Calculation of Number of Atoms in Unit cell

Problem 1: Many metals pack in cubic unit cells. The density of a metal and length of the unit cell can be used to determine the type for packing. For example, sodium has a density of 0.968 g/cm³ and a unit cell side length (a) of 4.29 Å

a. How many sodium atoms are in 1 cm³?
b. How many unit cells are in 1 cm³?
c. How many sodium atoms are there per unit cell?

Solution for Calculation of Number of Atoms in Unit cell:

1) Calculate the average mass of one atom of Na:

22.99 g mol¯¹ ÷ 6.022 x 10²³ atoms mol¯¹ = 3.82 x 10¯²³ g/atom

2) Determine atoms in 1 cm³:

0.968 g / 3.82 x 10¯²³ g/atom = 2.54 x 10²² atoms in 1 cm³

3) Determine volume of the unit cell:

(4.29 x 10¯⁸ cm)³ = 7.89 x 10¯²³ cm³

4) Determine number of unit cells in 1 cm³:

1 cm³ / 7.89 x 10¯²³ cm³ = 1.27 x 10²² unit cells

5) Determine atoms per unit cell:

2.54 x 10²² atoms / 1.27 x 10²² unit cells = 2 atoms per unit cell

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