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Born Haber Cycle

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The lattice enthalphy for an ionic compound is the enthalphy change when 1 mole of the solid in its standard state is formed from its ions in the gaseous state.

Na+(g) + Cl-(g) $\rightarrow$ NaCl(s) 

Lattice enthalpies for ionic compounds are a good measure of the strength of ionic bonding in the lattice and are needed in solubility. Tey cannot be measured directly but can be calculated from experimental data using a Born-Haber cycle. This splits up the enthalpy of formation into individual enthalpy changes, one of which is the lattice enthalpy.

What is the Born–Haber Cycle?

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"The lattice energy of an ionic compound is the energy required to separate the solid into gas-phase ions. It may be estimated using Hess's Law from a sequence of steps known as a Born-Haber cycle."

The Born-Haber cycle is an energy level diagram that has the similar usuage to an energy cycle. The Born-Haber cycle is useful in determining lattice energy (L.E) of ionic compounds from experimentally determined data since L.E values cannot be obtained directly. 

Born-Haber Cycle for: NaCl

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The Born Haber cycle is applicable in the calculation of lattice energy of ionic crystalline solids. The energy terms involved in building a crystal lattice such as NaCl is shown in steps.

Na(s) + $\frac{1}{2}$Cl2(g) $\rightarrow$ NaCl(crystal)

involves the following three steps.
  • The elements are converted into gaseous atoms.
Na(s) $\xrightarrow[\Delta H_{s}]{sublimation}$ Na(g)

$\frac{1}{2}$Cl2 $\xrightarrow[\frac{1}{2}\Delta H_{d}]{dissociation}$ Cl(g)
  • The gaseous atoms are converted into the ions.
Na(g) $\xrightarrow[(+I)]{ionization}$ Na+(g) + e-

Cl(g) $v\xrightarrow[Addition\ of\ electrons -E]{Electron\ affinity}$ Cl-(g)
  • The ions are packed together forming the NaCl crystal.
Na+(g) + Cl-(g) $\xrightarrow[(Lattice\ Energy)]{Crystal\ formation}$ NaCl (crystal)

Where,

$\Delta$Hs = Heat of sublimation of the sodium atom
$\Delta$Hd = Heat of dissociation of the molecular chlorine.
I = Ionization energy of the sodium
E = Electron affinity of the chlorine (g) atoms.
U = Lattice energy of sodium chloride.

Thus, the entropy of the formation ($\Delta H_{f}$) of NaCl (crystal) is the sum of the terms going around the cycle.

-$\Delta H_{f}$ = $\Delta H_{s}$ + I + $\frac{1}{2}\Delta H_{d} - E - U$

Therefore the Born Haber cycle for the formation of the NaCl crystal is shown below. 

Born Haber Cycle of NACL

Lattice Energy

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When ions combine to form an ionic solid there is a huge release of energy. The reaction is highly exothermic. The energy given out when ions of opposite charges come together to form a crystalline lattice is called the lattice energy.
"Lattice energy is the enthalpy change when 1 mole of an ionic compound is formed from its gaseous ions under standard conditions."

Lattice energy is more accurately called the lattice enthalphy. However, the term lattice energy is commonly applied to lattice enthalphy as well. Lattice energy is the internal energy change when 1 mole of an ionic compound is formed from its gaseous ions at 0K. Lattice enthalphy values are very close to the corresponding lattice energy values. 

The lattice energy of ionic compound is shown below. 

Lattice Energy of Ionic Compound

Lattice Energy Calculations via Born-Haber Cycle

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The Born-Haber cycle is an important application for calculating the lattice energy for a crystal from other parameters that can be measured directly. The lattice energy is defined as the change in energy that takes place when gaseous ions are packed together to form an ionic solid. 

This is represented by the following equation.

M+(g) + X-(g) $\rightarrow$ MX(s)

This is specific to monovalent ions. Since a crystal represents a highly stable ordered state a tremendous amount of energy will be released when an ionic solid forms from its ions. This implies that the reaction will be exothermic and have a negative sign.

The lattic energy cannot be experimentally determined; it can be found only by using the Born-Haber cycle. Nevertheless this value is called the experimental value of $\Delta$Hlatt since the Born-Haber cycle uses experimentally determined values. The value thus obtained is the lattice energy which the compound actually possesses; it is the real lattice energy.

Born Haber Cycle Example

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The lattice enthalphy of an ionic solid is the enthalphy change when one mole of the solid in its standard state is formed from its ions in the gaseous state. Lattice enthalphy is the enthalphy change when one mole of an ionic solid is formed its gaseous ion. Lattice enthalphy can also be defined as the enthalphy change when one mole of an ionic solid is separated into its gaseous ions. 

The energy changes associated with the formation of an ionic solid are usually shown in an energy cycle.

Born Haber Cycle

A Born-Haber cycle is similar to a Hess's cycle, enabling calculation of changes that cannot be measured directly.

Born Haber Cycle Problems

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Some of the solved problems based on Born-Haber cycle is given below.

Solved Examples

Question 1: Caculate the lattice energy of NaCl crystals from the following data.
Madelung constant A = 1.748
ro = 0.2814 nm
Born exponent n = 8
$\varepsilon_{o}$ = 8.854 $\times$ 10-12 C2 m-1J-1
e = 1.602 $\times$ 10-19C
Solution:
Step 1: asd

Step 2:
The Born-Lande expression for lattice energy is 

= $\frac{NA(Z)(Z)e^{2}}{4\pi\varepsilon_{o}r_{o}}\left(1-\frac{1}{n} \right)$

N = 6.022$\times$1023, Z = 1, A = 1.748, e = 1.602 $\times$ 10-19C
$\varepsilon_{o}$ = 8.854 $\times$ 10-12, n =8, ro = 0.2814, nm = 0.2814 $\times$ 10-9m, $\pi$ = 3.1416

= - $\frac{6.022 \times 10^{23} \times 1.748 \times (1.602 \times 10^{-19})^{2}}{4 \times 3.1416 \times 8.854 \times 10^{-12} \times 0.2814 \times 10^{-9}}\left(1-\frac{1}{n}\right)$
 
= - 862.7 $\times$7/8

= - 754.9 kJ mol-1


Question 2: Calculate the lattice energy for the formation of K+Cl- from the following data
  • Enthalphy of sublimation of potassium $\Delta H_{s}$
  • Ionization energy of potassium IE = 418.7 kJ mol-1
  • Enthalphy of dissociation of chlorine $\Delta H_{d}$ = 240 kJ mol-1
  • Electron affinity of chlorine EA = -348.7kJ mol-1
  • Enthalphy of formation of KCl, $\Delta H_{f}$ = -440.3 kJ mol-1

Solution:
The formation of K+Cl- may be represented as

K(s) + 1/2Cl2(g) $\rightarrow$ K+Cl-(S);  $\Delta H_{f}$ = -440.3 kJ mol-1

and the formation of lattice as 

K+(g) + Cl-(g) $\rightarrow$ K+Cl-(s)

From Born-Haber cycle for the process we have

$\Deta H_{f} = \Delta H_{s} + \frac{1}{2}\Delta H_{d} + IE - EA + U

Substituting the values we get

-440.3 = 90.9 + 1/2(241) + 418.7 - 348.7 + U

U = -721.1 kJ mol-1