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

Question

What is the wavelength of the radiation emitted produced in a line in the Lyman series when an electron falls from fourth stationary state in hydrogen atom? \(\left(\mathrm{R}_{\mathrm{H}}=1.1 \times 10^{7} \mathrm{~m}^{-1}\right)\) (a) \(96.97 \mathrm{~nm}\) (b) \(969.7 \mathrm{~nm}\) (c) \(9.697 \mathrm{~nm}\) (d) none

Step-by-Step Solution

Answer

The wavelength of the radiation is 9.697 nm, matching option (c).

Step 1: Identify Initial and Final States

The exercise requires determining the wavelength of radiation in the Lyman series. In the Lyman series, an electron falls to the first stationary state ( =1) from a higher energy level. Here, the electron descends from the fourth ( =4) state to the first ( =1) state.

Step 2: Apply the Rydberg Formula

The wavelength of radiation emitted when an electron falls between two energy levels can be determined using the Rydberg formula: \ \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant (\(1.1 \times 10^7 \ \text{m}^{-1}\)), \( n_1 \) is the final state (1 for Lyman series), and \( n_2 \) is the initial state (4 in this case).

Step 3: Substitute Values into the Formula

Substitute the known values into the Rydberg formula: \ \[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{4^2} \right) \] Simplifying inside the parenthesis: \ \( \frac{1}{1} = 1 \) and \( \frac{1}{16} = 0.0625 \).

Step 4: Calculate the Difference

Subtract the fractions inside the equation: \ \[ \frac{1}{\lambda} = 1.1 \times 10^7 \left(1 - 0.0625 \right) \] This results in: \ \[ \frac{1}{\lambda} = 1.1 \times 10^7 \times 0.9375 \].

Step 5: Solve for Wavelength

Continue by solving for \( \lambda \): \ \[ \frac{1}{\lambda} = 1.03125 \times 10^7 \] Therefore, the wavelength \( \lambda \) is: \ \[ \lambda = \frac{1}{1.03125 \times 10^7} \].

Step 6: Compute the Wavelength

Calculate \( \lambda = \frac{1}{1.03125 \times 10^7} \approx 9.697 \times 10^{-9} \ \text{m} \), converting the result to nanometers, which is \( 9.697 \ \text{nm} \).

Key Concepts

Rydberg Formula

The Rydberg formula is a key tool in understanding the wavelengths of spectral lines for the hydrogen atom. It helps calculate the wavelength of light resulting from an electron transition between two energy levels in an atom. Here's a simple breakdown:

• It describes how photons are emitted or absorbed during the transition of electrons between energy levels.
• The general formula is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
• In this equation, \( \lambda \) represents the wavelength, \( R_H \) stands for the Rydberg constant, \( n_1 \) is the principal quantum number of the final level, and \( n_2 \) is the principal quantum number of the initial energy level.

Using the Rydberg formula, one can predict the possible wavelengths of radiation produced when an electron moves from one orbit to another. This formula is particularly important in the Lyman, Balmer, and Paschen series of hydrogen.

Electron Transition

Electron transition refers to the movement of an electron from one energy level, or orbit, to another within an atom. When an electron jumps from a higher energy level to a lower one, energy is released in the form of light. This phenomenon is what we observe as radiation.

• Electrons occupy discrete energy levels or shells around the nucleus of an atom.
• When they absorb energy, they can jump to a higher energy level.
• Conversely, when they release energy, they fall back to a lower energy level.
• The specific energy difference between these levels determines the frequency and wavelength of the emitted or absorbed light.

In the case of the Lyman series, electrons fall to the innermost orbit, the first energy level. This is crucial to note because such transitions influence the type of electromagnetic radiation emitted, which typically falls in the ultraviolet range.

Hydrogen Atom Energy Levels

The hydrogen atom is the simplest atom, consisting of just one proton in its nucleus and a single electron. Its energy levels are quantized, meaning electrons can only occupy certain energy states. These distinct energy levels are crucial in understanding the atom's spectral lines.

• The energy levels in a hydrogen atom are represented by principal quantum numbers labeled as \( n \), such as \( n = 1, 2, 3, \) etc.
• The lowest energy level, or ground state, is when \( n = 1 \).
• Excited states are higher energy levels, such as \( n = 2 \), \( n = 3 \), and beyond.
• Electrons can transition between these levels, and the energy differences correspond to specific spectral lines.

Understanding the energy levels of hydrogen helps explain phenomena like the Lyman and Balmer series, where electron transitions produce distinct light emissions. For instance, when an electron falls from \( n = 4 \) to \( n = 1 \), it results in a photon with a specific wavelength as seen in the Lyman series.