Gallium crystallizes in a primitive cubic unit cell. The length of an edge of this cube is 362 pm. What is the radius of a gallium atom?

The face-centered gold crystal has an edge length of 407 pm. Based on the unit cell, calculate the density of gold.

Type of atom Fraction in unit cell
corner 1/8
face 1/2
body 1

To find the radius of a gallium atom, we need to first determine the length of the diagonal of the primitive cubic unit cell.

In a primitive cubic unit cell, the length of the diagonal (d) is equal to the edge length multiplied by the square root of 3:

d = a * √3

Given that the edge length (a) is 362 pm, we can calculate the diagonal (d):

d = 362 pm * √3

Now that we have the diagonal length, we can find the radius of a gallium atom. The radius (r) is equal to half of the diagonal length (d/2):

r = d/2

By substituting the value of d we calculated earlier:

r = (362 pm * √3) / 2

Simplifying further, we get the radius of a gallium atom.

To calculate the density of gold, we need to know the mass and volume of the unit cell.

The volume of a face-centered cubic unit cell can be calculated using the formula:

V = a^3 * (1 + 2^0.5)

Where a is the edge length of the unit cell, which in this case is equal to 407 pm.

Now that we have the volume of the unit cell, we need to determine the mass of one unit cell. This can be done by multiplying the atomic mass of gold by the fraction of gold atoms in the unit cell.

Given that gold has an atomic mass of 196.97 g/mol, and the fraction of gold atoms in the unit cell is 1 (since all atoms in the face-centered cubic unit cell are gold), we can calculate the mass of one unit cell.

Next, we can calculate the density of gold by dividing the mass of one unit cell by the volume of the unit cell.

Density = Mass / Volume

By substituting the values we calculated earlier, we can find the density of gold.

To determine the radius of a gallium atom, we need to find the relationship between the edge length of the unit cell and the atomic radius.

In a primitive cubic unit cell, the atoms are located at the corners of the cube. The edge length of the cube is equal to four times the atomic radius (since the atom is touching the corners of the cube). Therefore, we can use this relationship to calculate the atomic radius.

Given:
Edge length of the cube (a) = 362 pm

Let's calculate the atomic radius:

Atomic radius (r) = a/4
= 362 pm / 4
= 90.5 pm

Therefore, the radius of a gallium atom is 90.5 pm.

Now, let's move on to calculating the density of gold.

Given:
Edge length of the face-centered gold crystal (a) = 407 pm

The face-centered gold crystal has one atom on each of the corners and one atom in the center of each face. Therefore, there are a total of 4 corners atoms (each contributing 1/8) and 6 face atoms (each contributing 1/2) in each unit cell.

Fraction of corner atoms: 1/8
Fraction of face atoms: 1/2
Fraction of body atoms: 1

To calculate the density, we need to know the molar mass of gold (Au), which is approximately 196.97 g/mol.

Let's perform the calculations:

1. Calculate the number of atoms in each unit cell:
Number of corner atoms = 4 * (1/8) = 1/2
Number of face atoms = 6 * (1/2) = 3
Number of body atoms = 1

2. Calculate the total mass of atoms in each unit cell:
Total mass = (Number of corner atoms * atomic mass of gold) + (Number of face atoms * atomic mass of gold) + (Number of body atoms * atomic mass of gold)
= (1/2 * 196.97 g/mol) + (3 * 196.97 g/mol) + (1 * 196.97 g/mol)
= 588.775 g/mol + 590.91 g/mol + 196.97 g/mol
= 1376.655 g/mol

3. Calculate the volume of the unit cell:
Volume = a^3
= (407 pm)^3
= 67138023 pm^3

4. Convert the volume from pm^3 to cm^3:
Volume = 67138023 pm^3 * (1 cm/10^10 pm)^3
= 6.7138023 * 10^-2 cm^3

5. Calculate the density:
Density = Total mass / Volume
= 1376.655 g/mol / 6.7138023 * 10^-2 cm^3
≈ 20504 g/cm^3

Therefore, the density of gold in the face-centered gold crystal is approximately 20504 g/cm^3.

According to the information I have, gallium crystallizes in the orthorhombic system so that a is not = to b is not = to c. If you mean the primitive unit cell to be a SIMPLE cubic cell, then a = 2*radius but I don't believe that is correct.

For Au, it crystallizes in the face centered cubic system, FCC.
density = mass of unit cell/volume of unit cell.
mass of unit cell:
For a fcc crystal, there are 4 atoms per unit cell.
4 Ga atoms x atomic mass Ga atom/6.022 x 10^23 atoms = mass of 1 Ga atom.
volume of unit cell:
a^3 where a is the length of the edge which is listed in the problem. I suggest you change pm to cm before making the calculation so that volume comes out in cubic centimeters (cc). That way the density will be in g/cc.
Post your work if you get stuck.