Which has the highest lattice energy?

Lattice energy is inversely proportional to ion size and directly proportional to ion charge. It is given by the Born-Landé equation:

#DeltaH_"L" = -(z_(+)z_(-)e^2)/(4piepsilon_0r_0)(1 - 1/n)N_AM#

where:

• #z_(pm)# is the magnitude of charge of the ion.
• #e = 1.60217662 xx 10^(-19) "C"# is the charge of a proton.
• #epsilon_0 = 8.854187817 xx 10^(-12) "F"cdot"m"^(-1)# is the vacuum permittivity.
• #M# is the Madelung constant, where #M_i = sum_j (z_j)/(r_(ij)//r_0)# for the #i#th ion.
• #n# is the Born exponent, an experimentally-derived quantity between #5# and #12#.
• #N_A# is Avogadro's number.

Here we find:

• The larger the ions, the smaller the amount of energy released due to forming the lattice from the gaseous ions (since they are interacting more weakly from a longer distance), i.e. the smaller the lattice energy.
• The larger the charge on the ions, the more strongly the ions interact (higher charges interact more...), and thus, the larger the lattice energy.

Each of these factors affect lattice energy together, but here we have isolated factors.

• #"Al"_2"O"_3# has the higher lattice energy than that of #"Ga"_2"O"_3#, because #"Al"^(3+)# is a smaller cation than #"Ga"^(3+)# (one less quantum level, so the electrons are attracted in by more). All other factors are the same.
• #"Na"_2"O"# has the higher lattice energy than #"NaOH"# because the charge of #"O"^(2-)# is larger than that of #"OH"^(-)#. All other factors are the same.