Calculate the radius of a vanadium atom, given that V has a BCC crystal structure, a density of 5.96 g/cm3, and an atomic weight of 50.9 g/mol.
I used this equation:
density = (2)*(atomic weight) / (volume of unit cell)*(Avagodro's #)
rearranged to find the volume of the unit cell, V_c.
Then I used the relation:
V_c = a^3 = (4R/sqrt(3))^3
V_c = 64R^3/3sqrt(3)
to find the radius R.
I got R = 1.3x10^-8
They want answers in nm so I should get 0.13nm?
To calculate the radius of a vanadium atom in a body-centered cubic (BCC) crystal structure, we can use the formula you mentioned:
density = (2 * atomic weight) / (volume of unit cell * Avogadro's number)
First, rearrange the equation to solve for the volume of the unit cell (V_c):
V_c = (2 * atomic weight) / (density * Avogadro's number)
Substitute the given values:
V_c = (2 * 50.9 g/mol) / (5.96 g/cm^3 * 6.022 x 10^23 atoms/mol)
Calculate V_c:
V_c = 0.225 x 10^-23 cm^3/atom
Next, use the relation between the volume of a BCC unit cell (V_c) and the radius of the atom (R):
V_c = (4R / √3)^3
Rearrange the equation:
R^3 = (3√3 * V_c) / 4
R = (3√3 * V_c)^(1/3) / 4^(1/3)
Substitute the value of V_c:
R = (3√3 * 0.225 x 10^-23 cm^3)^(1/3) / 4^(1/3)
Calculate R:
R ≈ 1.3114 x 10^-8 cm
Since the answer is in centimeters, we need to convert it to nanometers. There are 10^7 nanometers in one centimeter:
R ≈ 1.3114 x 10^-8 cm * 10^7 nm/cm ≈ 0.13114 nm
So, the radius of the vanadium atom in a BCC crystal structure is approximately 0.13114 nm, which can be rounded to 0.13 nm.
To find the radius of the vanadium (V) atom using the given information, let's go step-by-step.
1. Begin with the provided equation:
density = (2 * atomic weight) / (volume of unit cell * Avogadro's number)
2. Rearrange the equation to solve for the volume of the unit cell:
(volume of unit cell) = (2 * atomic weight) / (density * Avogadro's number)
3. Substitute the given values into the equation:
(volume of unit cell) = (2 * 50.9 g/mol) / (5.96 g/cm^3 * (6.022 x 10^23/mol))
4. Calculate the volume of the unit cell:
(volume of unit cell) ≈ 4.424 x 10^-23 cm^3
5. Use the relation V_c = a^3 = (4R/√3)^3, where V_c represents the volume of the unit cell, and a represents the length of the cubic unit cell.
6. Rearrange the relation equation to solve for the radius R:
R = (√3/4) * (V_c)^(1/3)
7. Substitute the calculated volume of the unit cell into the equation:
R = (√3/4) * (4.424 x 10^-23 cm^3)^(1/3)
8. Calculate the radius:
R ≈ 1.29 x 10^-8 cm
9. Convert the radius to nm:
R ≈ 1.29 x 10^-1 nm
Therefore, the radius of the vanadium (V) atom is approximately 0.129 nm.