Problem 86 - A Complete Resource Book in Chemistry for JEE Main

Question

The shortest wavelength in hydrogen spectrum of Lyman series when \(\mathrm{R}_{H}=109678 \mathrm{~cm}^{-1}\) is (a) \(1002.7 \AA\) (b) \(1215.67 \AA\) (c) \(1127.30 \AA\) (d) \(911.7 \AA\)

Step-by-Step Solution

Answer

The shortest wavelength in the Lyman series is 911.7 Å.

Step 1: Understanding the Lyman Series

The Lyman series of hydrogen includes transitions where the electron falls to the n=1 level from higher energy levels. To find the shortest wavelength, we need the transition from infinity (n = ∞) to n = 1, as this involves the maximum energy change.

Step 2: Using the Rydberg Formula

The Rydberg formula to find the wavelength \( \lambda \) of a transition is \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \], where \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and upper energy levels, respectively.

Step 3: Plugging Values into the Formula

For n=1 and n=∞, the formula becomes \[ \frac{1}{\lambda} = R_H \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R_H(1) = R_H \]. This simplifies to \[ \frac{1}{\lambda} = 109678 \text{ cm}^{-1} \].

Step 4: Solving for Wavelength

To find \( \lambda \), take the reciprocal: \[ \lambda = \frac{1}{109678} \text{ cm}^{-1} \]. Converting to Angstroms (since 1 cm = 10^8 Å), we find \[ \lambda = \frac{10^8}{109678} \AA\].

Step 5: Calculating the Final Wavelength

Perform the calculation: \[ \lambda = \frac{10^8}{109678} \approx 911.7 \AA\].

Key Concepts

Hydrogen Spectrum

The hydrogen spectrum is a fascinating presentation of colors and invisible radiation created when the hydrogen atom transitions between different energy levels. When the electrons in a hydrogen atom absorb energy, they can jump to higher orbits, and when they fall back to lower orbits, they release energy in the form of light. This emitted light forms distinct lines that we observe as a spectrum.

The hydrogen spectrum is divided into various series, such as Lyman, Balmer, and Paschen. Each series corresponds to a different "home level" to which the electron returns. The Lyman series, for example, involves electrons transitioning to the n=1 level.

• **Lyman series**: Involves transitions to the n=1 energy level, emitting ultraviolet light.
• **Balmer series**: Involves transitions to the n=2 level, resulting in visible light.
• **Paschen series**: Involves transitions to the n=3 level and emits infrared radiation.

This segmentation allows scientists to explore the structure of atoms and understand their properties, like energy levels and ionization energies.

Rydberg Formula

The Rydberg formula is a powerful tool used to predict the wavelengths of spectral lines in the hydrogen atom. It tells us where the lines will appear in the spectrum for different electron transitions.

The formula is given by \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where:

• \( \lambda \) is the wavelength of the emitted light.
• \( R_H \) is the Rydberg constant for hydrogen, approximately \( 109678 \ \text{cm}^{-1} \).
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels, respectively.

This formula highlights how the position of spectral lines depends on the differences in the squares of the reciprocals of integers \( n_1 \) and \( n_2 \), reflecting the energy gap between those levels.

The beauty of the Rydberg formula lies in its simplicity and precision, enabling students and scientists alike to calculate wavelengths easily, such as determining the shortest wavelength in the Lyman series from the given data.

Energy Levels

In the context of the hydrogen atom, energy levels refer to the specific orbits or shells where the electron can reside. Each energy level is associated with a principal quantum number \( n \), starting from the lowest energy level (n=1) to higher ones like n=2, n=3, and so on.

These levels are quantized, meaning that electrons can only exist in these specific orbits and nowhere in between. When electrons move between these levels, they absorb or emit energy in discrete amounts, leading to the spectral lines observed in the hydrogen spectrum.

Understanding energy levels is crucial because:

• They dictate the wavelength and frequency of emitted or absorbed light.
• They help explain why atoms emit specific patterns of light, essential for spectroscopy.
• They illustrate the quantized nature of atomic energy states.

The greatest jump in energy, such as from level n=∞ to n=1, results in a significant release of energy, and thus, shorter wavelengths. This is why the Lyman series' shortest wavelength stems from this complete transition, which aligns with high-energy ultraviolet light.