Problem 42 - Chemistry The Central Science

Question

The hydrogen atom can absorb light of wavelength \(2626 \mathrm{nm}\). (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

The absorption of light with a wavelength of \(2626 nm\) falls in the infrared region of the electromagnetic spectrum. By using the Rydberg formula and trial and error method, we can determine that the initial energy level (n_i) is 1, and the final energy level (n_f) is 2.

Step 1: (a) Determine the region of the electromagnetic spectrum corresponding to 2626 nm)

To determine the region of the electromagnetic spectrum, we first need to know the wavelength in meters. Given the wavelength of 2626 nm, we can convert it to meters using the following conversion factor: 1 nm = 1 x 10^(-9) m So, 2626 nm = 2626 x 10^(-9) m = 2.626 x 10^(-6) m Now, we can check the electromagnetic spectrum to find the region corresponding to this wavelength. The different regions and their approximate wavelength ranges are: - Radio waves: > 1 x 10^(-1) m - Microwaves: 1 x 10^(-3) - 1 x 10^(-1) m - Infrared: 7 x 10^(-7) - 1 x 10^(-3) m - Visible light: 4 x 10^(-7) - 7 x 10^(-7) m - Ultraviolet: 1 x 10^(-8) - 4 x 10^(-7) m - X-rays: 1 x 10^(-11) - 1 x 10^(-8) m - Gamma-rays: < 1 x 10^(-11) m Looking at these ranges, it can be concluded that the wavelength (2.626 x 10^(-6) m) falls in the infrared region of the electromagnetic spectrum.

Step 2: (b) Determine the initial and final values of n)

To find the initial and final energy levels, we will use the Rydberg formula for the hydrogen atom: \(1/λ = R_H(1/n^2_i - 1/n^2_f)\) Where: - λ = wavelength (in meters) = 2.626 x 10^(-6) m - R_H = Rydberg constant for hydrogen ≈ 1.097 x 10^7 m^(-1) - n_i = initial energy level - n_f = final energy level We are given the wavelength and asked to find n_i and n_f values. Since the hydrogen atom is absorbing light energy, the electron must be transitioning from a lower energy level (n_i) to a higher energy level (n_f). First, let's isolate \(n^2_i\) term in the Rydberg formula: \(n^2_i = 1/(λR_H + 1/n^2_f)\) Now we'll use the trial and error method to find the closest integer values of n_i and n_f that satisfy the equation: Let's start with n_f = 2. \(n^2_i = 1/(2.626\cdot10^{-6}\cdot1.097\cdot10^7 + 1/4)\) \(n^2_i ≈ 1/(4.86 + 0.25)\) \(n^2_i ≈ 1/5.11 = 0.195\) Now since we know that ni must be an integer value, we can deduce that ni = 1, as it is the closest integer value to 0.195. So, the initial energy level (n_i) is 1. To find the final energy level (n_f), we can use the rearranged Rydberg formula: \(n^2_f = 1/((1/λ) - (1/R_H)(1/n^2_i))\) \(n^2_f = 1/((1/(2.626\cdot10^{-6})) - (1/(1.097\cdot10^7))(1/1^2))\) \(n^2_f ≈ 1/(3.81\cdot10^5 - 1.097\cdot10^7)\) \(n^2_f ≈ 4.02\) Now, since n_f must also be an integer value, we can deduce that n_f = 2, as it is the closest integer value to 4.02. Therefore, the initial energy level n_i is 1 and the final energy level n_f is 2.

Key Concepts

Hydrogen Atom Energy Levels

The energy levels of an atom correspond to the specific energies that an electron can have within the atom. In the case of the hydrogen atom, which consists of only one electron, these energy levels are quantized. This means that the electron can only exist in certain, discrete energy states. When the electron transitions between these energy levels, it either absorbs or emits energy in the form of light or electromagnetic radiation.

For example, when a hydrogen atom absorbs energy, the electron moves from a lower energy level, denoted as \( n_i \) (initial level), to a higher one, \( n_f \) (final level). These energy levels are labeled with quantum numbers (n), where \( n = 1 \) is the ground state (lowest energy level), and \( n = 2, 3, 4,\ldots\) are excited states (higher energy levels).

• At \( n = 1 \), the electron is closest to the nucleus and has the lowest energy.
• As the quantum number increases (\( n = 2, 3,\ldots\)), the energy level increases, meaning the electron is further from the nucleus and has higher energy.

Understanding these energy levels is crucial for explaining how hydrogen atoms emit or absorb specific wavelengths of light, as described by the Rydberg formula.

Rydberg Formula

The Rydberg formula is a mathematical equation used to predict the wavelengths of light emitted or absorbed by electrons in a hydrogen atom. It helps us determine the possible transitions between different energy levels of the electron. This formula is especially useful because it provides a quantitative relationship between the wavelengths of light and the quantum numbers of the initial and final energy levels.

The Rydberg formula is written as:\[\frac{1}{\lambda} = R_H\left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right)\]Where:

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

When a hydrogen electron absorbs light, it moves from a lower energy level (\( n_i \)) to a higher one (\( n_f \)). Conversely, when emitting light, the transition is from a higher to a lower energy level. By substituting known wavelengths into the Rydberg formula, we can calculate the corresponding values of \( n_i \) and \( n_f \), helping us understand the nature of the electronic transitions.

Infrared Region

The infrared region of the electromagnetic spectrum includes wavelengths longer than visible light but shorter than microwaves, roughly spanning from \( 700 \) nm to \( 1 \) mm. Infrared light is generally experienced as heat and is not visible to the human eye.

In the context of hydrogen atom transitions, wavelengths that fall within the infrared range often result from electrons moving between lower energy levels. This is because transitions involving larger energy changes typically produce wavelengths in the visible or ultraviolet regions, while those involving smaller energy changes result in infrared light.

For example, if the wavelength is \( 2626 \) nm, as in the original exercise, it clearly falls in the infrared region. Here, the energy transition is enough to move the electron between nearby energy levels, such as from \( n_i = 1 \) to \( n_f = 2 \), without requiring the higher energy typical of ultraviolet or visible light transitions.

Understanding the infrared region's characteristics helps in identifying the type of electronic transitions occurring in an atom and is essential in spectroscopy and other fields that involve analyzing light interactions.