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Chapter 10: Problem 47

A topaz crystal has an interplanar spacing \((d)\) of \(1.36 \AA^{\circ}(1 \AA=\) \(\left.1 \times 10^{-10} \mathrm{~m}\right) .\) 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 approximately \(7.20 \times 10^{-11}\) m.

Step by step solution

01

Convert angstroms to meters

We are given the interplanar spacing d in units of angstroms (Å), but we need to convert this to meters for our calculation. Recall that 1 Å = 1 × 10⁻¹⁰ m. \[ d = 1.36 \ Å × \frac{1 \times 10^{-10} \ m}{1 \ Å} = 1.36 \times 10^{-10} \ m \]
02

Convert degrees to radians

We are given the angle θ in degrees, but we need to convert this to radians for our calculation. Recall that 1° = π/180 radians. \[ \theta = 15.0 \ ° × \frac{\pi \ rad}{180 \ °} = \frac{15\pi}{180} \ \text{radians} \]
03

Apply Bragg's Law

Now that we have the interplanar spacing in meters and the angle in radians, we can plug these values into Bragg's Law to solve for the wavelength λ. Recall that Bragg's Law is: \[ n\lambda = 2d \sin{\theta} \] Plugging in the values (n = 1, d = 1.36 × 10⁻¹⁰ m, θ = 15.0°): \[ \lambda = \frac{2 \times (1.36 \times 10^{-10} \ m) \times \sin{\frac{15\pi}{180}}}{1} \]
04

Calculate the wavelength λ

Now, we can calculate the wavelength λ: \[ \lambda \approx 7.20 \times 10^{-11} \ \text{m} \] Therefore, the wavelength of the X-ray that should be used is approximately \(7.20 \times 10^{-11}\) m.

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Most popular questions from this chapter

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