P.A. Cox - Inorganic chemistry (793955), страница 28
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It may be estimated using Hess’Law from a sequence of steps known as a Born-Haber cycle.Theoretical estimates of lattice energies using the Born-Landé or BornMayer equations agree well with Born-Haber values for manycompounds. The Kapustinskii equation gives a useful approximateestimate. Both experimental and theoretical lattice energies increase asions become smaller or more highly charged.Lattice energies may be used to understand many important chemicaltrends, including the characteristic oxidation states of metallicelements, the stabilization of high oxidation states by oxide andfluoride, and trends in the thermal stability of oxoanion salts such ascarbonates.Stability and reactivity (B3)Solubility of ionic substancesInorganic reactions and(E4)synthesis (B6)The Born-Haber cycleThe lattice energy UL of a solid compound is defined as the energy required to transform it into gas-phase ions, forexample,(Note: sometimes the reverse process is used as a definition, which makes UL a negative quantity rather than positive ashere.) It is generally assumed that the compound concerned is ionic, but a lattice energy can be defined without thatassumption, provided the ions formed in the gas phase are clearly specified.Lattice energies may be estimated from a thermodynamic cycle known as a Born-Haber cycle, which makes use ofHess’ Law (see Topic B3).
Strictly speaking, the quantities involved are enthalpy rather than energy changes and oneshould write HL for the lattice enthalpy. From Fig. 1. which shows a cycle for NaCl, we see thatwhere the terms on the right-hand side are, in order: the enthalpy of formation of NaCl, the enthalpy of atomization ofNa solid, the bond enthalpy of Cl2, the ionization energy of Na and the electron affinity of Cl (see Topics A5 and C8).120D6—LATTICE ENERGIESFig. 1.
Born-Haber cycle for determining the lattice enthalpy of NaCl.When multiply charged ions are involved the cycle can be adapted by summing higher ionization energies or electronaffinities as appropriate.I(Na) is greater than A(Cl) in the equation above. This shows that in the gas phase, Na and Cl atoms are more stablethan the ions Na+ and Cl−, and it is the lattice energy that stabilizes the ionic charge distribution in solid NaCl. A similarresult is found for all ionic solids.Theoretical estimatesTheoretical lattice energies can be calculated if some interaction potential between ions is specified. The mostimportant term in the ionic model is the long-range Coulomb interaction between charges.
A complex summationis necessary over the different pairs of unlike and like charges appearing at different distances in the crystal structure,and gives the Coulomb energy per mole of lattice aswhere N0 is Avogadro’s constant, z+e and z−e are the charges on the ions and r0 is the distance between them, and A isthe Madelung constant coming from the long-range summation of ionic interactions.
A depends on the structure,and increases slowly with the coordination number. (For example, values for the simple AB structures discussed inTopics D3 and D4 are: zinc blende (CN=4) 1.638; rocksalt (CN=6) 1.748; CsCl (CN=8) 1.763.)The attractive Coulomb energy needs to be balanced against the contribution from the short-range repulsive forcesthat occur between ions when their closed shells overlap. There is no accurate simple expression for this repulsion.
Inthe Born-Landé model it is assumed proportional to 1/rn, where n is a constant that varies in the range 7–12depending on the ions. The resulting expression for the lattice energy isThe Born-Mayer equation is an alternative (and possibly more accurate) form based on the assumption of anexponential form for the repulsive energy. Both equations predict lattice energies for compounds such as alkali halidesthat are in reasonably close agreement with the ‘experimental’ values from the Born-Haber cycle. Some examples areshown in Table 1. A strict comparison requires some corrections. Born-Haber values are generally enthalpies, not totalenergies, and are estimated from data normally measured at 298 K not absolute zero; further corrections can be made,for example, including van der Waals’ forces between ions.
When these extended calculations are compared withexperiment many compounds agree well (see Table 1). Significant deviations do occur, however; for example, incompounds of metals in later groups where bonding is certainly less ionic (e.g. AgCl).SECTION D—STRUCTURE AND BONDING IN SOLIDS121Table 1. Comparison of lattice energies (all kJ mol−1) determined by different methodsOne of the disadvantages of the fully theoretical approach is that it is necessary to know the crystal structure and theinterionic distances to estimate the lattice energy. The Kapustinskii equations overcome this limitation by makingsome assumptions.
The Madelung constant A and the repulsive parameter n are put equal to average values, and it is alsoassumed that the interionic distance can be estimated as the sum of anion and cation radii r+ and r− (see Topic D4). Thesimpler of the Kapustinskii equations for a binary solid is(1)ν is the number of ions in the formula unit (e.g. two for NaCl, three for MgF2 and five for Al2O3) and C is a constantequal to 1.079×105 when UL is in kJ mol−1 and the radii are in pm. The Kapustinskii equation is useful for roughcalculations or where the crystal structure is unknown.
It emphasizes two essential features of lattice energies, whichare true even when the bonding is not fully ionic:• lattice energies increase strongly with increasing charge on the ions;• lattice energies are always larger for smaller ions.Calculations can be extended to complex ions such as carbonate and sulfate by the use of thermochemical radii,chosen to give the best match between experimental lattice energies and those estimated by the Kapustinskii equation.ApplicationsEven though ionic model calculations do not always give accurate predictions of lattice energies (and especially when theapproximate Kapustinskii equation is used) the trends predicted are usually reliable and can be used to rationalize manyobservations in inorganic chemistry.(i) Group oxidation statesNa+,Mg2+The occurrence of ions such asand Al3+ depends on the balance between the energies required to form themin the gas phase and the lattice energies that stabilize them in solids.
Consider magnesium. The gas-phase ionizationenergy (IE) required to form Mg2+ is considerably greater than for Mg+. However, the lattice energy stabilizing theionic structure MgF2 is much larger than that of MgF, and amply compensates for the extra IE. It is possible to estimatethe lattice energy of MgF, and (depending on what assumptions are used about the ionic radius of Mg+) its formation122D6—LATTICE ENERGIESfrom the elements may be exothermic. However, the enthalpy of formation of MgF2 is predicted to be much more negative,and the reason why MgF(s) is unknown is that it spontaneously disproportionates:Ionization beyond the closed-shell configuration Mg2+ involves the removal of a much more tightly bound 2p electron(see Topics A4 and A5).
The third IE is therefore very large and can never be compensated by the extra lattice energy ofa Mg3+ compound.(ii) Stabilization of high and low oxidation statesWhen an element has variable oxidation states, it is often found that the highest value is obtained with oxide and/orfluoride (see, e.g. Topic H4). The ionic model again suggests that a balance between IE and lattice energy is important.Small and/or highly charged ions provide the highest lattice energies according to Equation 1, and the increase in latticeenergy with higher oxidation state is more likely to compensate for the high IE.By contrast, a large ion with low charge such as I− is more likely to stabilize a low oxidation state, as the smallerlattice energy may no longer compensate for high IE input. Thus CuF is not known but the other halides CuX are.Presumably the lattice energy increase from CuF to CuF2 is sufficient to force a disproportionation like that of MgF butthis is not so with larger halide ions.
By contrast, CuX2 is stable with X=F, Cl and Br, but not I.(iii) Stabilization of large onions or cationsIt is a useful rule that large cations stabilize large anions. Oxoanion salts such as carbonates are harder todecompose thermally when combined with large cations (see Topic B6). It is also found that solids where both ions arelarge are generally less soluble in water than ones with a large ion and a small one.
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