If the radius of octahedral voids and radius of the atoms in close-packing are r and R, respectively. Derive the relation between r and R. What is the ratio of 100× (r/R). (Give answer to the nearest integer.)

Question

If the radius of octahedral voids and radius of the atoms in close-packing are r and R, respectively. Derive the relation between r and R. What is the ratio of 100× (r/R). (Give answer to the nearest integer.) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Derivation of relation between r and R
A sphere fitted into the octahedral void is shown by shaded circle. The spheres present in other layers are not shown in the figure
Solution

∴ ΔABC is a right angled triangle.
∴ We apply pythagoras theorem.
AC² = AB² + BC²
(2R)² = (R + r) ² + (R + r)² = 2 (R + r)²
4R² = 2 (R + r)²
2 R² = (R + r)²
2 R² = (R + r)²

(2 R)^{2} = (R + r)^{2}

2 R = R + r

r = 2 R - R

r = (2 - 1) R

r = (1.414 – 1) R
r = 0.414 R

\frac{100 r}{R} = 41

Q. If the radius of octahedral voids and radius of the atoms in close-packing are r and R, respectively. Derive the relation between r and R. What is the ratio of 100×Rr​. (Give answer to the nearest integer.)

Q. If the radius of octahedral voids and radius of the atoms in close-packing are r and R, respectively. Derive the relation between r and R. What is the ratio of $100 \times \frac{r}{R} .$ (Give answer to the nearest integer.)