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Chapter 10: Problem 49

A topaz crystal has an interplanar spacing \((d)\) of \(1.36 \AA\) \(\left(1 \mathrm{~A}=1 \times 10^{-10} \mathrm{~m}\right) .\) Calculate the wavelength of the \(\mathrm{X}\) ray that should be used if \(\theta=15.0^{\circ}\) (assume \(n=1\) ).

Short Answer

Expert verified
The wavelength of the X-ray that should be used is \(6.85 \times 10^{-11}\ m\).

Step by step solution

01

Convert units of interplanar spacing

First, we need to convert the interplanar spacing from Angstroms (A) to meters (m). We are given that 1 A = 1 x \(10^{-10}\) m. Therefore: \( d = 1.36\ A \times \frac{1 \cdot 10^{-10}\ m}{1\ A} = 1.36 \times 10^{-10}\ m \)
02

Formula for Bragg's Law

Now, we can rewrite the Bragg's law equation using the information given: \( n\lambda = 2d\sin\theta \) We know that n = 1, d = \(1.36 \times 10^{-10}\ m\), and θ = \(15.0^\circ\). We need to find the value of λ.
03

Plug the known values into the equation

We can now plug the known values into the equation: \( (1) \lambda = 2(1.36 \times 10^{-10}\ m) \sin(15.0^\circ) \)
04

Solve for the wavelength

Lastly, we will solve for λ: \( \lambda = 2(1.36 \times 10^{-10}\ m) \sin(15.0^\circ) \) \( \lambda = 6.85 \times 10^{-11}\ m \) So the wavelength of the X-ray that should be used is \(6.85 \times 10^{-11}\ m\).

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