sodium has a body-centered cubic structure with a unit cell that is 430 picometers wide. estimate the radius of a sodium atom. please help me, I don't know how to get the solution. THank you
for a cube u could use Pythagorean a square plus b square equal c square were a and b are equal and they both equal to 430 pm. and the answer u get is 4 radii and you want one radii so divide it by 4.
To estimate the radius of a sodium atom, we can make a few assumptions and use the given information about the unit cell width.
Step 1: Understand the body-centered cubic (BCC) structure
A BCC structure consists of atoms at each corner and one atom in the center of the cube. The atom at the center is shared by eight unit cells, so it contributes only 1/8th to a single unit cell.
Step 2: Calculate the effective volume of the unit cell
Since the width of the unit cell is given as 430 picometers, we can represent it as the edge length of a cube. The volume of a cube is given by V = l^3, where l is the edge length.
V = (430 pm)^3
Step 3: Calculate the volume of the atoms in the unit cell
In a BCC structure, there are two 1/8th atoms at the corners and one full atom at the center. So, the total volume contributed by the atoms in a unit cell is equal to the volume of two corner atoms plus the volume of the central atom:
Volume of atoms = (2 * (1/8) * V_atom) + (1 * V_atom)
= (1/4) * V_atom + V_atom
= (5/4) * V_atom
Here, V_atom represents the volume of a single sodium atom.
Step 4: Calculate the radius of a sodium atom
The volume of an atom can be given as V_atom = (4/3) * π * r^3, where r is the radius of the atom.
(5/4) * V_atom = (5/4) * (4/3) * π * r^3
= (5/3) * π * r^3
Now, solve for the radius (r):
(5/3) * π * r^3 = Volume of atoms
r^3 = (3/5π) * Volume of atoms
r = ([(3/5π) * Volume of atoms])^(1/3)
Substituting the calculated value of the Volume of atoms from Step 2 into the equation will give you an estimate of the radius of a sodium atom.