Ch.6 - Electronic Structure of Atoms

Brown - Chemistry: The Central Science 14th Edition

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Brown 14th Edition

Ch.6 - Electronic Structure of Atoms

Problem 44b

Chapter 6, Problem 44b

The hydrogen atom can absorb light of wavelength 1094 nm. (b) Determine the final value of n associated with this absorption.

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Identify the formula to use: The Rydberg formula for hydrogen absorption 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 (1.097 \(\times\) 10^7 \(\text{ m}\)^{-1}), \( n_1 \) is the initial energy level, and \( n_2 \) is the final energy level.

Convert the wavelength from nanometers to meters: Since 1 nm = 1 \(\times\) 10^{-9} m, convert 1094 nm to meters.

Assume the initial energy level \( n_1 \) is 1, as the problem does not specify it, and this is a common assumption for absorption from the ground state.

Rearrange the Rydberg formula to solve for \( n_2 \): \( n_2 = \sqrt{\frac{1}{\frac{1}{n_1^2} - \frac{\lambda}{R_H}}} \).

Substitute the known values into the equation: Use the converted wavelength and the Rydberg constant to calculate \( n_2 \).

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Key Concepts

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

Energy of Photons

The energy of a photon is directly 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. For the hydrogen atom, when it absorbs a photon, the energy from the photon is used to excite an electron to a higher energy level.

Quantum Energy Levels

In a hydrogen atom, electrons occupy discrete energy levels, denoted by the principal quantum number n. The energy associated with each level can be calculated using the formula E_n = -13.6 eV/n². When a photon is absorbed, the electron transitions from a lower energy level (n_initial) to a higher one (n_final), and the difference in energy corresponds to the energy of the absorbed photon.

Balmer and Rydberg Formulas

The Rydberg formula allows for the calculation of the wavelengths of spectral lines in hydrogen and is given by 1/λ = R_H(1/n_final² - 1/n_initial²). This formula is essential for determining the final energy level (n_final) after absorption, as it relates the wavelength of light absorbed to the transition between quantum states of the hydrogen atom.

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