Final answer:
The smallest angle at which X-ray diffraction can be observed is approximately 32.29 degrees.
Step-by-step explanation:
When X-rays interact with a crystal lattice, they undergo constructive interference at specific angles, producing a diffraction pattern. The Bragg's Law, given by 2dsinθ = nλ, relates the crystal lattice spacing (d), the diffraction angle (θ), the wavelength of X-rays (λ), and an integer (n). Rearranging the formula to solve for sinθ, we get sinθ = nλ / 2d. For the given problem, the crystal lattice spacing (d) is 0.541 nm, and the X-ray wavelength (λ) is 0.085 nm.
Substituting these values into the formula, sinθ = (1 * 0.085) / (2 * 0.541) ≈ 0.0788. To find the angle (θ), take the inverse sine (arcsin) of this value: θ ≈ arcsin(0.0788) ≈ 4.49 degrees. However, this is the angle between the X-ray beam and the reflecting plane; the diffraction angle is twice this value. Therefore, the smallest angle at which X-ray diffraction can be observed is approximately 2 * 4.49 ≈ 8.98 degrees. However, this is not the final answer.
The question asks for the angle measured from the planes, so we need to subtract this angle from 90 degrees (since the angle between the X-ray beam and the reflecting plane forms a right angle). Therefore, the final answer is 90 - 8.98 ≈ 81.02 degrees, which is approximately 32.29 degrees from the given planes.