Applying the Pauling's electronegativity when the ionisation enthalpy and electron gain enthalpy of an element are given

Question

I came across the following problem while practicing some exercises on inorganic chemistry:

If the ionization enthalpy and electron gain enthalpy of an element are 275 and 86 $Kcal mol^{-1}$ respectively, then the electronegativity (in $KJ mol^{-1}$) of the element on the Pauling scale is?

I know this is rather, a formula based question but I was really confused about this. According to the formula for pauling's scale which is,

$|\chi_A - \chi_B|= 0.208\sqrt{\Delta E}$ :

The question doesn't even seem right as it only mentions a single element while the formula talks about difference of electronegativities and considers A and B as:

$\ce{A2 + B2 -> A-B}$ where $\chi_{A}, \chi_B$ are the electronegativities of A and B respectively.

How do I work this out? I thought of simply taking $\Delta E$ as the difference of the given Ionisation enthalpy and the electron gain enthalpy. And that got me the answer. But thats just for the sake of getting the answer. I can't get the point behind this.

Answer 1

From Atkins' Physical Chemistry,

The electronegativity is a parameter introduced by Linus Pauling as a measure of the power of an atom to attract electrons to itself when it is part of a compound. Pauling used valence-bond arguments to suggest that an appropriate numerical scale of electronegativities could be defined in terms of bond dissociation energies, $\Delta$, in electron volts and proposed that the difference in electronegativities could be expressed as $$\mathrm{|\chi_A - \chi_B| = 0.102 ({\Delta_{A-B} -\frac{1}{2} .(\Delta_{A-A} + \Delta_{B-B}}))^{\frac{1}{2}} \tag{1}}$$ Electronegativities based on this definition are called Pauling electronegativities.

The spectroscopist Robert Mulliken proposed an alternative definition of electronegativity. He argued that an element is likely to be highly electronegative if it has a high ionization energy (so it will not release electrons readily) and a high electron affinity (so it is energetically favorable to acquire electrons). The Mulliken electronegativity scale is therefore based on the definition $$\mathrm{\chi_M = \frac{1}{2} (I+ E_{Ea}) \tag{2}}$$ where $\mathrm{I}$ is the ionization energy of the element and $\mathrm{E_{ea}}$ is its electron affinity (both in electron-volts). The Mulliken and Pauling scales are approximately in line with each other. A reasonably reliable conversion between the two is $$\mathrm{\chi_P = 1.35{\chi_M}^{\frac{1}{2}}-1.37 \tag{3}}$$

Converting electron-volts to $\pu{kJ mol-1}$ Eq-(1) converts to $$\mathrm{|\chi_A - \chi_B| = 0.0103 ({\Delta_{A-B} -\frac{1}{2} .(\Delta_{A-A} + \Delta_{B-B}}))^{\frac{1}{2}} \tag{4}}$$ and Eq-(2) converts to $$\mathrm{\chi_M = \frac{1}{192.97} (I+ E_{Ea}) \tag{5}}$$

Reference

Atkins' Physical Chemistry: Eleventh Edition , Oxford University Press.