A metal crystallizes in a body-centered unit cell with an edge length of 2.00 $\times$ 10² pm.Assume the atoms in the cell touch along the cube diagonal.The percentage of empty volume in the unit cell will be: A)0% B)26.0% C)32.0% D)68.0% E)none of these

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Quizplus · Accepted Answer

The body-centered unit cell has one atom in the center and 8 atoms at each corner of the cube. The radius of the atom can be calculated using the length of the diagonal of the cube (2.00 x 10^(10) pm) and the Pythagorean theorem:

d = sqrt(3)a
where d is the diagonal length of the cube
a is the edge length of the cube

So, the radius of each atom is:

r = d/2 * sqrt(3)
  = 2.00 x 10^(10) pm/2 * sqrt(3)
  = 1.16 x 10^(10) pm

The volume of the atom can be calculated using the formula:

V = 4/3 * πr^3
  = 4/3 * π * (1.16 x 10^(10) pm)^3
  = 6.06 x 10^(-23) cm^3

The volume of the unit cell can be calculated using the formula:

V_cell = a^3
       = (2.00 x 10^(10) pm)^3
       = 8.00 x 10^(-23) cm^3

The total volume occupied by the atoms in the unit cell can be calculated by multiplying the volume of one atom by the total number of atoms in the cell:

V_atoms = 9 * 6.06 x 10^(-23) cm^3
        = 5.46 x 10^(-22) cm^3

Therefore, the percentage of empty volume in the unit cell is:

% empty volume = 100% - (V_atoms/V_cell) x 100%
               = 32.0%

A Metal Crystallizes in a Body-Centered Unit Cell with an Edge