Problem 123 - Chemistry

Question

Scientists have found interstellar hydrogen atoms with quantum number \(n\) in the hundreds. Calculate the wavelength of light emitted when a hydrogen atom undergoes a transition from \(n=236\) to \(n=235 .\) In what region of the electromagnetic spectrum does this wavelength fall?

Step-by-Step Solution

Answer

The wavelength of light emitted when a hydrogen atom undergoes the transition from \(n=236\) to \(n=235\) and the corresponding region in the electromagnetic spectrum will be calculated based on the Rydberg formula and the data for electromagnetic spectrum.

Step 1: Understand the question and write down given values

The question is asking for the wavelength of the light emitted when a hydrogen atom transitions from \(n = 236\) to \(n = 235\). These `n` values are given.

Step 2: Apply the Rydberg formula

The Rydberg formula is \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\] where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant, and \( n_1, n_2 \) are the principal quantum numbers, respectively. Here, \( n_1 = 235 \) and \( n_2 = 236 \). The Rydberg constant \( R_H \) is \(1.097373 * 10^7 m^{-1}\). Plugging these values into the Rydberg formula allows us to calculate the wavelength.

Step 3: Find the wavelength

We can now solve the equation for the wavelength \( \lambda \). After inserting the given values and the Ryberg constant, we'll get a result.

Step 4: Identify the region in the electromagnetic spectrum

Using the calculated wavelength, we can then refer it with values of different regions of the electromagnetic spectrum to identify in which region our calculated wavelength falls.

Key Concepts

Hydrogen Atom Transition

When a hydrogen atom undergoes a transition, it moves from one energy level to another. This transition involves the atom's electron jumping between orbits or quantum levels, specifically defined by principal quantum numbers, denoted as \(n\). Each orbit corresponds to a certain energy state of the electron in the atom. These transitions are what lead to the emission or absorption of light.
In this context, the atom transitions from quantum level \(n = 236\) to \(n = 235\). While the change might seem minor, even small transitions in high quantum levels can result in significant energy changes. This energy change is released as a photon, which we observe as light.

• The principal quantum number \(n\) determines the size and energy of the electron orbit.
• A decrease in \(n\) signifies that the electron is dropping to a lower energy level, releasing energy in the form of light.
• The emitted light's wavelength is related to the difference in energy between these two levels.

Wavelength Calculation

The calculation of the wavelength of light emitted during an atom's electron transition is achieved through the Rydberg formula. This formula is especially useful for hydrogen atoms due to their simple structure, having only one electron. The Rydberg formula is expressed as:\[\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]Here, \(\lambda\) represents the wavelength of emitted light. \(R_H\) is the Rydberg constant, a vital factor in these calculations, valued at \(1.097373 \times 10^7 \text{ m}^{-1}\). In our situation:

• The initial quantum level \(n_2 = 236\) and the final level \(n_1 = 235\).
• Substitute these values into the formula to solve for \(\lambda\).

By calculating \(\lambda\), we find the precise wavelength and can further explore its implications in different spectrums.

Electromagnetic Spectrum Regions

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength. When calculating a wavelength, it's crucial to identify which part of the spectrum it fits into, as each region has unique characteristics and applications.
The spectrum ranges from gamma rays, with the shortest wavelengths, to radio waves, which have the longest. Between these extremes lies visible light, but there are also ultraviolet, infrared, microwave, X-rays, and more.
After determining the wavelength using the Rydberg formula, we can pinpoint the specific category it falls into:

• Wavelengths shorter than visible light fall into categories like X-rays or ultraviolet.
• Wavelengths longer than visible light may be infrared, microwaves, or radio waves.
• Knowing a wavelength's position helps in understanding its potential uses and dangers.

In our scenario, once the wavelength is calculated from the hydrogen transition, you match this value with known spectrum regions to identify where it belongs. This helps understand the nature of the radiation emitted.