Question

In the emission spectrum from hydrogen gas, what type of energy is released when electrons drop from the fifth to the first energy level?

Short answer

Ultraviolet radiation is released, with an energy of approximately \(2.0928 \times 10^{-18}\) Joules.

Step-by-step solution

Understanding Electron Transition

When electrons move between energy levels in an atom, they release energy in the form of electromagnetic radiation. In this problem, the transition is from the fifth energy level to the first energy level in a hydrogen atom.

Identifying the Series

The energy transitions in hydrogen are categorized into series based on the final energy level. For transitions ending at the first energy level, it is part of the Lyman series, which releases ultraviolet radiation.

Calculating Energy Released

The energy released during an electronic transition can be calculated using the Rydberg formula: \[E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]where \(R_H\) is the Rydberg constant \(2.18 \times 10^{-18} \text{ J}\), \(n_1\) is the final energy level (1), and \(n_2\) is the initial energy level (5).

Substituting Values into the Formula

Substitute the values from step 3 into the Rydberg formula: \[E = 2.18 \times 10^{-18} \text{ J} \left( \frac{1}{1^2} - \frac{1}{5^2} \right) = 2.18 \times 10^{-18} \text{ J} \left( 1 - 0.04 \right)\]This simplifies to:\[E = 2.18 \times 10^{-18} \times 0.96 \text{ J}\]

Final Calculation

Calculate the energy released: \[E = 2.0928 \times 10^{-18} \text{ J}\]This confirms that as the electron transitions from the fifth to the first energy level, ultraviolet radiation is emitted, releasing approximately \(2.0928 \times 10^{-18}\) Joules of energy.

Key concepts

Hydrogen Emission Spectrum

Hydrogen emission spectrum is a set of electromagnetic waves emitted by hydrogen atoms when their electrons transition between different energy levels. Each transition corresponds to a specific wavelength of light. Unlike a continuous spectrum, the hydrogen spectrum consists of distinct lines, each representing an electronic transition.

• When an electron in a hydrogen atom absorbs energy, it jumps to a higher energy level, becoming excited.
• Due to the natural tendency of physical systems to seek lower energy states, the electron soon returns to a lower energy level.
• As it does, it releases energy in the form of photons, contributing to the emission spectrum.

These lines form series named after the scientists who discovered them, such as the Lyman, Balmer, and Paschen series. Understanding these series helps in comprehending the different energetic transitions occurring in hydrogen.

Lyman Series

The Lyman series is a segment of the hydrogen emission spectrum observed when electrons transition to the first energy level, or ground state, of a hydrogen atom. First discovered by Theodore Lyman, this series falls entirely in the ultraviolet region of the electromagnetic spectrum.

• Key for understanding is that all transitions in the Lyman series involve electrons dropping to the first energy level, which is the lowest possible level.
• Because the energy released is so high, the corresponding photons fall within the ultraviolet range, invisible to the naked eye.
• This series is the most energetic among the hydrogen spectral series, indicating the large energy changes when dropping to the ground state.

Comprehending the Lyman series and its characteristics can broaden one's understanding of hydrogen's electronic structure and energy emission behavior.

Rydberg Formula

The Rydberg formula is a crucial mathematical expression used to predict the wavelengths of photons emitted from hydrogen as electrons transition between energy levels. Formulated by Johannes Rydberg, it relates energy release to electron movement between orbitals.

• The formula is: \[E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]
• Here, \(E\) denotes the energy difference between two levels, and \(R_H\) is the Rydberg constant, approximately \(2.18 \times 10^{-18} \text{ J}\) for hydrogen.
• The variables \(n_1\) and \(n_2\) refer to the final and initial energy levels, respectively.

This formula is essential in calculating the energy associated with specific electronic transitions in hydrogen and is applicable in a variety of spectroscopic analyses.

Ultraviolet Radiation

Ultraviolet (UV) radiation is a form of electromagnetic radiation with a wavelength shorter than that of visible light but longer than X-rays. It is a key component of the hydrogen emission spectrum, particularly within the Lyman series.

• UV radiation is produced when electrons in an atom drop to a significantly lower energy level, releasing substantial energy.
• This radiation has applications not only in natural processes but also in technology, such as medical sterilization and fluorescent lighting.
• Understanding UV radiation involves recognizing its role in chemical reactions and biological effects, like its capacity to cause skin damage with prolonged exposure.

The study of ultraviolet radiation also contributes to unraveling the mysteries of atomic interactions and energy transitions.