Problem 57 - Chemistry: An Atoms-Focused Approach

Question

What is the wavelength of the photons emitted by hydrogen atoms when they undergo \(n=4 \rightarrow n=3\) transitions? In which region of the electromagnetic spectrum does this radiation occur?

Step-by-Step Solution

Answer

Answer: The wavelength of the emitted photons is approximately \(1.88\times10^{-6} \, \text{m}\) and it belongs to the infrared region of the electromagnetic spectrum.

Step 1: Understand the Rydberg formula for hydrogen

The Rydberg formula for hydrogen 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 radiation, \(R_H\) is the Rydberg constant for hydrogen (\(R_H = 1.097373\times10^7 \, \text{m}^{-1}\)), and \(n_1\) and \(n_2\) are the initial and final energy levels, respectively.

Step 2: Plug in the values for the transition

In our case, the transition is \(n=4 \rightarrow n=3\). Hence, \(n_1 = 4\) and \(n_2 = 3\). Plug these values into the equation: \( \frac{1}{\lambda} = R_H\left(\frac{1}{3^2}-\frac{1}{4^2}\right) \)

Step 3: Calculate the wavelength

Now, we calculate the wavelength by solving the equation: \( \frac{1}{\lambda} = R_H\left(\frac{1}{9}-\frac{1}{16}\right) \) \( \frac{1}{\lambda} = 1.097373\times10^7 \, \text{m}^{-1} \times \left(\frac{7}{144}\right) \) \( \lambda = \frac{1}{1.097373\times10^7 \, \text{m}^{-1} \times \frac{7}{144} } \) \( \lambda \approx 1.88\times10^{-6} \, \text{m} \)

Step 4: Identify the region in the electromagnetic spectrum

Now that we have obtained the wavelength of the emitted radiation, we can identify which region of the electromagnetic spectrum it belongs to. Wavelengths in the range of \(7\times10^{-7}\) to \(10^{-3}\) meters belong to the infrared region. Since \(1.88\times10^{-6} \, \text{m}\) falls within this range, the emitted radiation belongs to the infrared region of the electromagnetic spectrum.

Key Concepts

Rydberg Formula

The Rydberg Formula is key to understanding how hydrogen atoms emit light during electron transitions between energy levels. Initially formulated by Johannes Rydberg in the late 19th century, this formula calculates the wavelength of light emitted when an electron jumps between orbits in a hydrogen atom. The formula is expressed as:\[ \frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) \]Where:

• \(\lambda\) is the wavelength of emitted radiation,
• \(R_H\) is the Rydberg constant, approximately \(1.097373 \times 10^7 \, \text{m}^{-1}\),
• \(n_1\) and \(n_2\) represent the principal quantum numbers of the initial and final energy states.

For a transition from a higher energy level \(n_1\) to a lower level \(n_2\), the emitted photon has energy, and hence wavelength, determined by the difference in these energy levels. This formula applies specifically to hydrogen, but the concept can be adjusted for other elements.
The formula's simplicity hides the complexity and beauty of quantum mechanics governing the micro-world of atoms.

Electromagnetic Spectrum

The Electromagnetic Spectrum encompasses all types of electromagnetic radiation. Radiation in this spectrum travels in waves characterized by their wavelength and frequency. The spectrum includes a variety of radiation types from radio waves with the longest wavelengths to gamma rays with the shortest.

• Radio Waves: Used for communication, travel long distances and penetrate through obstacles.
• Microwaves: Ideal for cooking and radar technology.
• Infrared (IR): Experienced as heat, used in heaters and night-vision equipment.
• Visible Light: The only part of the spectrum visible to the human eye, consists of all the colors we see.
• Ultraviolet (UV): Responsible for sunburns, used in sterilization.
• X-rays: Penetrates soft tissue, crucial in medical imaging.
• Gamma Rays: Highly penetrating, used in treating cancer.

Each type of radiation has unique uses and interactions with matter, with applications spanning multiple fields from medicine to communication. Understanding this spectrum enables control and utilization across many technologies.

Infrared Radiation

Infrared Radiation is a type of electromagnetic radiation with wavelengths longer than visible light but shorter than microwaves. This places it in the wavelength range of approximately \(7 \times 10^{-7}\) to \(10^{-3}\) meters.

• Role: Often associated with heat, because objects at room temperature emit most of their thermal radiation in the infrared spectrum.
• Uses: This form of radiation is frequently used in night-vision technology, meteorology, and astronomical observations.
• Subdivisions: The infrared spectrum is sometimes divided further into near, mid, and far-infrared.
• Detection: Special sensors are required to "see" infrared light, as it is not visible to the naked eye.

In the hydrogen emission spectrum, infrared radiation occurs when electrons fall to lower energy states, emitting photons in this wavelength range. These transitions account for much of the heat we feel from distant sources, demonstrating how integral infrared radiation is to both technology and our natural environment.