Ch.8 - The Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 6th Edition

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Tro 6th Edition

Ch.8 - The Quantum-Mechanical Model of the Atom

Problema 76

Capitolo 8, Problema 76

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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Identify the formula to calculate the frequency of light emitted during an electron transition: \( \nu = \frac{\Delta E}{h} \), where \( \nu \) is the frequency, \( \Delta E \) is the change in energy, and \( h \) is Planck's constant.

Use the Rydberg formula to calculate the change in energy \( \Delta E \) for each transition: \( \Delta E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_H \) is the Rydberg constant, \( n_1 \) is the final energy level, and \( n_2 \) is the initial energy level.

For each transition, substitute the values of \( n_1 \) and \( n_2 \) into the Rydberg formula to find \( \Delta E \).

Substitute the calculated \( \Delta E \) and Planck's constant \( h \) into the frequency formula to find the frequency \( \nu \) for each transition.

Repeat the process for each transition: a. \( n = 4 \rightarrow n = 3 \), b. \( n = 5 \rightarrow n = 1 \), c. \( n = 5 \rightarrow n = 4 \), d. \( n = 6 \rightarrow n = 5 \).

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Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by quantum numbers (n). The energy associated with each level increases with n, and transitions between these levels result in the emission or absorption of light. The difference in energy between two levels determines the wavelength and frequency of the emitted light.

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Hydrogenation Reactions

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of spectral lines in hydrogen. It relates the wavelengths of emitted light to the initial and final energy levels of the electron transition. The formula is given by 1/λ = R_H(1/n1² - 1/n2²), where R_H is the Rydberg constant, and n1 and n2 are the principal quantum numbers of the lower and higher energy levels, respectively.

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Skeletal Formula

Frequency and Wavelength Relationship

The frequency (ν) of light is inversely related to its wavelength (λ) through the equation c = λν, where c is the speed of light. This relationship allows us to calculate the frequency of light emitted during electron transitions by first determining the wavelength using the Rydberg formula and then applying this equation to find the corresponding frequency.

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Frequency-Wavelength Relationship

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