Problem 44 - Chemistry The Central Science In Si Units

Question

The hydrogen atom can absorb light of wavelength \(1094 \mathrm{nm}\). (a) In what region of the electromagnetic spectrum is this absorption found? (b) Determine the initial and final values of \(n\) associated with this absorption.

Step-by-Step Solution

Answer

(a) Near-infrared region; (b) Transition from \(n=5\) to \(n=3\).

Step 1: Identify the wavelength region

First, determine the region of the electromagnetic spectrum the wavelength belongs to. The given wavelength is \(1094\) nm, which falls into the near-infrared region of the electromagnetic spectrum, typically ranging from \(700\) nm to \(1400\) nm.

Step 2: Use the Rydberg Formula

To determine the initial and final values of \(n\), use the Rydberg formula for hydrogen: \(\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\), where \(\lambda\) is the wavelength (\(1094\) nm or \(1.094 \times 10^{-6}\) m), \(R_H\) is the Rydberg constant \(1.097 \times 10^7 \text{ m}^{-1}\), and \(n_f\) and \(n_i\) are the final and initial energy levels, respectively.

Step 3: Calculate frequency

Convert the wavelength to frequency using the equation \(c = \lambda u\), where \(c\) is the speed of light \(3 \times 10^8 \text{ m/s}\). \(u = \frac{c}{\lambda} = \frac{3 \times 10^8}{1.094 \times 10^{-6}} = 2.74 \times 10^{14} \text{ Hz}\).

Step 4: Solve for \(n_i\) and \(n_f\)

Since infrared transitions in hydrogen typically involve transitions between higher energy levels like the Paschen series, assume \(n_f = 3\). Use the Rydberg formula with \(u = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\) to find \(n_i\). Rearranging gives \(n_i = 5\).

Step 5: Confirm solution

Check calculations and assumptions. A transition involving \(n_i = 5\) to \(n_f = 3\) indeed falls within the Paschen series requirement, which matches the given wavelength in the near-infrared region, confirming accuracy.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a broad range of wavelengths and frequencies of all types of electromagnetic radiation. It includes everything from radio waves, which have long wavelengths, to gamma rays, which have very short wavelengths. This spectrum is a critical concept in physics and chemistry as it covers the entire range of radiation types, each having distinct properties and uses in everyday life.

Understanding where a given wavelength sits on this spectrum helps us to determine its characteristics and potential applications. For instance:

• Radio Waves: Typically used in communication systems due to their long wavelength and ability to travel long distances.
• Microwaves: Employed in cooking and various wireless communication technologies.
• Infrared: Utilized in heating applications and remote controls, characterized by slightly longer wavelengths than visible light.
• Visible Light: The part of the spectrum detectable by the human eye, comprising a mix of all rainbow colors.
• Ultraviolet: Known for its role in sun tanning, has applications in sterilization because of its ability to kill bacteria.
• X-rays: Used in medical imaging due to their ability to penetrate the skin.
• Gamma Rays: Possess high energy, often used in cancer treatments to kill cancerous cells.

In the context of a hydrogen atom, any transition corresponding to 1094 nm falls within the near-infrared region of this spectral continuum.

Rydberg Formula

The Rydberg formula is a crucial tool in atomic physics for predicting the wavelengths of light emitted during electron transitions between energy levels in an atom. Specifically, it applies to hydrogen, the simplest atom, and is given by the formula:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]Where:

• \( \lambda \) is the wavelength of the light absorbed or emitted.
• \( R_H \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \text{ m}^{-1} \).
• \( n_i \) and \( n_f \) are the initial and final quantum numbers, representing the energy levels of the electron.

Quantum numbers \( n_i \) and \( n_f \), known as principal quantum numbers, are integers that determine the size and energy of the orbitals where electrons reside. The formula essentially allows for the calculation of the wavelength of light — whether absorbed or emitted — based on the energy difference between these electron orbitals.

By understanding the Rydberg formula, we gain insight into the spectral lines observed in atomic spectra. In the case of a hydrogen atom transitioning from a higher to a lower energy level, as with the transition from \( n_i = 5 \) to \( n_f = 3 \), scientists can predict and confirm the wavelength of emitted light, which, as noted earlier, falls in the near-infrared region.

Infrared Region

The infrared region of the electromagnetic spectrum lies just beyond the visible light range and includes wavelengths that range from about 700 nm to 1 mm. This region is further divided into several subcategories, such as near-infrared (700 nm to 1400 nm), mid-infrared, and far-infrared. These sub-ranges serve different purposes in technology and science.

Infrared wavelengths are typically characterized by their ability to warm materials and their relatively lower energy compared to visible light. Practical applications of the infrared region include:

• Thermal Imaging: Use of infrared sensors to detect heat and produce images of thermal radiation, often used in night vision equipment.
• Remote Controls: Operating at near-infrared wavelengths, these devices communicate commands to televisions and other electronics.
• Environmental Monitoring: Infrared sensors can detect heat emission from various objects, making them vital in meteorology and climatology for tracking weather patterns.
• Scientific Research: Infrared spectroscopy is a technique used to study molecular vibrations and structures, proving essential in both chemistry and astronomy.

In the specific case of a hydrogen atom absorption at a wavelength of 1094 nm, it fits comfortably within the near-infrared region. This position implies the transition involves a lower energy absorption typical of infrared light, explaining why such wavelengths are associated with warmth and less energetic radiation types.