Rubidium metal, Rb, crystallizes in a body-centered cubic arrangement. What is the radius of one atom of Rb, if its density is 1.5320 g/cm3?

To find the radius of one atom of Rb, we need to use the formula for the density of a substance and the formula for the volume of a body-centered cubic arrangement.

The density formula is:

density = mass / volume

The formula for the volume of a body-centered cubic arrangement is:

volume = (4/3) * π * r^3

Where:
- density is the given density of Rb (1.5320 g/cm^3)
- mass is the mass of one atom of Rb
- volume is the volume occupied by one atom of Rb
- r is the radius of one atom of Rb

To find the radius, we need to find the mass of one atom of Rb. The molar mass of Rb is approximately 85.4678 g/mol.

To find the mass of one atom of Rb, we divide the molar mass by Avogadro's number (6.022 x 10^23):

mass = molar mass / Avogadro's number

mass = 85.4678 g/mol / (6.022 x 10^23 mol^-1)

Now, we can rearrange the density formula to solve for volume:

volume = mass / density

Substituting the values:

volume = (85.4678 g/mol / (6.022 x 10^23 mol^-1)) / 1.5320 g/cm^3

Finally, we can rearrange the formula for the volume of a body-centered cubic arrangement to solve for the radius (r):

r = [(3 * volume) / (4 * π)]^(1/3)

Substituting the calculated value for volume:

r = [(3 * (85.4678 g/mol / (6.022 x 10^23 mol^-1)) / 1.5320 g/cm^3) / (4 * π)]^(1/3)

By evaluating this equation, we can find the result for the radius of one atom of Rb.

To find the radius of one atom of Rb, we can use the formula that relates the density, atomic mass, and atomic radius of an element.

First, we need to find the atomic mass of Rb. The atomic mass of Rubidium is approximately 85.468 amu.

Next, we need to calculate the volume of one atom of Rb. In a body-centered cubic (BCC) arrangement, there are two atoms per unit cell, where each atom contributes 1/8 to its own volume and 1/2 to the volume of the unit cell. The volume of a BCC unit cell can be calculated using the formula:

V_unit cell = (4/3) * π * r^3

where r is the radius of one atom.

The volume of one atom can be calculated using the formula:

V_atom = (1/2) * V_unit cell

Now we can find the radius (r) by rearranging the equation:

r = (3 * V_atom) / (4 * π)^(1/3)

Finally, substitute the known values into the equation:

r = (3 * (1/2) * V_unit cell) / (4 * π)^(1/3)

Now calculate the volume of the unit cell by rearranging the equation for volume:

V_unit cell = (4 * M) / (ρ * N_A)

where M is the atomic mass, ρ is the density, and N_A is Avogadro's number (6.0221 x 10^23 mol^-1).

Substituting the known values into the equation:

V_unit cell = (4 * 85.468) / (1.5320 * 6.0221 x 10^23)

Now substitute the calculated value of V_unit cell into the equation for the radius to find the radius of one atom of Rb.

2 atoms/unit cell

mass unit cell = 85.47*2/6.022e23 =?
volume = mass/density = ?
a^3 = volume
Solve fora.
Then 4r = a(3)^1/2
Solve for r.