A metal crystallizes in a face-centered cubic (FCC) lattice with an edge length of 500 pm . If the density of the metal is 4.0 g/cm ^3 , the total number of atoms present in a 100 g sample of this metal is x 10²³ . The value of x is ___ .

Step-by-step solution

For a face-centered cubic (FCC) lattice, the number of atoms per unit cell is Z = 4 . Given edge length, a = 500 pm = 5 10⁻⁸ cm Volume of one unit cell, V = a^3 = (5 10⁻⁸)^3 = 125 10⁻²⁴ cm ^3 Mass of one unit cell can be calculated using the density: Mass = Density Volume Mass = 4.0 125 10⁻²⁴ = 500 10⁻²⁴ g = 5 10⁻²² g The number of unit cells in a 100 g sample is: Number of unit cells = 100 5 10⁻²² = 20 10²¹ = 2 10²³ Since each FCC unit cell contains 4 atoms, the total number of atoms in the sample is: Total atoms