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 of these

Short answer

The wavelength is 96.97 nm, which corresponds to option (a).

Step-by-step solution

Understanding the Lyman Series

The Lyman series refers to the set of spectral lines representing transitions where an electron falls to the ground state ( = 1) from any higher energy level ( = 2, 3, 4, ...). In this problem, an electron falls from the fourth stationary state ( _2 = 4) to the ground state ( _1 = 1).

Using the Rydberg Formula

To find the wavelength of the emitted radiation, we use the Rydberg formula for hydrogen : \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \(n_1\) and \(n_2\) are the lower and higher energy levels respectively, and \(R_H\) is the Rydberg constant \((1.1 \times 10^7 \, m^{-1})\).

Plug in the Quantum Levels

Substitute the quantum levels into the formula: \(n_1 = 1\) and \(n_2 = 4\). This gives: \[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{4^2} \right) \].

Calculate the Energy Difference

Calculate \( \frac{1}{1^2} - \frac{1}{4^2} \): \[ \frac{1}{1} - \frac{1}{16} = 1 - 0.0625 = 0.9375 \].

Calculate the Wavelength Inverse

Substituting the calculated energy difference into the Rydberg formula: \[ \frac{1}{\lambda} = 1.1 \times 10^7 \times 0.9375 = 1.03125 \times 10^7 \, m^{-1} \].

Find the Wavelength

Find the wavelength by taking the reciprocal of the calculated value: \[ \lambda = \frac{1}{1.03125 \times 10^7} = 9.697 \times 10^{-8} \, m \].

Convert Wavelength to Nanometers

Convert the wavelength from meters to nanometers (1 m = \(10^9\) nm): \[ 9.697 \times 10^{-8} \, m = 96.97 \, nm \].

Key concepts

Rydberg Formula

The Rydberg formula is a vital mathematical tool used to calculate the wavelengths of spectral lines in the electromagnetic spectrum of hydrogen. This formula helps us understand how electrons transition between different quantum levels, emitting or absorbing light energy that appears as spectral lines. The formula is written as:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]Here, \( \lambda \) represents the wavelength of the light emitted or absorbed, \( R_H \) denotes the Rydberg constant for hydrogen (approximately \(1.1 \times 10^7 \, m^{-1}\)), and \(n_1\) and \(n_2\) are the principal quantum numbers of the lower and higher energy levels, respectively. Since the transitions between these levels involve significant changes in energy states, the formula plays a pivotal role in predicting and explaining these spectral phenomena.

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe but profound in its contributions to quantum physics. It consists of a single proton in its nucleus, surrounded by one orbiting electron. Understanding hydrogen is crucial because its electron can occupy various energy levels or states.

• The electron transitions between these levels correspond to spectral lines that we can observe.
• Each transition is associated with a specific amount of energy, often visualized using an energy level diagram called the Bohr model.

Due to its simplicity, hydrogen serves as the benchmark for studying atomic spectra and other more complex atoms. Scientists use hydrogen's emissions to decode the elementary processes in quantum mechanics and atomic structure.

Spectral Lines

Spectral lines are unique patterns of light emission or absorption produced when electrons transition between energy levels in atoms. These lines are visible because each transition emits or absorbs specific wavelengths.

• In the case of hydrogen, the spectral lines result from the electron's movement between quantum levels.
• Spectral lines are not only vital for understanding atomic structure but also for identifying elements in stars and galaxies.

When applying the Rydberg formula, the lines arise as distinct wavelengths that can help in determining the electronic transitions in an atom. Each series of spectral lines, such as the Lyman series for hydrogen, corresponds to electron transitions ending at a specific lower energy level.

Quantum Levels

Quantum levels, also known as quantum states or energy levels, refer to the discrete energy positions that an electron can occupy in an atom. These levels are quantized, meaning an electron must jump from one level to another rather than existing in-between states.

• For hydrogen, these levels are defined by quantum numbers, denoted by \(n\).
• The lowest energy state is the ground state, and any higher state is called an excited state.

Transitions between these quantum levels involve the emission or absorption of energy in the form of photons. The energy difference between levels matches the energy of the emitted or absorbed light, corresponding to the spectral lines we observe.