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Chapter 9: Problem 49

A topaz crystal has an interplanar spacing \((d)\) of \(1.36 Å\) \(\left(1Å =1 \times 10^{-10} \mathrm{m}\right) .\) Calculate the wavelength of the 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 approximately 3.99 Å.

Step by step solution

01

Convert the given information into the appropriate units

We are given d as 1.36 Å. We need to convert this into meters. 1 Å = 1 × 10⁻¹⁰ m d = 1.36 Å × 10⁻¹⁰ m/Å = 1.36 × 10⁻¹⁰ m
02

Convert the angle θ from degrees to radians

To use the sine function in our calculations, we must first convert the angle θ from degrees to radians using the following conversion factor: 1 rad = 180°/π θ = 15.0° × π/180° = \(15.0\pi/180 = \pi/12\) radians
03

Use Bragg's Law to find the wavelength of the X-ray

Bragg's Law: \(n\lambda = 2d\sin\theta\) We have n = 1, d = 1.36 × 10⁻¹⁰ m, and θ = \(\pi/12\) radians. Plug these values into Bragg's Law and solve for λ: \(1\cdot\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12)\) Divide both sides by 1 (which doesn't affect the value): \(\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12)\) Now, calculate the result: \(\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12) \approx 3.99 \times 10^{-10} \mathrm{m}\)
04

Convert the wavelength back to Å

Our final step is to convert the result back to Å for easier readability: \(\lambda = 3.99 \times 10^{-10} \mathrm{m} \times \frac{1Å}{1\times 10^{-10}\mathrm{m}} \approx 3.99 Å\) The wavelength of the X-ray that should be used is approximately 3.99 Å.

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