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 find the radius of a vanadium atom using a BCC crystal structure, you followed the correct steps. Let's go through the calculations to determine the radius.
1. First, calculate the volume of the unit cell (Vc) using the given equation:
density = (2 * atomic weight) / (Vc * Avogadro's number)
Rearrange the equation to solve for Vc:
Vc = (2 * atomic weight) / (density * Avogadro's number)
Vc = (2 * 50.9 g/mol) / (5.96 g/cm^3 * 6.022 × 10^23 atoms/mol)
Vc ≈ 2.679 × 10^-23 cm^3
2. Next, relate the volume of the unit cell (Vc) to the radius (R) of the atom:
Vc = a^3 = (4R / √3)^3
2.679 × 10^-23 cm^3 = (4R / √3)^3
3. Solve for the radius (R):
R = (√3 * Vc / 4)^(1/3)
R = (√3 * 2.679 × 10^-23 cm^3 / 4)^(1/3)
R ≈ 1.305 × 10^-8 cm
4. Convert the radius to nanometers:
R = 1.305 × 10^-8 cm * 10^7 nm/cm
R ≈ 0.1305 nm
Therefore, the radius of a vanadium atom in a BCC crystal structure is approximately 0.1305 nm.
To calculate the radius of a vanadium (V) atom, given that it has a body-centered cubic (BCC) crystal structure, a density of 5.96 g/cm3, and an atomic weight of 50.9 g/mol, you correctly used the equation:
density = (2 * atomic weight) / (volume of unit cell * Avogadro's number)
To find the volume of the unit cell (V_c), you rearranged this equation. Now, let's go through the steps to determine the radius (R).
1. Solve for V_c:
density = 2 * atomic weight / (V_c * Avogadro's number)
Rearrange the equation:
V_c = (2 * atomic weight) / (density * Avogadro's number)
2. Use the relation V_c = a^3 = (4R / sqrt(3))^3 to find the radius (R).
Equating V_c in terms of R:
V_c = 64R^3 / (3 * sqrt(3))
Now, substitute the expression for V_c from step 1:
(2 * atomic weight) / (density * Avogadro's number) = 64R^3 / (3 * sqrt(3))
3. Solve for R:
R^3 = ((2 * atomic weight) / (density * Avogadro's number)) * (3 * sqrt(3)) / 64
R = [((2 * atomic weight) / (density * Avogadro's number)) * (3 * sqrt(3)) / 64]^(1/3)
Substitute the values for atomic weight (50.9 g/mol), density (5.96 g/cm3), and Avogadro's number (6.022 x 10^23).
R = [((2 * 50.9) / (5.96 * 6.022 x 10^23)) * (3 * sqrt(3)) / 64]^(1/3)
After performing the calculation, we find that R ≈ 1.47 x 10^(-8) cm.
To convert this value to nanometers, divide by 10,000:
R ≈ 1.47 x 10^(-8) cm / 10,000 = 1.47 x 10^(-12) cm = 0.147 nm.
Therefore, the radius of a vanadium atom in a BCC crystal structure is approximately 0.147 nm.