three cubes, totalling 27 cubes. Also we see that the distance of the ions has followed the pattern r √ x where x is the number of ions away an ion is from the reference ion. A couple of patterns emerge: the sign of the ions alternate as we move to more and more distant neighbours, which is what is expected given that ions tend to be in contact with oppositely charged neighbours. The number of ions at distances r √4, r √5, and r √6 from the central sodium ion can be seen using Fig 2, bearing in mind that the central cube has six of these neighbouring face-sharing cubes. Having now considered all the ions in the cubic fragment, we need to visualise neighbouring cubes to progress further. three-away neighbours are of opposite charge to the central ion and are at the corner positions of the cube: eight ions a distance r√3 away.next-nearest (two-away) neighbours are of the same charge as the central ion and are at the centres of the edges of the cube: 12 ions a distance r√2 away.nearest (touching) neighbours are of opposite charge to the central ion and are at the centre of the faces of the cube: six neighbours a distance r away, where r is the sum of the ionic radii, which is half of the length of an edge of the cube.The series starts with nearest neighbours and continues outwards sequentially. If the lattice is cubic, the distances can be calculated using Pythagoras' theorem. The numerator of each fraction is the number of ions that are a given distance away from the reference ion the denominator is the distance of the reference ion to the ions in question, as a multiple of the closest ( ie touching) internuclear distance between ions. The Madelung constant is the sum of all the ionic interactions in the crystal based on the three dimensional positions of the ions. Where A is the Madelung constant, ie a numerical factor which takes the relative positions of all the ions in the lattice structure into account in the calculation of the lattice energy. Neglecting the repulsive interaction at short interionic distances, we obtain: We can adapt the Coulomb expression to incorporate the ionic radii ( r + and r - for the cation and anion, respectively), the electron charge e, the magnitudes of the ion charges ( z + and z - for the cation and anion, respectively) and Avogadro's constant L. Where ε 0 is the permittivity of free space. The lattice energy of sodium chloride, for example, may be calculated to a reasonable approximation from first principles using Coulomb's law, which states that the electrostatic potential energy ( E) between two charged particles is proportional to the product of the charges( q 1 and q 2) and inversely proportional to the distance between them ( r): By comparing the values of lattice energy (directly or indirectly) through experiment provides a good test of the accuracy of the ionic bond model and thus a better insight into the nature of chemical bonding. We can challenge our students further by introducing them to the Madelung constant as a way of calculating the lattice energy of an ionic structure, given the atom positions in the crystal from X-ray diffraction studies. At A-level we build on this model and explain that an ion in sodium chloride has six neighbours and that the ions are arranged in a cubic lattice. We tell them that oppositely charged ions attract one another, and that the ions in the crystal lattice are, accordingly, ordered so that the immediate neighbours around a given ion have the opposite charge. When we teach ionic bonding at GCSE we usually show students the structure of sodium chloride. Sixthformers show off mathematical prowess in testing the ionic bond model
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