Lattice enrgy :Lattice energy and Born-Haber cycle

Lattice energy is the change in energy that occurs when an ionic solid is seperated into isolated  ions in the gas phase.For sodium

NaCl (s)  ------------>  Na(g) + Cl(g)

The distance between ions in the crystal are continously enlarged until the ions are very far apart.The energy required for this process is the lattice energy of NaCl.

Born- Haber cycle : The Born-Haber Cycle is Hess' Law applied to an ionic solid.

Using Born-Habe cycle:  The values used in the Born-Haber Cycle are all predetermined changes in enthalpy for the process like Ionisation energy,electron effinity,dissociation,sublimation etc . Hess' Law allows us to add or subtract these values, which allows us to determine the lattice energy.

To understand about the lattice energy we will consider the example of Solid sodium chloride and will find its lattice energy.As direct determination of lattice of an ionic solid is difficult we determine this quantity indirectly from the experiment by means of a thermochemical "cycle" based on Hess's law.

To obtain the lattice energy of NaCl, think NaCl solid being formed form the elements by two different routes, as shown in the figure.In one route NaCl (s) is formed directly from elements, Na(s) + 1/2Cl (g).The energy change for this is ΔH0 , which is given in the table of standard enthalpy of formation  as -411 kj per mole of NaCl.The second route consists of following steps: 

  1. Sublimation of sodium : Mettalic sodium is vaporaized to gas of sodium atoms.Sublimation is transformation of a solid to a gas).The enthalpy change for this process,measured experimentally, is 108 kj per mole of sodium.
  2. Dissociation of chlorine : Chlorine molecules are dissociated to atoms,The enthalpy change for this equals the Cl---Cl bond dissociation energy,which is 240 kj per mole of bonds, or 120 kj per mole of Cl atoms.
  3.  Ionisation of sodium : Sodium atoms are ionized to Na+ ions.The enthalpy change is essentially the ionization energy of atomic sodium, which equals 496 kj per mole of Na.
  4. Formation of chloride ion ;The electrons from the ionization of sodium atoms are transferred to chlorine atoms.The enthalpy change for this is the electron affinity of atomic chlorine,which equals  - 349 kj per mole of Cl atoms.
  5.  Formation of NaCl (s) from ions : The ions Na+ and Cl- formed in the steps 3 and 4 the combine to give solid NaCl.Becasue this process is just reverse of the one corresponding to the lattice energy  (breaking of solid into ions),the enthalpy change is the negative of the lattice energy.If we let U be the lattice energy, the enthalpy change for step 5 is -U.

Lets write these five steps and add them..We also add enthalpy changes,following the Hess's law.

Na (s)       -------------->   Na (g)                     ΔH1   =       108 kj

1/2 Cl2(g)     ----------------> Cl (g)                ΔH2 =           102 kj

Na (g)         ----------------> Na+ (g) + e- (g)  ΔH3 =          496 kj

Cl (g) +  e- (g) ------------> Cl- (g)                ΔH4 =         -349 kj

Na+ (g) + Cl- (g) --------->  NaCl (s)           ΔH5 =          - U

---------------------------------------------------------------------------------

Na (s)  +  1/2 Cl2 (g)  --------------> NaCl         ΔH0f  = 375 - U

while summing these equation ,we cancel terms that appear on both the left and right sides of the arrows.The final equation is simply the formation reaction for the NaCl(s).Adding the enthalpy changes,we find that the enthalpy change for this formation reaction is 375 kj - U . But the enthalpy change for this formation reaction has been determined calorimetrically adn equals -411 kj.Equating these two values,we get

     375 kj - U = - 411 kj

Solving for U yields the lattice energy of NaCl :

                  U = (375 + 411 ) kj = 786 kj

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