Ch.7 - Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.7 - Quantum-Mechanical Model of the Atom

Problem 71

Chapter 7, Problem 71

An electron in the n = 7 level of the hydrogen atom relaxes to a lower-energy level, emitting light of 397 nm. What is the value of n for the level to which the electron relaxed?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134112831

Verified step by step guidance

Identify the given values: initial energy level (n_i = 7) and wavelength of emitted light (λ = 397 nm).

Use the Rydberg formula for hydrogen to relate the initial and final energy levels to the wavelength of the emitted light: \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), where \( R_H \) is the Rydberg constant (approximately 1.097 x 10^7 m^-1).

Rearrange the Rydberg formula to solve for the final energy level (n_f): \( \frac{1}{n_f^2} = \frac{1}{n_i^2} + \frac{1}{\lambda R_H} \).

Calculate the value of \( \frac{1}{n_f^2} \) using the given wavelength and the Rydberg constant.

Take the square root of the reciprocal of the calculated value to find the final energy level (n_f).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by the principal quantum number n. The energy levels are quantized, meaning electrons can only exist in specific states. The energy difference between these levels determines the wavelength of light emitted or absorbed when an electron transitions between them.

Wavelength and Energy Relationship

The energy of a photon emitted during an electron transition is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. A shorter wavelength corresponds to higher energy, while a longer wavelength indicates lower energy. This relationship is crucial for calculating the energy difference between the initial and final states of the electron.

Rydberg Formula

The Rydberg formula provides a way to calculate the wavelengths of spectral lines in hydrogen and is given by 1/λ = R_H (1/n1² - 1/n2²), where R_H is the Rydberg constant, n1 is the lower energy level, and n2 is the higher energy level. By rearranging this formula, one can determine the principal quantum number of the lower energy level (n1) when the wavelength of emitted light is known.

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Textbook Question

Determine whether each transition in the hydrogen atom corresponds to absorption or emission of energy. a. n = 3 → n = 1 b. n = 2 → n = 4 c. n = 4 → n = 3

The human eye contains a molecule called 11-cis-retinal that changes shape when struck with light of sufficient energy. The change in shape triggers a series of events that results in an electrical signal being sent to the brain that results in vision. The minimum energy required to change the conformation of 11-cis-retinal within the eye is about 164 kJ/mol. Calculate the longest wavelength visible to the human eye.

According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength: 3p → 2s or 4p → 3p?

An argon ion laser puts out 5.0 W of continuous power at a wavelength of 532 nm. The diameter of the laser beam is 5.5 mm. If the laser is pointed toward a pinhole with a diameter of 1.2 mm, how many photons travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. (1 W = 1 J/s)

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Textbook Question

Calculate the frequency of the light emitted when an electron in a hydrogen atom makes each transition: a. n = 4 → n = 3 b. n = 5 → n = 1 c. n = 5 → n = 4 d. n = 6 → n = 5

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