Question

which of the following series in the spectrum of hydrogen atom lies in the visible legion of the electro magnetic spectrum? (A) Paschen (B) Lyman (C) Brakett (D) Balmer

Short answer

The series in the spectrum of the hydrogen atom that lies in the visible region of the electromagnetic spectrum is the (D) Balmer series.

Step-by-step solution

Find the formula for the wavelength of energy transitions in the Hydrogen spectrum

We can use the Rydberg formula to find the wavelength of transitions in the hydrogen spectrum: \(\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\) where λ is the wavelength, R_H is the Rydberg constant for hydrogen (\(1.097 \times 10^7 m^{-1}\)), n_1 and n_2 are the initial and final energy states of the electron (with \(n_1 < n_2\)), respectively.

Calculate the wavelength for the given series

We will calculate the wavelength for each series by substituting the n values. (A) Paschen series - Transitions from n ≥ 4 to n = 3: \(\frac{1}{\lambda_{Paschen}} = R_H \left(\frac{1}{3^2} - \frac{1}{4^2} \right)\) The shortest wavelength is when the electron transitions from n = 4 to n = 3 which corresponds to the highest energy transition. Any other transition will have a longer wavelength. (B) Lyman series - Transitions from n ≥ 2 to n = 1: \(\frac{1}{\lambda_{Lyman}} = R_H \left(\frac{1}{1^2} - \frac{1}{2^2} \right)\) The shortest wavelength is when the electron transitions from n = 2 to n = 1 which corresponds to the highest energy transition. Any other transition will have a longer wavelength. (C) Brackett series - Transitions from n ≥ 5 to n = 4: \(\frac{1}{\lambda_{Brackett}} = R_H \left(\frac{1}{4^2} - \frac{1}{5^2} \right)\) The shortest wavelength is when the electron transitions from n = 5 to n = 4, which corresponds to the highest energy transition. Any other transition will have a longer wavelength. (D) Balmer series - Transitions from n ≥ 3 to n = 2: \(\frac{1}{\lambda_{Balmer}} = R_H \left(\frac{1}{2^2} - \frac{1}{3^2} \right)\) The shortest wavelength is when the electron transitions from n = 3 to n = 2, which corresponds to the highest energy transition. Any other transition will have a longer wavelength.

Determine which series falls in the visible region

Calculate the smallest wavelength for each series and compare it to the bounds of the visible spectrum (400 nm to 700 nm): (A) Paschen series: \(\lambda_{Paschen} = \frac{1}{1.097 \times 10^7(1/9 - 1/16)} = 8.21 \times 10^{-7}\, m\) or 821 nm, which falls outside the visible region. (B) Lyman series: \(\lambda_{Lyman} = \frac{1}{1.097 \times 10^7(1 - 1/4)} = 1.215 \times 10^{-7}\, m\) or 121.5 nm, which falls outside the visible region. (C) Brackett series: \(\lambda_{Brackett} = \frac{1}{1.097 \times 10^7(1/16 - 1/25)} = 1.45 \times 10^{-6}\, m\) or 1450 nm, which falls outside the visible region. (D) Balmer series: \(\lambda_{Balmer} = \frac{1}{1.097 \times 10^7(1/4 - 1/9)} = 6.563 \times 10^{-7}\, m\) or 656.3 nm, which falls within the visible region. Since the shortest wavelength of the Balmer series falls within the visible region of the electromagnetic spectrum, the correct answer is (D) Balmer series.

Key concepts

Rydberg Formula

The Rydberg formula is a pivotal equation in understanding the spectral lines emitted by hydrogen atoms. It allows us to calculate the wavelengths of light resulting from electron transitions between different energy levels of the hydrogen atom. The formula is given as:\[ \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where:

• \( \lambda \) is the wavelength of the emitted light.
• \( R_H \) is the Rydberg constant for hydrogen, approximately \( 1.097 \times 10^7 \text{ m}^{-1} \).
• \( n_1 \) and \( n_2 \) are the principal quantum numbers, representing the initial and final energy levels of the electron, respectively, with \( n_1 < n_2 \).

This formula is essential as it predicts the wavelength of light that corresponds to an electron jump between two orbits of a hydrogen atom. Understanding these transitions is crucial for identifying the various series in the hydrogen spectrum.

Balmer Series

The Balmer series is one of the hydrogen spectral series that lies within the visible light spectrum, making it particularly significant. It consists of the wavelengths of electromagnetic radiation emitted during electron transitions from higher energy levels (\( n \geq 3 \)) to the second energy level (\( n = 2 \)). The Balmer series is characterized by its four visible spectral lines, typically observed as:

• Red line at approximately 656 nm (H-alpha)
• Blue-green line at approximately 486 nm (H-beta)
• Aqua line at approximately 434 nm (H-gamma)
• Violet line at approximately 410 nm (H-delta)

These lines are within the visible region, making them easily observable through a spectroscope. Understanding the Balmer series helps us analyze and study hydrogen's emission spectrum more effectively.

Visible Electromagnetic Spectrum

The visible electromagnetic spectrum refers to the portion of the electromagnetic spectrum that is visible to the human eye. It ranges from about 400 nm to 700 nm in wavelength, corresponding to violet and red light, respectively. Some key points about the visible spectrum include:

• It consists of various colors: violet, indigo, blue, green, yellow, orange, and red.
• These colors arise due to different wavelengths associated with light emitted by sources like the sun or the hydrogen atoms.
• Understanding where the spectral lines of elements like hydrogen fall within this range helps in the study of atomic spectra and can be used to identify substances or elemental composition within stars.

The Balmer series is a noteworthy example of spectral lines that reside in the visible spectrum, as these transitions can be directly observed.

Energy Transitions

Energy transitions in a hydrogen atom occur when an electron moves between different energy levels, releasing or absorbing photons in the process. These transitions are responsible for the spectral lines observed in hydrogen's emission spectra. Key aspects of energy transitions include:

• The release of energy occurs when an electron moves to a lower energy level, emitting a photon of light.
• The energy of the emitted photon corresponds to the difference in energy between two levels.
• This energy is given by the Rydberg formula, facilitating calculations of the resulting wavelength.
• Each hydrogen spectral series, including the Lyman, Balmer, and others, corresponds to transitions ending at a specific energy level.

Appreciating these transitions provides critical insight into quantum mechanics and the behavior of electrons in atoms.

Wavelength Calculations

Wavelength calculations are fundamental to determining which spectral lines fall within certain regions of the electromagnetic spectrum. These calculations are primarily achieved using the Rydberg formula, as it precisely evaluates the wavelength from energy level transitions.Here's how they work:

• Identify the initial and final energy states: For instance, the Balmer series involves transitions from \( n \geq 3 \) to \( n = 2 \).
• Use the Rydberg formula to compute the wavelength of emitted light.
• Compare calculated wavelengths with known spectrum regions, such as the visible range of 400 nm to 700 nm.
• This comparison determines the visibility of certain spectral lines, like distinguishing the Balmer series from other non-visible series.

Mastering these calculations enables scientists to analyze and understand the properties and behaviors of elements through their spectral lines.