a metal crystallizes in the face centred cubic cubic crustal structure with a unit cell edge of 5.48 x10 power of -8 cm.the density of the metal is 6.90g per cubic cm.

a. what is the mass in grams, of a single atom of this element.
b. what is the atomic weight of the element (g per mol).
c. what is the radius, in cm, of an atom of an element.

thank you..

To find the answers to these questions, we need to follow a series of steps. Let's go through each question one by one:

a. What is the mass in grams of a single atom of this element?
First, we need to find the volume of the unit cell. Since the structure is face-centered cubic (FCC), there are a total of 4 atoms per unit cell. The volume of an FCC unit cell can be calculated using the formula: V = (a^3)/4, where "a" is the edge length of the unit cell.

Given: a = 5.48 x 10^(-8) cm (unit cell edge length)
V = (5.48 x 10^(-8))^3/4

Next, we need to find the mass of the unit cell. Density is defined as mass divided by volume. The volume of the unit cell can be converted to cubic centimeters by multiplying by 1 cm^3 / (10^(-8) cm)^3 (conversion factor from cm to cm^3).

Density = 6.90 g/cm^3
V_cm^3 = V * (1 cm^3 / (10^(-8) cm)^3)

Now, we can calculate the mass of the unit cell:
Mass = Density * V_cm^3

Since there are 4 atoms in the unit cell, we can finally calculate the mass of a single atom by dividing the total mass by 4.

b. What is the atomic weight of the element (g per mol)?
The atomic weight is the weighted average mass of all the atoms in a sample of the element. A mole of a substance contains Avogadro's number of particles (6.022 x 10^23). To find the atomic weight, we need to convert the mass of a single atom to grams per mole by multiplying by Avogadro's number.

c. What is the radius, in cm, of an atom of the element?
The radius of an atom can be determined using the formula for the face-centered cubic system:
Radius = 0.5 * sqrt(2) * a

Let's plug in the values and calculate each of the answers step by step.

Sure, I can help you with that!

a. To find the mass of a single atom of the element, we need to calculate the mass of the unit cell. The unit cell volume in cubic centimeters is given by:

Volume of unit cell = (Edge length)^3
= (5.48 x 10^-8 cm)^3
= 1.99 x 10^-23 cm^3

The density of the metal is given as 6.90 g/cm^3, so the mass of the unit cell would be:

Mass of unit cell = Volume of unit cell x Density
= (1.99 x 10^-23 cm^3) x (6.90 g/cm^3)
= 1.37 x 10^-22 g

Since a unit cell contains 4 atoms (in a face-centered cubic structure), the mass of a single atom would be:

Mass of single atom = Mass of unit cell / 4
= (1.37 x 10^-22 g) / 4
= 3.43 x 10^-23 g

Therefore, the mass of a single atom of this element is 3.43 x 10^-23 grams.

b. To find the atomic weight of the element, we need to convert the grams per atom to grams per mole. The molar mass (atomic weight) of an element is the mass of one mole of atoms. One mole of any substance contains Avogadro's number of entities, which is approximately 6.022 x 10^23.

Atomic weight = Mass of single atom x Avogadro's number
= (3.43 x 10^-23 g) x (6.022 x 10^23 atoms/mol)
= 20.66 g/mol

Therefore, the atomic weight of the element is 20.66 grams per mole.

c. To find the radius of an atom of the element, we can use the face-centered cubic arrangement. In a face-centered cubic structure, the distance from the center of a cube to the corner is equal to the radius of the atom in that structure.

The diagonal of the face diagonal of a face-centered cubic unit cell can be calculated using the Pythagorean theorem:

Face diagonal = sqrt(2) x Edge length
= sqrt(2) x (5.48 x 10^-8 cm)
= 7.76 x 10^-8 cm

Since the face diagonal of the cube is equal to twice the radius (2r), we can find the radius using:

2r = Face diagonal
r = Face diagonal / 2
= (7.76 x 10^-8 cm) / 2
= 3.88 x 10^-8 cm

Therefore, the radius of an atom of the element is 3.88 x 10^-8 cm.

I hope this helps! Let me know if you have any further questions.