× (B atoms)/(Si atoms)? B atoms Si atoms × Be sure to answer all parts. Silicon

Question

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(B atoms)/(Si atoms)?

B atoms

Si atoms

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Be sure to answer all parts.

Silicon used in computer chips must have an impurity level below 109 (that is, fewer than one impurity atom for every 109 atoms). Silicon has a diamond crystal structure. Each cubic unit cell (edge length a = 543 pm) contains eight Si atoms. If there are 1.0

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1013 boron atoms per cubic centimeter in a sample of pure silicon, what is the ratio

(B atoms)/(Si atoms)?

B atoms

Si atoms

=  

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10
Enter your answer in scientific notation.

Explanation / Answer

We need to calculate the number of Si atoms in 1 cm3 . We can arrive at the answer by carrying out the following three steps:
(1) Determine the volume of an Si unit cell in cubic centimeters,
(2) Determine the number of Si unit cells in 1 cm3 , and
(3) Multiply the number of unit cells in 1 cm3 by 8, the number of Si atoms in a unit cell.

Step 1: The volume of the unit cell, V, is

V = a3

= (543 pm)3 x [ { (1 x 10-12 m) / 1 pm} x { 1 cm / (1 x 10-12 m)]3

= 1.60 x 10-22 cm3

Step 2: The number of cells per cubic centimeter is given by:

number of unit cells = 1 cm3 x [ ( 1 unit cell) / (1.60 x 10-22 cm3) ]
= 6.25 x 1021 unit cells

Step 3: Since there are 8 Si atoms per unit cell, the total number of Si atoms is:

number os Si atoms = [ (8 Si atom) / (1 unit cell) ] x (6.25 x 1021 unit cells)
   = 5.0 x 1022 Si atoms

Finally, to calculate the purity of the Si crystal, we write:

B atoms / Si atoms = (1.0 x 1013 B atoms) / (5.0 x 1022 Si atoms)
   = 2.0 x 10-10

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