Problem 163 - A Complete Resource Book in Chemistry for JEE Main

Question

The wavelength of the radiation emitted, when in a hydrogen atom electron falls from infinity to station ary state 1 , would be (Rydberg constant \(=1.097 \times 10^{7}\) \(\left.\mathrm{m}^{-1}\right)\) (a) 9 lnm (b) \(192 \mathrm{~nm}\) (c) \(406 \mathrm{~nm}\) (d) \(9.1 \times 10^{-8} \mathrm{~nm}\)

Step-by-Step Solution

Answer

The wavelength is 91.1 nm. The closest option is (d) \(9.1 \times 10^{-8} \text{ nm}\).

Step 1: Understand the concept

The exercise involves calculating the wavelength of radiation emitted when an electron falls from infinity to the ground state (n=1) in a hydrogen atom. This is a transition to the Lyman series.

Step 2: Use the formula for wavelength in the Lyman Series

The wavelength (\(\lambda\)) of the photon emission in the hydrogen atom can be calculated using the Rydberg formula: \[\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]where \(\mathbf{R} = 1.097\times10^7 \text{ m}^{-1}\), \(\mathbf{n_1} = 1\), and \(\mathbf{n_2} = \infty\).

Step 3: Substitute the values

Replace \(n_1\) and \(n_2\) in the formula: \[\frac{1}{\lambda} = 1.097\times10^7 \left(\frac{1}{1^2} - \frac{1}{\infty^2}\right) \]Simplifying gives \[\frac{1}{\lambda} = 1.097\times10^7 \].

Step 4: Solve for wavelength

Solve the equation \(\frac{1}{\lambda} = 1.097\times10^7 \) to find λ: \[\lambda = \frac{1}{1.097\times10^7} \approx 9.11 \times 10^{-8} \text{ m} \].

Step 5: Convert wavelength from meters to nanometers

Since the wavelength is typically expressed in nanometers, convert 9.11 × 10⁻⁸ m to nm (1 m = 10⁹ nm): \[9.11 \times 10^{-8} \text{ m} = 91.1 \text{ nm}\].

Key Concepts

Rydberg Formula

In the realm of atomic physics, the Rydberg formula is an essential tool used to determine the wavelengths of photons emitted during electron transitions in hydrogen atoms. The Rydberg formula is expressed as: \[ \frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \] where:

• \(\lambda\) is the wavelength of the emitted photon,
• \(R\) is the Rydberg constant, which equals \(1.097\times10^7 \text{ m}^{-1}\),
• \(n_1\) and \(n_2\) are the principal quantum numbers of the electron's orbit before and after the transition.

This formula is extremely useful because it allows us to quantify the energy transitions as electrons move between defined energy levels. For example, in a hydrogen atom, one can find the precise wavelength of light emitted when an electron falls from a higher energy level to a lower one. This mechanism not only helps us understand atomic spectra but also proves fundamental in fields such as spectroscopy and quantum mechanics.

Lyman Series

The Lyman series is a set of spectral lines in the ultraviolet range, resulting from electron transitions in hydrogen atoms to the n=1 energy level or ground state. This series was first discovered by Theodore Lyman, which is why it is named after him. When an electron transitions from a higher energy level, like n=2, 3, etc., down to the n=1 level, it emits a photon of ultraviolet light. These emissions form the Lyman series:

• Transitions to n=1 result in the shortest wavelengths and highest energy photons in the hydrogen spectrum.
• This series provides insights into the quantized nature of atomic energy levels.
• It plays a significant role in fields like astronomy and chemistry, where understanding electron behavior in atoms is crucial.

By utilizing the Rydberg formula, one can predict the wavelengths of these emissions, aiding in the comprehension of atomic structures and spectra.

Photon Emission

Photon emission occurs when an electron in an atom transitions from a higher energy level to a lower one, releasing energy in the form of a photon. The energy of this photon is directly related to the difference between the two energy levels. In the case of the hydrogen atom, when an electron falls from infinity to the n=1 level, a photon is emitted. The energy and wavelength of this photon can be calculated using the Rydberg formula, as the principal quantum numbers involved define the energy states. Key points to remember include:

• The emitted photon carries energy proportional to the energy gap between the two levels.
• This phenomenon is responsible for the visible and UV spectrums observed in hydrogen emission spectra.
• Photon emission is pivotal in applications such as lasers, LEDs, and even in stars, affecting technological and natural phenomena alike.

Understanding photon emission is crucial, as it reveals fundamental processes of energy transfer at the atomic level, influencing our interpretation of both artificial and cosmic phenomena.