A compound of formula $$A_2B_3$$ has the HCP lattice. Which atom forms the HCP lattice and what fraction of the tetrahedral voids are occupied by the other atoms?

We have a compound whose simplest formula is $$A_2B_3$$. One kind of atom is arranged in a hexagonal close-packed lattice (hcp) and the other kind occupies only tetrahedral voids. We must identify which atom sits on the hcp points and what fraction of the tetrahedral voids the second atom occupies.

First, recall an essential fact about close packing:

In any hcp (or ccp) lattice containing $$N$$ atoms, the number of tetrahedral voids is $$2N$$.

Let us assume that atom $$B$$ constitutes the hcp lattice. Thus, the number of $$B$$ atoms present is

$$N_B = N.$$

The total number of tetrahedral voids created by these $$N$$ atoms equals

$$2N.$$

Suppose a fraction $$f$$ of these tetrahedral voids is occupied by atom $$A$$. Then the number of $$A$$ atoms is

$$N_A = f \times 2N = 2fN.$$

The compound’s overall stoichiometry gives the fixed ratio

$$\frac{N_A}{N_B} = \frac{2}{3}.$$

Substituting $$N_A = 2fN$$ and $$N_B = N$$ into this ratio, we obtain

$$\frac{2fN}{N} = \frac{2}{3} \;\;\Longrightarrow\;\; 2f = \frac{2}{3} \;\;\Longrightarrow\;\; f = \frac{1}{3}.$$

So exactly one-third of all the tetrahedral voids are filled by the $$A$$ atoms. The remaining two-thirds of the tetrahedral sites stay empty.

Now let us verify quickly that placing $$A$$ on the hcp lattice instead would fail. If $$A$$ were hcp, we would have $$N_A = N$$ and tetrahedral voids $$= 2N$$. To achieve the formula $$A_2B_3$$ we would need

$$N_B = \frac{3}{2}N = 1.5N,$$

which would mean filling $$\frac{1.5N}{2N} = 0.75$$ or $$75\%$$ of all tetrahedral voids with $$B$$. This scenario does not match any of the given options (they mention $$\frac{2}{3} = 0.667$$, not $$0.75$$). Hence our earlier assumption that $$B$$ is the hcp atom is the only one that fits an available choice.

Therefore, $$B$$ atoms form the hcp framework, and exactly $$\dfrac{1}{3}$$ of the tetrahedral voids are occupied by $$A$$ atoms.

Hence, the correct answer is Option C.