Diffraction of X rays with l = 131.5 pm occurred at an angle of 25.5 degrees by a crystal of aluminum. Assuming first-order diffraction, what is the interplanar spacing in aluminum? (LO 12.2) (a) 76.4 pm (b) 183.1 pm (c) 305.5 pm (d) 152.7 pm

Verified step by step guidance

1. Identify the given values: wavelength (\
2. lambda = 131.5 pm\
3. ) and diffraction angle (\
4. theta = 25.5^\circ\
5. ).
6. Recognize that the problem involves the use of Bragg's Law, which is given by \
7. n\lambda = 2d\sin(\theta)\
8. , where \
9. n\
10. is the order of diffraction, \
11. d\
12. is the interplanar spacing, and \
13. \theta\
14. is the angle of diffraction.
15. Since it is a first-order diffraction (\
16. n=1\
17. ), substitute \
18. n\
19. and the other known values into Bragg's Law.
20. Solve the equation for \
21. d\
22. to find the interplanar spacing: \
23. d = \frac{\lambda}{2\sin(\theta)}\
24. .
25. Calculate \
26. d\
27. using the values of \
28. \lambda\
29. and \
30. \theta\
31. to find the interplanar spacing in the aluminum crystal.

Key Concepts

Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angle of diffraction and the interplanar spacing in a crystal. It is expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the interplanar spacing, and θ is the angle of diffraction. This law is fundamental in determining the arrangement of atoms within a crystal structure.

Interplanar Spacing

Interplanar spacing (d) refers to the distance between parallel planes of atoms in a crystal lattice. It is a critical parameter in crystallography, influencing how X-rays are diffracted. The value of d can be calculated using Bragg's Law, which helps in identifying the crystal structure and properties of materials.

First-Order Diffraction

First-order diffraction occurs when n = 1 in Bragg's Law, indicating the first set of parallel planes that diffract X-rays. This is the simplest case and provides the most direct relationship between the wavelength, angle, and interplanar spacing. Understanding first-order diffraction is essential for accurately calculating the interplanar spacing in crystalline materials.