A sample of molybdenum (Mo) is analyzed by x-ray diffraction using NiKα radiation. Calculate the value of the
angle at which the lowest-angle reflection is observed.
Express your answer in degrees:
lambda = 2d*sin theta.
I assume you have d for Mo and Kalpha for Ni.
Ya. answer is 22 degrees. i calculated
Well, according to my calculations...ahem...*honk honk*...the value of the angle at which the lowest-angle reflection is observed is approximately "Ican'tbelievethisanglequestionagain" degrees.
To calculate the angle at which the lowest-angle reflection is observed, we need to use Bragg's Law, which states:
nλ = 2d sin(θ)
Where:
- n is the order of the reflection (in this case, n=1 as it is the lowest-angle reflection)
- λ is the wavelength of the x-ray radiation (in this case, λ = the wavelength of NiKα radiation, which is approximately 1.54 Ångstroms or 1.54 x 10^-10 meters)
- d is the spacing between the crystal planes of the sample
- θ is the angle between the incident x-ray beam and the crystal planes.
To solve for θ, we need to rearrange the equation:
sin(θ) = nλ / (2d)
Let's assume the spacing between the crystal planes (d) for molybdenum (Mo) is 0.314 nm or 3.14 x 10^-10 meters.
Plugging in the values, we have:
sin(θ) = 1(1.54 x 10^-10) / (2(3.14 x 10^-10))
simplifying,
sin(θ) = 0.494
To find the angle θ, we take the arcsin (inverse sine) of the result:
θ = arcsin(0.494)
Using a scientific calculator or online tool to simplify the arcsin, we find:
θ ≈ 29.3 degrees
Therefore, the angle at which the lowest-angle reflection is observed is approximately 29.3 degrees.
To calculate the value of the angle at which the lowest-angle reflection is observed, we need to use Bragg's Law.
Bragg's Law states that for constructive interference to occur between X-rays and a crystal lattice, the following condition must be met:
nλ = 2d sin(θ)
Where:
- n is the order of the reflection (for the lowest-angle reflection, n = 1)
- λ is the wavelength of the X-ray radiation (given as NiKα)
- d is the interplanar spacing of the crystal lattice planes
- θ is the angle of incidence or reflection
We are given that NiKα radiation is used, which has a wavelength of 1.54 Ångstroms (Å) or 1.54 x 10^-10 meters.
To determine the interplanar spacing, we need to use the Miller indices of the plane associated with the lowest-angle reflection for molybdenum (Mo). Let's assume it is the (100) plane.
The formula for calculating the interplanar spacing (d) for a cubic crystal lattice is:
d = a / √(h^2 + k^2 + l^2)
Where:
- a is the lattice constant (the edge length of the unit cell)
- h, k, l are the Miller indices of the plane
For molybdenum (Mo), the lattice constant is 3.14 Å.
Substituting the values into the equation:
d = 3.14 Å / √(1^2 + 0^2 + 0^2)
d = 3.14 Å / 1
d = 3.14 Å
Now we can solve for θ using Bragg's Law:
nλ = 2d sin(θ)
1(1.54 x 10^-10 m) = 2(3.14 x 10^-10 m) sin(θ)
Dividing both sides of the equation by 2d and taking the inverse sine (sin^-1) of both sides:
sin(θ) = (1.54 x 10^-10 m) / (2(3.14 x 10^-10 m))
θ = sin^-1((1.54 x 10^-10 m) / (2(3.14 x 10^-10 m)))
Calculating the value using a calculator:
θ ≈ 14.8 degrees
Therefore, the value of the angle at which the lowest-angle reflection is observed is approximately 14.8 degrees.