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प्रश्न
If the radius of the octahedral void is r and the radius of the atoms in close packing is R, derive the relation between r and R.
The radius of an octahedral void is r and the radius of the atom is R when these are in close-packing. Derive a relationship between the two.
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उत्तर
A sphere with center O is fitted into the octahedral void as shown in the above figure. It can be observed from the figure that ΔPOQ is right-angled
∠POQ = 90°
Now, applying the Pythagorean theorem, we can write:
PQ2 = PO2 + OQ2
⇒ (2R)2 = (R + r)2 + (R + r)2
⇒ (2R)2 = 2(R + r)2
⇒ 2R2 = (R + r)2
⇒ `sqrt 2R = R + r`
⇒ `r = sqrt 2R - R`
⇒ `r = (sqrt 2 - 1)R`
⇒ `r = (1.414 - 1)R`
⇒ r = 0.414 R
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