Find the radius of octahedral void

Question: The arrangement of X^- ions around A⁺ ion in solid AX is given in the figure (not drawn to scale). If the radius of X^- is 250pm, what is the radius of A⁺?

 

Solution:

According to given figure, A⁺ is present in the octahedral void of X-. An octahedral void is formed by six spheres which are arranged on the corner of two triangles. When these two triangles are placed with their apex faced opposite sides, octahedral void arises. 

The limiting radius in octahedral void is related to the radius of sphere as:

r_void = 0.414 r_sphere

                                        r _A+ = 0.414  r_X-

                                                   = 0.414 x 250 pm

                                                   = 103.5 pm

Thus, radius of A⁺ is 103.5 pm.