Ch.6 - Electronic Structure of Atoms

Brown - Chemistry: The Central Science 15th Edition

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Brown 15th Edition

Ch.6 - Electronic Structure of Atoms

Problem 89b

Chapter 6, Problem 89b

Consider a transition in which the electron of a hydrogen atom is excited from n = 1 to n = ∞. (b) What is the wavelength of light that must be absorbed to accomplish this process?

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Identify the initial and final energy levels of the electron transition. Here, the initial level is \( n_1 = 1 \) and the final level is \( n_2 = 3 \).

Use the Rydberg formula to calculate the energy difference between these two levels: \( \Delta E = -R_H \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) \), where \( R_H \) is the Rydberg constant for hydrogen, approximately \( 2.18 \times 10^{-18} \) J.

Calculate the energy \( \Delta E \) using the values of \( n_1 \) and \( n_2 \) in the Rydberg formula.

Convert the energy \( \Delta E \) to wavelength using the equation \( \lambda = \frac{hc}{\Delta E} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \) Js and \( c \) is the speed of light \( 3.00 \times 10^8 \) m/s.

Solve for \( \lambda \) to find the wavelength of light that must be absorbed for the electron transition from \( n = 1 \) to \( n = 3 \).

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Key Concepts

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

Quantum Energy Levels

In quantum mechanics, electrons in an atom occupy discrete energy levels, denoted by principal quantum numbers (n). For hydrogen, these levels are quantized, meaning an electron can only exist in specific states. The transition from a lower energy level (n=1) to a higher one (n=2, for example) involves the absorption of energy, which corresponds to the difference in energy between these levels.

Energy-Wavelength Relationship

The energy of a photon 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. This relationship indicates that shorter wavelengths correspond to higher energy photons. To find the wavelength of light absorbed during an electron transition, one must calculate the energy difference between the two quantum levels and then use this equation.

Rydberg Formula

The Rydberg formula provides a way to calculate the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms. It is expressed as 1/λ = R_H(1/n1² - 1/n2²), where R_H is the Rydberg constant, n1 and n2 are the principal quantum numbers of the lower and higher energy levels, respectively. This formula is essential for determining the wavelength of light absorbed when an electron transitions between specific energy levels.

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Textbook Question

In an experiment to study the photoelectric effect, a scientist measures the kinetic energy of ejected electrons as a function of the frequency of radiation hitting a metal surface. She obtains the following plot. The point labeled 'n0' corresponds to light with a wavelength of 542 nm. (a) What is the value of n0 in s - 1?

Textbook Question

The human retina has three types of receptor cones, each sensitive to a different range of wavelengths of visible light, as shown in this figure (the colors are merely to differentiate the three curves from one another; they do not indicate the actual colors represented by each curve):

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Textbook Question

Consider a transition in which the electron of a hydrogen atom is excited from n = 1 to n = ∞. (a) What is the end result of this transition?

Textbook Question

The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed.

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Textbook Question

The watt is the derived SI unit of power, the measure of energy per unit time: 1 W=1 J/s. A semiconductor laser in a CD player has an output wavelength of 780 nm and a power level of 0.10 mW. How many photons strike the CD surface during the playing of a CD 69 minutes in length?

Textbook Question

Consider a transition in which the electron of a hydrogen atom is excited from n = 1 to n = ∞. (d) How are the results of parts (b) and (c) related to the plot shown in Exercise 6.88?