All the noble gases except helium crystallize with fcc structures. Derive an equation relating the atomic radius to the density of an fcc solid of given molecular weight and apply it to deduce the atomic radii of the noble gases.

mass atom = #atoms/unit cell x atomic mass atom/6.022E23

Use density to convert mass to volume.
a(cell edge) = V1/3

4r = a*sqrt(2)

thank you!!i understand now

To derive an equation relating the atomic radius to the density of an fcc solid of a given molecular weight, we need to analyze the structure of an fcc solid and use some basic mathematical relationships. Here's how we can do it step by step:

Step 1: Understand the FCC structure of a solid
In the face-centered cubic (FCC) structure, each lattice point is shared by four neighboring atoms, forming a tetrahedral arrangement. This results in a coordination number of 12, meaning each atom is surrounded by 12 neighboring atoms.

Step 2: Determine the number of atoms per unit cell
In an FCC structure, there are 4 atoms per unit cell. This can be calculated by considering that each corner atom contributes 1/8 of its volume to the unit cell, and each face-centered atom contributes the remainder, which is 1/2. Therefore, total contribution per unit cell = (8 x 1/8) + (6 x 1/2) = 4 atoms.

Step 3: Relate atomic radius to unit cell length
In an FCC structure, the atoms touch along the face diagonal of the unit cell. Let's denote the length of this face diagonal as "d."

Using the Pythagorean theorem, we can relate it to the atomic radius "r" by:
d^2 = 4r^2 (since the face diagonal passes through two radii)

Step 4: Calculate the density of the solid
The density (ρ) of the solid can be calculated using the equation:
ρ = (molecular weight)/(volume of the unit cell)

Step 5: Express volume of the unit cell in terms of the atomic radius
The volume of the unit cell (Vc) can be expressed in terms of the atomic radius (r) as:
Vc = d^3 = (4r^2)^(3/2) = 4√2r^3

Step 6: Derive the equation relating atomic radius to density
By substituting the value of Vc in the density equation, we get:
ρ = (molecular weight)/[4√2r^3]

Solving this equation for r, the atomic radius, we get:
r = (molecular weight)/(4ρ√2)^(1/3)

Step 7: Apply the equation to deduce the atomic radii of the noble gases
To deduce the atomic radii of noble gases, we need the molecular weights (M) and densities (ρ) of the respective noble gases. Here are the values for the noble gases at standard conditions:

- Helium (He): M = 4.003 g/mol, ρ = 0.1785 g/L
- Neon (Ne): M = 20.180 g/mol, ρ = 0.9002 g/L
- Argon (Ar): M = 39.948 g/mol, ρ = 1.784 g/L
- Krypton (Kr): M = 83.798 g/mol, ρ = 3.733 g/L
- Xenon (Xe): M = 131.293 g/mol, ρ = 5.894 g/L
- Radon (Rn): M = 222.018 g/mol, ρ = 9.73 g/L

By plugging these values into the equation derived in step 6, we can calculate the atomic radii of the noble gases.