# Determine which of the following ions is the hardest acid? "Cs"^(+), "Ca"^(2+), "Ba"^(2+), "Li"^(+)

Jun 21, 2017

The strongest capacity to polarize, or the polarizing ability, is highest for the atom with the most concentrated electron density. That would be the smallest atomic radius, and (usually) be largest positive charge.

The absolute hardness $\eta$ of these ions (except for ${\text{Ba}}^{2 +}$) would be (Appendix B.5):

• ${\text{Cs}}^{+}$: $10.6$
• ${\text{Ca}}^{2 +}$: $19.52$
• ${\text{Li}}^{+}$: $35.12$

(higher $\eta$ $\to$ harder acid $\to$ more polarizing)

${\text{Ba}}^{2 +}$ has such a larger radius than ${\text{Li}}^{+}$ that it is no competition. ${\text{Li}}^{+}$ is the hardest acid here and is the most polarizing (highest capacity to polarize).

Further supporting data shown here says that ${\text{Cs}}^{+}$ is a soft acid, implying that ${\text{Ba}}^{2 +}$ is either soft or borderline soft. (And we have already settled whether ${\text{Ca}}^{2 +}$ or ${\text{Li}}^{+}$ was harder.)

From Hard-Soft Acid-Base (HSAB) theory, the hardness of the electron density for an atom or molecule allows us to predict which substances will react with each other.

It also allows us to predict how polarizing (how well they polarize) or polarizable (how easily they GET polarized) they might be. In this regard, we have the following trends:

• Harder atomic acids have smaller atomic radii, larger positive ionic charge magnitudes, and relatively low electronegativity.
• Harder atomic bases have smaller atomic radii, larger negative ionic charge magnitudes, and high electronegativity.

Clearly, these choices are all Lewis acids. The hardest Lewis acid would be the most polarizing. Hardness can be measured by the hardness parameter $\eta$ (Appendix B.5):

• ${\text{Cs}}^{+}$: $10.6$
• ${\text{Ca}}^{2 +}$: $19.52$
• ${\text{Li}}^{+}$: $35.12$

(${\text{Ba}}^{2 +}$ wasn't listed in the reference.)

Higher $\eta$ means more concentrated electron density. It appears that the atomic radius of lithium is so small that even though ${\text{Ca}}^{2 +}$ has the higher positive charge, it still is not as hard as ${\text{Li}}^{+}$.

And of course, since ${\text{Ba}}^{2 +}$ has a larger atomic radius than ${\text{Ca}}^{2 +}$, there is no way that ${\text{Ba}}^{2 +}$ is harder than ${\text{Li}}^{+}$, because ${\text{Ba}}^{2 +}$ has a poorer hold on its own electron density (more polarizable $\to$ less able to polarize).

Further supporting data shown here says that ${\text{Cs}}^{+}$ is a soft acid, implying that ${\text{Ba}}^{2 +}$ is either soft or borderline (neither soft nor hard).

Therefore, $L {i}^{+}$ is the hardest acid and has the greatest polarizing ability here.