Beryllium oxide (BeO) may form a crystal structure that consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions. If the ionic radius of \(\mathrm{Be}^{2+}\) is \(0.035 \mathrm{nm}\), then (a) Which type of interstitial site will the \(\mathrm{Be}^{2+}\) ions occupy? (b) What fraction of these available interstitial sites will be occupied by \(\mathrm{Be}^{2+}\) ions?

Interstitial sites exist in crystal structures where atoms or ions can occupy the spaces between atoms in the structure. For an HCP structure, two types of interstitial sites are common: tetrahedral (T) sites and trigonal-bipyramidal (C) sites. In an HCP structure: - A tetrahedral site is formed by a triangular arrangement of three \(\mathrm{O}^{2-}\) ions beneath and one \(\mathrm{O}^{2-}\) ion above for a total of 4 oxygen ions. - A trigonal-bipyramidal site is formed by a triangle of three \(\mathrm{O}^{2-}\) ions in one plane and two other \(\mathrm{O}^{2-}\) ions above and below the central ion in the triangle.

For HCP structures, the appropriate interstitial site for an ion is determined by the size ratio of the ions, defined as the ratio of the radius of the cation (smaller ion) to the anion (larger ion). The given radius of Beryllium ions (\(\mathrm{Be}^{2+}\)) is 0.035 nm. To calculate the size ratio, we first need the radius of the given anion, the oxygen ion (\(\mathrm{O}^{2-}\)). Since the values are not given, we can use standard values from an ionic radius table: The radius of \(\mathrm{O}^{2-}\) is approximately 1.40 Å, which is equal to 0.140 nm. Now calculate the size ratio: Size ratio = \(\frac{\text{radius of cation}}{\text{radius of anion}}\) Size ratio = \(\frac{0.035 \text{ nm}}{0.140 \text{ nm}} =\) 0.25

To determine which interstitial site the \(\mathrm{Be}^{2+}\) ions will occupy, we need to compare the calculated size ratio with the ideal size ratios for the different interstitial sites in an HCP structure. In general: - Tetrahedral sites are preferred when the size ratio is between 0.225 and 0.414. - Trigonal-bipyramidal sites are preferred when the size ratio is between 0.155 and 0.225. From our calculations, the size ratio is 0.25. Since this value falls within the ideal size ratio range for tetrahedral sites, the \(\mathrm{Be}^{2+}\) ions will occupy tetrahedral interstitial sites in the BeO crystal. Answer (a): The \(\mathrm{Be}^{2+}\) ions will occupy tetrahedral interstitial sites.

Now we need to find the fraction of tetrahedral interstitial sites occupied by \(\mathrm{Be}^{2+}\) ions. Since the chemical formula is BeO, there is an equal number of Beryllium and Oxygen ions in the crystal. In an HCP structure, there are two tetrahedral interstitial sites for each oxygen ion. Since the number of \(\mathrm{Be}^{2+}\) ions is equal to the number of \(\mathrm{O}^{2-}\) ions, the fraction of tetrahedral interstitial sites occupied by \(\mathrm{Be}^{2+}\) ions will be: Fraction of occupied sites = \(\frac{\text{Number of} \,\, \mathrm{Be}^{2+}\text{ ions}}{\text{Number of tetrahedral sites}} = \frac{\text{Number of} \,\, \mathrm{O}^{2-} \text{ ions}}{2 \cdot \text{Number of} \,\, \mathrm{O}^{2-} \text{ ions}}=\frac{1}{2}\). Answer (b): The fraction of tetrahedral interstitial sites occupied by \(\mathrm{Be}^{2+}\) ions is \(\frac{1}{2}\).