question_answer

Relationship between atomic radius r and the edge length a of a body centred cubic unit cell is

A) \[r=\frac{a}{2}\]

B)  \[r=\sqrt{\frac{a}{2}}\]

C)  \[r=\frac{\sqrt{3}}{4}a\]

D)  \[r=\frac{3a}{2}\]

Correct Answer: C

Solution :

 From the figure, In \[\Delta PQR,\] \[{{b}^{2}}={{a}^{2}}+{{a}^{2}}\] [From \[(A{{B}^{2}})={{(AC)}^{2}}+(B{{C}^{2}})\]] \[{{b}^{2}}=2{{a}^{2}}\] \[b=\sqrt{2}a\] In \[\Delta PRS,\] \[{{(4r)}^{2}}={{(SR)}^{2}}+{{(PR)}^{2}}\] \[={{(a)}^{2}}+{{(\sqrt{2}a)}^{2}}\] \[{{(4r)}^{2}}=3{{a}^{2}}\] \[4r=\sqrt{3}a\] \[r=\frac{\sqrt{3}}{4}a\]