Question

Niobium crystallises in body-centred cubic structure. If density is $ 8.55gc{m^{ - 3}} $ . Calculating the atomic radius of niobium using its atomic mass is $ 93u $ .

Answer 1

Hint :The mass of the unit cell is the number of atoms in the unit cell multiplied with the mass of each atom. They are represented as Z and m. where m is mass and Z is unit cell number $ (\dfrac{M}{{{N_A}}}) = \dfrac{{Zm}}{{{N_A}}} $ ,and Avogadro’s number is represented by $ {N_A} $ .

Complete Step By Step Answer:
Volume of the unit cell is represented $ {a^3} $ which is not given in the question. With help of $ {a^3} $ we can find radius.
No of atom in BCC unit cell of niobium $ Z = 2 $ atoms
Atomic mass $ = 93u $
 When we use the density formula which is:
Density of unit cell mass $ = \dfrac{{mass{\text{ of unit cell}}}}{{volume{\text{ of unit cell}}}} $
We, can get the value of $ {a^3} $
Density = $ \dfrac{{Zm}}{{{N_A}}} \times \dfrac{1}{{{a^3}}} $
Density, $ d = \dfrac{{Zm}}{{{a^3}{N_A}}}gc{m^{ - 3}} $
By substituting the above values, we get
 $ d = \dfrac{{Zm}}{{{a^3}{N_A}}}gc{m^{ - 3}} $
 $ 8.55{\text{ = }}\dfrac{{2 \times 93}}{{{a^3} \times 6.023 \times {{10}^{23}}}} $
 $ {{\text{a}}^3}{\text{ = }}\dfrac{{2 \times 93}}{{8.55 \times 6.023 \times {{10}^{23}}}} $
 $ {\text{a = }}{\left\langle {\dfrac{{2 \times 93}}{{8.55 \times 6.023 \times {{10}^{23}}}}} \right\rangle ^{\dfrac{1}{3}}} $
 $ a = 3.306 \times {10^{ - 8}}cm $
But we know that in BCC unit cell $ 4r = \sqrt 3 a $
 $ r = \dfrac{{\sqrt 3 }}{4}a $
 $ r = \dfrac{{\sqrt 3 }}{4} \times 3.306 \times {10^{ - 8}}cm $
 $ r = 14.32 \times {10^{ - 9}}cm $
 $ r = 14.32pm $
Hence, the radius value is $ 14.32pm $ .

Additional Information:
Once, when minerals were being examined in 1801 by Charles Hatchett, he was stunned by the specimen named columbite. He thought it could be having some unknown element, which was true. He found it by heating a sample of potassium carbonate, dissolved the product in water. Then after adding acid, he got precipitated. But he was unable to find the element further itself. Also, it was then named columbium by him and it was known for many years.

Note :
Columbium is another name for niobium and often found to occur with tantalum in nature. These are very difficult to separate. When a German chemist named Heinrich rose , in 1844 proved that columbite contains both elements and he renamed columbium niobium.