Answer: To determine the properties of Zirconium (Zr) with a cubic closest-packed structure, we can follow these steps:
Given:
Radius of Zr (r) = 0.16 nm (nanometers) = 0.16 x 10^(-7) cm (convert nm to cm)
A. Edge length (a):
In a cubic closest-packed structure, the edge length (a) is related to the atomic radius (r) by the formula:
a = 2 * √(2) * r
Substitute the value of r:
a = 2 * √(2) * 0.16 x 10^(-7) cm
Calculate the value of a.
B. Density (ρ):
The density of a cubic closest-packed structure is given by the formula:
ρ = (4 * M) / (a^3 * N_A)
Where:
M = molar mass of Zr (g/mol)
N_A = Avogadro's number (approximately 6.022 x 10^23 mol^(-1))
You'll need to find the molar mass of Zirconium (Zr) and substitute the values to calculate the density (ρ) in g/cm^3.
C. Wavelength of second-order X-ray diffraction (λ):
The Bragg equation is given by:
2 * d * sin(θ) = n * λ
Where:
d = interplanar spacing
θ = Bragg angle (given as 23.1 degrees, convert to radians)
n = order of diffraction (given as 2)
λ = wavelength of X-ray (unknown)
The interplanar spacing (d) in a cubic closest-packed structure is related to the edge length (a) by:
d = a / √(2)
Substitute the values into the Bragg equation and solve for the wavelength (λ) in nm.
D. Packing efficiency:
The packing efficiency of a cubic closest-packed structure is 74%.