a. Using Equation 6.5, calculate the energy of an electron in the hydrogen atom when n = 2 and when n = 6. Calculate the wavelength of the radiation released when an electron moves from n = 6 to n = 2.

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Identify the formula for the energy of an electron in a hydrogen atom: $E_n = -\frac{R_H}{n^2}$, where $R_H$ is the Rydberg constant (2.18 \times 10^{-18} \text{ J}) and $n$ is the principal quantum number.

Calculate the energy of the electron when $n = 2$ using the formula: $E_2 = -\frac{R_H}{2^2}$.

Calculate the energy of the electron when $n = 6$ using the formula: $E_6 = -\frac{R_H}{6^2}$.

Determine the energy difference between the two states: $\Delta E = E_2 - E_6$. This energy difference corresponds to the energy of the photon released.

Use the relationship between energy and wavelength: $\Delta E = \frac{hc}{\lambda}$, where $h$ is Planck's constant (6.626 \times 10^{-34} \text{ J s}) and $c$ is the speed of light (3.00 \times 10^8 \text{ m/s}), to solve for the wavelength $\lambda$.

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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 of an electron in these levels can be calculated using the formula E_n = -13.6 eV/n², where E_n is the energy at level n. This means that as n increases, the energy becomes less negative, indicating that the electron is less tightly bound to the nucleus.

Photon Emission and Wavelength

When an electron transitions between energy levels, it can emit or absorb a photon, which corresponds to the energy difference between the two levels. The energy of the emitted photon can be calculated using the equation ΔE = E_initial - E_final. The wavelength of the emitted radiation can then be determined using the equation λ = hc/ΔE, where h is Planck's constant and c is the speed of light.

Planck's Constant and the Speed of Light

Planck's constant (h) is a fundamental constant that relates the energy of a photon to its frequency, given by E = hf, where f is the frequency. The speed of light (c) is the speed at which light travels in a vacuum, approximately 3.00 x 10^8 m/s. These constants are essential for converting energy differences into wavelengths, allowing for the calculation of the wavelength of emitted radiation during electron transitions.

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Textbook Question

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? a. from n = 4 to n = 2

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Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? b. from an orbit of radius 2.12 Å to one of radius 8.46 Å

Textbook Question

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: a. from n = 3 to n = 6

Textbook Question

Consider a transition of the electron in the hydrogen atom from n = 4 to n = 9. b. Will the light be absorbed or emitted?

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Textbook Question

The visible emission lines observed by Balmer all involved nf = 2. (a) Which of the following is the best explanation of why the lines with nf = 3 are not observed in the visible portion of the spectrum: (i) Transitions to nf = 3 are not allowed to happen, (ii) transitions to nf = 3 emit photons in the infrared portion of the spectrum, (iii) transitions to nf = 3 emit photons in the ultraviolet portion of the spectrum, or (iv) transitions to nf = 3 emit photons that are at exactly the same wavelengths as those to nf = 2.

Textbook Question

The visible emission lines observed by Balmer all involved nf = 2. (b) Calculate the wavelengths of the first three lines in the Balmer series—those for which ni = 3, 4, and 5—and identify these lines in the emission spectrum shown in Figure 6.11.