The angle of reflection for first order monochromatic X-rays from a crystal whose atomic spacing is `2.5 Å` is `15^(@)`. Calculate the wavelength of X-rays.

Answer

To solve the problem of calculating the wavelength of X-rays using Bragg's law, we will follow these steps:

### Step 1: Understand Bragg's Law
Bragg's law is given by the equation:
\[ n\lambda = 2d \sin \theta \]
where:
- \( n \) is the order of diffraction,
- \( \lambda \) is the wavelength of the X-rays,
- \( d \) is the atomic spacing of the crystal,
- \( \theta \) is the angle of reflection.

### Step 2: Identify the Given Values
From the problem, we have:
- The atomic spacing \( d = 2.5 \, \text{Å} \) (which is \( 2.5 \times 10^{-10} \, \text{m} \)),
- The angle of reflection \( \theta = 15^\circ \),
- The order of diffraction \( n = 1 \) (since it is the first order).

### Step 3: Substitute the Values into Bragg's Law
Using the values in Bragg's law:
\[ 1\lambda = 2 \times (2.5 \, \text{Å}) \times \sin(15^\circ) \]

### Step 4: Calculate \( \sin(15^\circ) \)
Using a calculator or trigonometric tables:
\[ \sin(15^\circ) \approx 0.2588 \]

### Step 5: Substitute \( \sin(15^\circ) \) into the Equation
Now we substitute \( \sin(15^\circ) \) into the equation:
\[ \lambda = 2 \times 2.5 \, \text{Å} \times 0.2588 \]

### Step 6: Perform the Calculation
Calculating the right side:
1. First, calculate \( 2 \times 2.5 = 5 \, \text{Å} \).
2. Then, multiply by \( 0.2588 \):
\[ \lambda = 5 \, \text{Å} \times 0.2588 \approx 1.294 \, \text{Å} \]

### Step 7: Final Result
Thus, the wavelength of the X-rays is:
\[ \lambda \approx 1.294 \, \text{Å} \]

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