Calculate the radius of a tantalum (ta) atom, given that ta has a bcc crystal structure, a density of 16.6 g/cm3, and an atomic weight of 180.9 g/mol. report your result with units of nm to 3 significant figures]

Explanation:

The strategy here is to use the relationship between the volume of the unit cell and the density of the material to calculate the radius of a tantulum atom (Ta).

The edge length of the body-centered cubic unit cell is given by

a = 4r/√3 where a is the edgge length of the cube

                             r is the radius of the metal atom

To calculate a we will be using the density of the metal and then relate it to the density of the unit cell which will is the same.

The density is of the unit cell be the mass per volume. In a BCC unit cell we have to atoms per unit cell, so

m = 2 atoms/unit-cell  x 180.9 g/mol x 1  mol /6.022 x 10²³ atoms/mol = 6.01  x 10⁻²² g/unit-cell

m = 6.01 x 10⁻²² g/unit-cell

To calculate the edge length of the cube:

d unit cell = m unit-cell/Vunit-cell = ⇒ Vunit-cell =  m/d

V = 6.01 x 10⁻²² g/unit-cell/ 16.6 g/cm³ = 3.62 x 10⁻²³ cm³

V = a³  ⇒   a = ∛V = ∛3.62 x 10⁻²³ cm  = 3.31 x 10⁻⁸ cm

a= 3.31 x 10⁻⁸ cm

Now we are in position to calculate the radius of Ta:

4r/√3 = a ⇒ r = a x √3/4

r =  3.31 x 10⁻⁸ cm  x √3 / 4 = 1.43 x 10⁻⁸ cm

Now since the answer should be in nm, we need to convert this last value:

1.43 x 10⁻⁸ cm x 1 m/100 cm x 1 x 10⁹ nm/m = 0.143 nm

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