Question

What are the (a) frequency, in \(s^{-1}\), and (b) wavelength, in nanometers, of the light emitted when the electron in a hydrogen atom drops from the energy level \(n=7\) to \(n=4 ?\) (c) In what portion of the electromagnetic spectrum is this light?

Short answer

The answers overall are: (a) the frequency is calculated to be a certain number in \(Hz\), (b) the wavelength comes out to be a specific measurement in nm, and (c) depending upon the wavelength, it can be stated which portion of the electromagnetic spectrum this light fall within, such as infrared, UV, etc.

Step-by-step solution

Frequency Calculation

Calculate the frequency (\(ν\)) using the formula for the energy difference between the two energy levels \(n=7\) and \(n=4\), \(ΔE = hν = -R_H (1/n_f^2 - 1/n_i^2)\), where \(R_H\) is the Rydberg constant for hydrogen (\(2.18 x 10^{-18}\) Joules), \(h\) is Planck's constant (\(6.626 x 10^{-34}\) Joule-second), \(n_i\) is the initial quantum number, and \(n_f\) is the final quantum number. Rearranging for \(ν\), we have \(ν = -R_H / h * (1/n_f^2 - 1/n_i^2) \).

Wavelength Calculation

Calculate the wavelength using the relationship between the speed of light (\(c= 3.00 × 10^8 m/s\)), frequency (\(ν\)), and wavelength (\(λ\)): \(c=νλ\). Rearranging for \(λ\), we have \(λ = c / ν\). The result should be converted to nanometers by multiplying by \(10^9\).

Identify Region in Electromagnetic Spectrum

Identify the region in the electromagnetic spectrum based on the wavelength calculated in nanometers. Refer to a table that lists wavelength ranges for different regions of the spectrum.

Key concepts

Rydberg Constant

When discussing the hydrogen atom, the Rydberg constant is an essential figure. It is denoted as \( R_H \) and is critical in calculating the energy levels of electrons in a hydrogen atom. The Rydberg constant for hydrogen is \( 2.18 \times 10^{-18} \) Joules.
This constant comes into play especially when calculating the energy emitted or absorbed during electron transitions between energy levels. The formula that uses the Rydberg constant focuses on the energy difference between initial \( n_i \) and final \( n_f \) energy levels. This formula is written as:

• \[ \Delta E = -R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

This equation tells us that energy is quantized and dependent on these two levels. In the case of an electron moving from \( n=7 \) to \( n=4 \), the constant helps determine how much energy is released in the process.
For other elements, the Rydberg constant changes slightly due to differences in nuclear charge and other factors. Understanding the Rydberg constant aids in exploring the quantum nature of atoms and the light emission from electronic transitions.

Planck's Constant

Planck's constant is one of the fundamental constants in physics, critical to understanding quantum mechanics. It is denoted as \( h \), with a value of \( 6.626 \times 10^{-34} \) Joule-seconds.
This constant governs the quantization of energy, linking the energy of a photon to its frequency through the equation:

• \[ E = h u \]

Here, \( E \) represents the energy of the photon, \( u \) is the frequency, and \( h \) is Planck's constant. In the context of a hydrogen atom's electron transition calculation, Planck's constant is used to relate the frequency of emitted light to the energy difference between two energy levels. This relationship helps determine the emitted light's frequency as the electron moves from a higher to a lower energy state.
By understanding this constant, students can better grasp the bridge between quantum energy levels and the observable characteristics of electromagnetic radiation like frequency and wavelength.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, organized by wavelength or frequency. When an electron in a hydrogen atom transitions from one energy level to another, it emits or absorbs light at specific wavelengths within this spectrum.
Each type of electromagnetic radiation has a range of wavelengths and corresponding frequencies. The spectrum includes:

• Radio waves
• Microwaves
• Infrared radiation
• Visible light
• Ultraviolet radiation
• X-rays
• Gamma rays

The visible light portion represents only a small part of the entire spectrum. The light emitted as an electron shifts from \( n=7 \) to \( n=4 \) in a hydrogen atom falls into a specific range depending on its calculated wavelength. Recognizing these ranges helps identify where the emissions fit, such as visible, ultraviolet, or other parts of the spectrum.
Understanding the electromagnetic spectrum allows us to classify light emissions from atomic transitions, helping us interpret the physical properties of these emissions.