Problem 105 - A Complete Resource Book in Chemistry for JEE Main

Question

Calculate the \(\lambda\) of X-rays which give a diffraction angle \(2 \theta=16.80^{\circ}\) for a crystal. (Given interplanar distance \(=\) \(0.200 \mathrm{~nm} ;\) diffraction \(=\) first order \(; \sin 8.40^{\circ}=0.1461\) ) (a) \(58.4 \mathrm{pm}\) (b) \(5.84 \mathrm{pm}\) (c) \(584 \mathrm{pm}\) (d) \(648 \mathrm{pm}\)

Step-by-Step Solution

Answer

The wavelength \( \lambda \) is 58.4 pm, so option (a) is correct.

Step 1: Understand the Problem

We need to calculate the wavelength \( \lambda \) of X-rays that produce a diffraction angle \( 2\theta = 16.80^{\circ} \) for a given crystal with an interplanar distance \( d = 0.200 \ \text{nm} \). The diffraction is first order, meaning \( n = 1 \).

Step 2: Convert Units

Convert the interplanar distance from nanometers to picometers for consistency with the answer options. 1 nm = 1000 pm, so \( d = 0.200 \ \text{nm} = 200 \ \text{pm} \).

Step 3: Identify the Relevant Formula

The formula linking the wavelength, diffraction angle, and interplanar distance is Bragg's law: \[ n\lambda = 2d\sin\theta \] where \( n \) is the order of diffraction, \( \lambda \) is the wavelength, \( d \) is the interplanar distance, and \( \theta \) is the angle of incidence.

Step 4: Calculate \( \theta \)

Given that the diffraction angle \( 2\theta = 16.80^{\circ} \), we find \( \theta \) by dividing by 2. \[ \theta = \frac{16.80^{\circ}}{2} = 8.40^{\circ} \]

Step 5: Substitute Values into Bragg's Law

Using \( n = 1 \), \( d = 200 \ \text{pm} \), and \( \sin 8.40^{\circ} = 0.1461 \), substitute these values into Bragg's law: \[ 1 \times \lambda = 2 \times 200 \ \text{pm} \times 0.1461 \]

Step 6: Solve for \( \lambda \)

Calculate \( \lambda \):\[ \lambda = 2 \times 200 \ \text{pm} \times 0.1461 = 58.44 \ \text{pm} \]This rounds to 58.4 pm.

Key Concepts

X-ray Diffraction

X-ray diffraction is a technique used to study the structure of crystalline materials. It works based on the concept that X-rays have a wavelength similar to the spacing between atoms in a crystal. When X-rays encounter a crystal, they are scattered in specific directions. This phenomenon is utilized to determine the crystal's atomic and molecular structure.

The process of X-ray diffraction involves passing X-rays through a crystal. The crystal then diffracts these X-rays at various angles. The angles at which the X-rays are scattered, along with the intensity of the scattered rays, produce a diffraction pattern. This pattern can be analyzed to deduce the arrangement of atoms within the crystal.

After identifying the diffraction angle and knowing the geometry of atomic spacing in the crystal lattice, one can use the data in conjunction with Bragg's Law to find various properties, such as the interplanar distances and the wavelength of the X-rays.

Interplanar Distance

Interplanar distance, denoted as \( d \), measures the spacing between planes in a crystal lattice. In the context of X-ray diffraction, it is a critical parameter that helps link the crystal structure to the diffraction pattern.

The value of \( d \) influences the angles at which X-rays are diffracted, making it essential for calculating properties like the wavelength of the X-rays using Bragg's Law. It is typically measured in units such as nanometers (nm) or picometers (pm), where 1 nm equals 1000 pm.

For instance, in our exercise, the interplanar distance was initially given as 0.200 nm, which we converted into picometers (200 pm) for ease of calculation and consistency with other measurements. Knowing the exact interplanar distance is crucial to applying Bragg's Law effectively, as it directly affects the accuracy of determining the X-ray wavelength.

Wavelength Calculation

The calculation of an X-ray wavelength in diffraction studies usually employs Bragg's Law: \( n\lambda = 2d\sin\theta \). This fundamental equation connects the order of diffraction \( n \), wavelength \( \lambda \), interplanar spacing \( d \), and the angle of incidence \( \theta \).

To perform a wavelength calculation, you often follow these steps:

• Identify the order of diffraction \( n \): In many problems, a first-order diffraction (\( n = 1 \)) is used for simplicity.
• Measure the angle \( \theta \): If given a total diffraction angle \( 2\theta \), divide by two to find \( \theta \).
• Look up the interplanar distance: Know the \( d \) value in compatible units.
• Use the sine function: Calculate \( \sin\theta \) or use provided values.
• Substitute into Bragg's Law: Insert all known values into the equation and solve for \( \lambda \).

For instance, in the given exercise, after substituting \( n = 1, \) \( d = 200 \, \text{pm}, \) and \( \sin 8.40^{\circ} = 0.1461 \) into Bragg's Law, the calculation results in a wavelength of 58.4 pm. This simple yet powerful equation can help unravel the intricate details of a crystal structure by providing precise wavelength calculations of the X-rays used.