The metal potassium crystallizes in a body-centered cubic lattice.If the density of potassium is 0.856 g/cm³,what is the unit cell volume?

Question

Quizplus · Accepted Answer

The unit cell of a body-centered cubic lattice has one atom at each of the eight corners and one atom at the center of the cube. The atoms at the corners are shared among eight adjacent unit cells, while the atom in the center is entirely within the unit cell. Thus, the total number of atoms in the unit cell is 1 + (1/8)*8 = 2.

The density of potassium is given as 0.856 g/cm3. We know that:

density = mass / volume

Rearranging this equation gives:

volume = mass / density

The mass of a unit cell can be calculated as follows:

mass = number of atoms x atomic mass

The atomic mass of potassium (K) is 39.1 g/mol. Therefore, the mass of a single K atom is 39.1 g/mol / 6.022 x 1023 atoms/mol = 6.49 x 10-23 g/atom.

The number of atoms in a unit cell is 2, as we calculated earlier. Therefore, the mass of a unit cell is:

mass = 2 x 6.49 x 10-23 g/atom = 1.298 x 10-22 g

Now we can use the equation for volume:

volume = mass / density = (1.298 x 10-22 g) / (0.856 g/cm3) = 1.52 x 10-22 cm3

This is the volume of a single unit cell. We can express it in picometers by converting cm to pm:

1 cm = 1 x 1010 pm

Therefore,

volume = 1.52 x 10-22 cm3 = 1.52 x 10-22 cm3 x (1 x 1010 pm/cm)3 = 1.52 x 108 pm3

Thus, the best choice is C, which represents the unit cell volume as 1.52 x 10^8 pm3.

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