Problem 58 - Chemistry: An Atoms-Focused Approach

Question

What is the frequency of the photons emitted by hydrogen atoms when they undergo \(n=5 \rightarrow n=3\) transitions? In which region of the electromagnetic spectrum does this radiation occur?

Step-by-Step Solution

Answer

Answer: The frequency of the emitted radiation is approximately \(1.92 \times 10^{14} Hz\), and it falls within the infrared region of the electromagnetic spectrum.

Step 1: Write down the Rydberg's formula for frequency

The Rydberg's formula for frequency can be written as: $$ \nu = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) $$ where \(R_H\) is the Rydberg constant for hydrogen (approximately \(3.29 \times 10^{15} Hz\)), \(n_1\) is the initial energy level, and \(n_2\) is the final energy level.

Step 2: Substitute the given energy levels into the formula

For our problem, we are given that the electron transitions from the \(n=5\) level to the \(n=3\) level. Substitute \(n_1 = 5\) and \(n_2 = 3\) into the formula: $$ \nu = R_H \left(\frac{1}{5^2} - \frac{1}{3^2}\right) $$

Step 3: Calculate the frequency

Now, plug in the value of \(R_H\) and perform the calculations: $$ \nu = (3.29 \times 10^{15}) \left(\frac{1}{25} - \frac{1}{9}\right) = (3.29 \times 10^{15}) \left(\frac{9-25}{225}\right) \approx 1.92 \times 10^{14} Hz $$

Step 4: Convert the frequency to wavelength

To identify the region of the electromagnetic spectrum, we need to find the wavelength of the radiation. Use the formula \(c = \lambda \nu\), where \(c\) is the speed of light (\(3 \times 10^8 m/s\)) and \(\lambda\) is the wavelength. Solve for \(\lambda\): $$ \lambda = \frac{c}{\nu} = \frac{3 \times 10^8}{1.92 \times 10^{14}} \approx 1.56 \times 10^{-6}m = 1560 nm $$

Step 5: Identify the region of the electromagnetic spectrum

The calculated wavelength of the emitted radiation (1560 nm) falls within the infrared region of the electromagnetic spectrum, which typically spans from 700 nm to 1 mm. The emitted radiation has a frequency of approximately \(1.92 \times 10^{14} Hz\) and falls within the infrared region of the electromagnetic spectrum.

Key Concepts

Understanding Hydrogen Atom Transitions

When electrons in a hydrogen atom transition between energy levels, they emit or absorb photons. These transitions are key to understanding how atoms release energy. The electron starts in an initial energy level, often labeled as \(n_1\), and moves to a final energy level, labeled as \(n_2\). Each transition correlates to a specific amount of energy.

• If \(n_1 > n_2\), the electron emits energy in the form of a photon.
• If \(n_1 < n_2\), the electron absorbs a photon and gains energy.

In our example, the electron transitions from the \(n=5\) level to the \(n=3\) level, emitting a photon with a specific frequency. This channeled energy is what provides insights into the spectral lines of hydrogen.

Exploring the Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, from gamma rays to radio waves. Each type of radiation has a unique wavelength and frequency, defining its place in the spectrum.

• Shorter wavelengths, like gamma rays, have higher frequencies and more energy.
• Longer wavelengths, like radio waves, have lower frequencies and less energy.

In the context of our problem, the calculated wavelength is 1560 nm. This places it in the infrared region of the spectrum.
Infrared radiation is often associated with heat and is invisible to the naked eye, but it plays a crucial role in various scientific applications.

Photon Frequency Calculation Using Rydberg Formula

To find the frequency of the photon emitted during a transition, we use the Rydberg formula. This formula is specifically used for hydrogen and involves the Rydberg constant \(R_H\). The formula is written as:
\[ u = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
By substituting \(n_1 = 5\) and \(n_2 = 3\) into this equation and using \(R_H = 3.29 \times 10^{15} \, \text{Hz}\), we calculated the frequency \(u\) to be approximately \(1.92 \times 10^{14} \, \text{Hz}\).
This frequency falls within the infrared region of the electromagnetic spectrum. Calculating frequencies helps us understand the type of radiation involved and has implications for fields like astronomy and quantum physics.