The structures of beryllium chloride in solid state and vapour phase, are :

The correct answer is Chain and dimer, respectively.

To understand the structures of beryllium chloride (${\mathrm{BeCl}}_2$) in the solid state and the vapour phase, we can analyze the hybridization and bonding behavior of beryllium ($\mathrm{Be}$).

1. Solid State Structure:
In the solid state, beryllium chloride exists as a polymeric chain structure. Beryllium is electron-deficient (having only 4 valence electrons in ${\mathrm{BeCl}}_2$ monomer). To complete its octet, each beryllium atom accepts electron pairs from the chlorine atoms of adjacent molecules, forming coordinate covalent bonds (bridge bonds). This results in a polymeric chain where each beryllium atom is tetrahedrally coordinated and undergoes ${\mathrm{sp}}^3$ hybridization.

2. Vapour Phase Structure:
In the vapour phase (at temperatures below 1200 K), beryllium chloride exists predominantly as a dimer, ${\mathrm{Be}}_2{\mathrm{Cl}}_4$. In this dimeric form, chlorine atoms act as bridges between the two beryllium atoms, and the beryllium atoms exhibit ${\mathrm{sp}}^2$ hybridization. At very high temperatures (above 1200 K), the dimer dissociates into linear monomeric ${\mathrm{BeCl}}_2$ molecules with $\mathrm{sp}$ hybridization. Therefore, in the vapour phase, the primary structure is a dimer.

Thus, the structures of beryllium chloride in the solid state and vapour phase are chain and dimer, respectively.