The dissociation energy, that is, the energy needed to break a NaCl crystal into individual sodium and chlorine atoms, is determined experimentally to be 764.4 kJ/mol at standard temperature and pressure (STP) (see Table 10.2). Use this value to calculate the range parameter ρ for NaCl. Strategy

Question

The dissociation energy, that is, the energy needed to break a NaCl crystal into individual sodium and chlorine atoms, is determined experimentally to be 764.4 kJ/mol at standard temperature and pressure (STP) (see Table 10.2). Use this value to calculate the range parameter ρ for NaCl.

Strategy First divide the experimental dissociation energy by Avogadro’s number (the number of ion pairs per mole) to obtain a value of [latex]1.269 imes 10^{-18}[/latex] J/ion pair. Then with the equilibrium position at [latex]r=r_{0}, ext { we have } V\left(r=r_{0} ight)=-1.269 imes 10^{-18}[/latex] J. With this numerical value of [latex]V\left(r=r_{0} ight)[/latex], Equation (10.21) can be solved for [latex] ho / r_{0}[/latex].

[latex]V\left(r=r_{0} ight)=-\frac{\alpha e^{2}}{4 \pi \epsilon_{0} r_{0}}\left[1-\left( ho / r_{0} ight) ight][/latex] (10.21)

Accepted Answer

We solve Equation (10.21) for [latex] ho / r_{0}[/latex] in terms of [latex]V\left(r=r_{0} ight)[/latex] and obtain

[latex]\begin{aligned}\frac{ ho}{r_{0}} &=1+\frac{4 \pi \epsilon_{0} r_{0} V\left(r=r_{0} ight)}{\alpha e^{2}} \quad \&=1+\frac{(0.282 nm )\left(-1.269 imes 10^{-18} J ight)}{\left(8.988 imes 10^{9} N \cdot m ^{2} / C ^{2} ight)(1.7476)\left(1.602 imes 10^{-19} C ight)^{2}} \&=0.112\end{aligned}[/latex]

Therefore [latex] ho=0.112 r_{0}=0.0316[/latex] nm, which is in agreement with the value listed in Table 10.2. Indeed, this shows that the repulsive potential is very short range.

Table 10.2 Properties of Salt Crystals with the NaCl Structure
	Nearest-	Repulsive Range	Dissociation
Salt	Neighbor	Parameter	Energy
Crystal	Separation (nm)	[latex] ho[/latex] (nm)	(kJ/mol)
LiF	0.214	0.029	1014
LiCl	0.257	0.033	832.6
LiBr	0.275	0.034	794.5
LiI	0.3	0.037	743.8
NaF	0.232	0.029	897.5
NaCl	0.282	0.032	764.4
NaBr	0.299	0.033	726.7
NaI	0.324	0.035	683.2
KF	0.267	0.03	794.5
KCl	0.315	0.033	694
KBr	0.33	0.034	663.5
KI	0.353	0.035	627.5
RbF	0.282	0.03	759.3
RbCl	0.329	0.032	666.8
RbBr	0.345	0.034	638.8
RbI	0.367	0.035	606.6
From C. Kittel, Introduction to Solid State Physics, 5th ed., New York: Wiley (1976), p. 92.

Question 10.4: The dissociation energy, that is, the energy needed to break...

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