Pauling electronegativity (${\chi }_{\mathrm{P}}$)

Pauling electronegativity is a relative, dimensionless scale motivated by bond-dissociation energies. The interesting observation is that a chemical bond between unlike chemical atoms can be stronger than a simple baseline inferred from the corresponding like-atom chemical bonds. An empirical excess energy can then be associated with an electronegativity difference. The 1932 paper The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms built such a map; its original coordinates used hydrogen as zero, so those coordinates should not be mistaken for the modern tabulated Pauling electronegativities.^[ref]

One arithmetic-baseline version of the Pauling scale relation isHere $D_{AB}$, $D_{AA}$ and $D_{BB}$ are consistently defined bond-dissociation energies per molecule, and the arithmetic mean is the chosen baseline. With molar energies instead, write $\sqrt{{\Delta }_D/(96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$, approximately $0.102\sqrt{{\Delta }_D/(1\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$. The denominator makes the result dimensionless; the factor $0.102$ is not a unit-independent magic constant. This is an empirical relation with a specified baseline, not a definition valid for every chemical bond.^[ref]

For an explicitly synthetic example, suppose $D_{AA}=D_{BB}=300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and $D_{AB}=396.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ with the same temperature and dissociation convention. Then ${\Delta }_D=96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and the model gives an electronegativity difference of $1$. Replacing $D_{AB}$ by $300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ gives zero; a negative excess would make this real square-root model inapplicable, not produce a negative electronegativity difference. These invented inputs illustrate the formula and are not measurements of named chemical substances.

Only differences appear, so adding the same constant to every Pauling electronegativity leaves this relation unchanged. A reference value must fix the origin. The bond-dissociation energies alone also do not tell which endpoint attracts electron density more strongly, because interchanging $A$ and $B$ leaves the formula unchanged. Chemical comparisons fix that direction. Thus Pauling electronegativity is useful as a compact ordering, while the underlying measured energies and approximations remain essential.

A Pauling bond-energy excess subtracts a chosen like-atom baseline from the unlike-atom bond-dissociation energy. For an unequal-input arithmetic example, stipulate $D_{AA}=200$, $D_{BB}=450$ and $D_{AB}=421.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, all with one dissociation convention. The arithmetic mean baseline and excess areThe arithmetic-baseline Pauling scale model therefore gives $|{\chi }_{\mathrm{P}}(A)-{\chi }_{\mathrm{P}}(B)|=\sqrt{96.485/96.485}=1$. The bond-dissociation energy $421.485$ is not itself the excess: subtracting the baseline is essential. All three numbers are stipulated, not experimental values for named chemical elements.

Changing the baseline changes the model. For the same stipulated inputs, a geometric mean baseline would be $\sqrt{200\times 450}=300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, yielding an excess of $121.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and a difference of approximately $1.122$. It is a different construction, even though both start from the same bond-dissociation energies. State which baseline is used before comparing electronegativity differences.

The arithmetic-baseline Pauling scale also has a useful inverse: a difference of $0.5$ corresponds to an excess of $24.12125\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, and a difference of $2$ corresponds to $385.94\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Doubling the electronegativity difference quadruples the modeled excess. Interchanging the endpoint labels changes neither quantity; the square root alone cannot choose the direction of bond polarization.

In The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms (printed pp. 3576–3578, Table II and equation (1)), Linus Pauling organized excess energies for chemical bonds into an approximately additive map of their square roots. He also identified a limitation: his H–F map separation of $2$ predicted an excess of $4.00\mathrm{eV}$, exceeding his table's value by $1.23\mathrm{eV}$. Those are historical values and a historical model discrepancy, not modern recommended inputs.^[ref]