4
$\begingroup$

I have this table here. \begin{array} {|r|r|}\hline \text{Radius Ratio} & \text{Coordination number} & \text{Type of Void} \\ \hline <0.155 & 2 & \text{Linear} \\ \hline 0.155-0.225 & 3 & \text{Triangular Planar} \\ \hline 0.225-0.414 & 4 & \text{Tetrahedral} \\ \hline 0.414-0.732 & 6 & \text{Octahedral} \\ \hline 0.732-1 & 8 & \text{Cubic} \\ \hline \end{array}

I get the ionic radius for $\ce{Cl-}, r_{a} = \pu{181 pm}$
I get the ionic radius for $\ce{K+}, r_{c} = \pu{133 pm}$

For $\ce{KCl}$, Cation-anion radius ratio = $\frac{r_c}{r_a} = \frac{133}{181} \approx 0.735$

This value falls within the range of $0.732 \; \text{to}\; 1$, so I would expect $\ce{KCl}$ to have coordination number $8$, which corresponds to BCC structure similar to $\ce{CsCl}$. But in reality, it takes FCC structure like $\ce{NaCl}$. Why is this so?

$\endgroup$
3
  • $\begingroup$ Note that the CsCl structure is not bcc. It's simple cubic because identical ions (such as the chloride ions) form a simple cubic array. $\endgroup$ May 13, 2023 at 11:39
  • 5
    $\begingroup$ Ionic radii are not real things but an approximation to the truth - in reality all ions are not hard spheres but at least slightly "squishy" and close to the boundaries given by the radius ration rules there are exceptions. Also note KCl transforms to the CsCl structure at relatively low pressure - e.g. iopscience.iop.org/article/10.1088/0953-8984/14/44/377/meta Also as another example of the radius ratio rules not being golden note NaCl also transforms to CsCl under pressure pubs.aip.org/aip/jap/article/39/1/319/504587/… $\endgroup$
    – Ian Bush
    May 13, 2023 at 11:46
  • 2
    $\begingroup$ @OscarLanzi My apologies. After researching for a while, I discovered you're right. The material I'm reading from suggests that $\ce{CsCl}$ forms "BCC structure $\ce{Cl-}$ ions at the corners of a cube". $\endgroup$ May 13, 2023 at 11:51

1 Answer 1

7
$\begingroup$

ChemLibreTexts describes the radius ratio rule as follows.

(We) consider that the anion is the packing atom in the crystal and the smaller cation fills interstitial sites ("holes"). Cations will find arrangements in which they can contact the largest number of anions. If the cation can touch all of its nearest neighbor anions, as shown at the right for a small cation in contact with larger anions, then the fit is good. If the cation is too small for a given site, that coordination number will be unstable and it will prefer a lower coordination structure.

Note that this interpretation allows for the radius of cation to be slightly larger spacing the anions out. This is depicted below.

stable and unstable cation and anion packing with stability limit or critical stability

Say the left and right limits of the ratio for stability limit ratio or critical ratio corresponding to maximum packing efficiency are $r_\text{min}$ and $r_\text{max}$, respectively. The (more) accurate interpretation of the radius ratio rule is:

$$ \dfrac{r_\text{cation}}{r_\text{anion}} \in [r_\text{min}, r_\text{max}+\epsilon) $$

where $\epsilon$ is a small number. Now consider the octahedral (FCC) range.

$$ \dfrac{r_\text{cation}}{r_\text{anion}} \in \pu{[0.414, 0.732+\epsilon)} $$

Since the radius ratio of $\ce{KCl}\ \pu{(0.735)}$ only differs from the upper limit of the octahedral coordination range by $\pu{0.003}$; therefore, it can show both FCC and BCC structures.


The radius ratio rule is really just a geometric rule, and, that too, a rule of thumb in solid-state chemistry. It helps make a priori predictions regarding the structures, that seem (mostly) work while making a bunch of assumptions and not considering a lot of factors. At the end of the day, the structure is what it is.

Some Assumptions of Radius Ratio Rules

  1. Ions are hard spheres.
  2. Only Electrostatic attraction occurs.
  3. Ions interact only to minimize electrostatic potential.
  4. No Polarization or Covalent bond formation occurs.

The list goes on...

If the assumptions of the radius ratio rule were accurate, square planer geometry shouldn't exist at all, because every square planer geometry can also exist as an octahedral one while minimizing electrostatic potential.


Do note that FCC to BCC phase transitions$^1$ of $\ce{KCl}$ have also been observed under (not too) high pressures ($\pu{GPa}$).

References

  1. Mashimo, Tsutomu & Nakamura, K & Tsumoto, K & Zhang, Y & Ando, Shinji & Tonda, H. (2002). Phase transition of KCl under shock compression. Journal of Physics: Condensed Matter. 14. 10783. 10.1088/0953-8984/14/44/377.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.