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In the Bohr model of the hydrogen atom, what is the de Broglie wavelength of the electron when it is in (a) the \(n\) = 1 level and (b) the \(n\) = 4 level? In both cases, compare the de Broglie wavelength to the circumference 2\(\pi{r_n}\) of the orbit.

Short Answer

Expert verified
The de Broglie wavelengths are 3.32 × 10⁻¹⁰ m for n=1 and 1.32 × 10⁻⁹ m for n=4. The wave matches the circumference for n=1, but is shorter for n=4.

Step by step solution

01

Understanding the Bohr Model

The Bohr model of the hydrogen atom describes electrons in quantized orbits around the nucleus. Each orbit corresponds to a specific energy level, denoted by the principal quantum number \(n\).
02

de Broglie Wavelength Formula

The de Broglie wavelength \(\lambda\) of a particle is given by \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\), \(m\) is the mass of the electron \(9.109 \times 10^{-31} \text{ kg}\), and \(v\) is the velocity of the electron.
03

Calculating Electron Velocity for Orbit

For the Bohr model, the velocity \(v\) of an electron in orbit \(n\) is given by \(v = \frac{e^2}{2\varepsilon_0 h}\cdot\frac{1}{n}\), where \(e\) is the elementary charge \(1.602 \times 10^{-19} \text{ C}\), and \(\varepsilon_0\) is the vacuum permittivity \(8.854 \times 10^{-12} \text{ C}^2/\text{N} \cdot m^2\).
04

Electron Velocity at n=1

For \(n=1\), substitute the known values into the formula to calculate \(v\). \[v_{n=1} = \frac{1.602^2 \times 10^{-38}}{2 \times 8.854 \times 10^{-12} \times 6.626 \times 10^{-34}} = 2.18 \times 10^6 \text{ m/s}.\]
05

de Broglie Wavelength for n=1

Use the calculated velocity to find the de Broglie wavelength for \(n=1\). Substitute values in \(\lambda = \frac{h}{mv}\).\[\lambda_{n=1} = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 2.18 \times 10^6} \approx 3.32 \times 10^{-10} \text{ m}.\]
06

Circumference for n=1

The circumference for \(n=1\) is given by \(2\pi r_n\), where \(r_n = \frac{n^2 h^2 \varepsilon_0 }{\pi m e^2}\). Calculate value when \(n=1\).\[r_1 = \frac{1^2 \times 6.626^2 \times 10^{-68} \times 8.854 \times 10^{-12}}{\pi \times 9.109 \times 10^{-31} \times 1.602^2 \times 10^{-38}} \approx 5.29 \times 10^{-11} \text{ m}\]\[C_{n=1} = 2\pi \times 5.29 \times 10^{-11} = 3.32 \times 10^{-10} \text{ m}.\]
07

Electron Velocity at n=4

For \(n=4\), apply the velocity formula.\[v_{n=4} = \frac{1.602^2 \times 10^{-38}}{2 \times 8.854 \times 10^{-12} \times 6.626 \times 10^{-34} \times 4} = 5.45 \times 10^5 \text{ m/s}.\]
08

de Broglie Wavelength for n=4

Calculate the de Broglie wavelength for \(n=4\).\[\lambda_{n=4} = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 5.45 \times 10^5} \approx 1.32 \times 10^{-9} \text{ m}.\]
09

Circumference for n=4

Calculate the circumference for \(n=4\). \[r_4 = \frac{4^2 \times 6.626^2 \times 10^{-68} \times 8.854 \times 10^{-12}}{\pi \times 9.109 \times 10^{-31} \times 1.602^2 \times 10^{-38}} \approx 8.468 \times 10^{-10} \text{ m}\]\[C_{n=4} = 2\pi \times 8.468 \times 10^{-10} \approx 5.316 \times 10^{-9} \text{ m}.\]
10

Comparing Wavelength and Circumference

For both \(n=1\) and \(n=4\), compare the de Broglie wavelength to the orbit's circumference. For \(n=1\), the wavelength and circumference approximately match, as expected in the Bohr model. For \(n=4\), the wavelength is much smaller than the circumference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie Wavelength
The de Broglie wavelength is a fundamental concept in quantum mechanics. It describes the wave-like nature of particles such as electrons. According to de Broglie, every moving particle or object has an associated wavelength, given by the formula \( \lambda = \frac{h}{mv} \). Here, \( \lambda \) represents the wavelength, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\), \( m \) is the mass of the particle, and \( v \) is its velocity.

This concept is crucial in understanding how particles can exhibit properties of both particles and waves. In the Bohr model, the de Broglie wavelength helps explain why electrons occupy specific orbital paths around the nucleus. These paths correspond to integer multiples of the electron's wavelength, ensuring the wave pattern is complete and stable. This integrative view accounts for the electron's stability and the discrete energy levels observed in atoms.
Principal Quantum Number
The principal quantum number, denoted by \( n \), is a key factor in the Bohr model and quantum mechanics. It represents the different energy levels available to an electron within an atom and essentially dictates the electron's orbit and energy.

The value of \( n \) can be a positive integer (1, 2, 3, etc.), with each number indicating a different energy level or "shell." The lower the value of \( n \), the lower the energy level, and the closer the electron is to the nucleus. The higher the \( n \), the higher the energy and the further the electron's orbit from the nucleus. For instance, when an electron is in the \( n = 1 \) level, it's in the ground state, which is its lowest energy state. Conversely, \( n = 4 \) represents a higher energy level.

The principal quantum number also influences the electron's velocity and the radius of its orbit. It is a critical component in understanding electron arrangement and behavior in atoms.
Electron Velocity
Electron velocity in the Bohr model is vital in determining the behavior of the electron within its orbit. The velocity of an electron orbiting the nucleus is influenced by the principal quantum number \( n \). It can be calculated using the formula:

\[ v = \frac{e^2}{2\varepsilon_0 h}\cdot\frac{1}{n} \]

Here, \( e \) is the elementary charge, \( \varepsilon_0 \) is the vacuum permittivity, and \( h \) is Planck's constant.

From this formula, we observe that the electron's velocity decreases as the principal quantum number \( n \) increases. In simpler terms, electrons in higher energy levels \( (n = 4) \) move slower compared to those in lower levels \( (n = 1) \). This change in speed reflects how the attractive force from the nucleus lessens with increased distance, affecting the electron's kinetic energy and stability within its orbit.
Electron Orbit
Electron orbit refers to the specific path or "shell" that an electron follows around the nucleus in the Bohr model. This model visualizes electrons as occupying fixed orbits, much like planets revolving around the sun. Each orbit corresponds to a quantized energy level, determined by the principal quantum number \( n \).

The electron orbit’s radius is not arbitrary but follows from the formula:

\[ r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} \]

Here, \( r_n \) is the radius of the orbit for a given \( n \). It shows that the radius increases with the square of \( n \), meaning higher energy levels are further from the nucleus.

Moreover, the circumference of the electron's orbit at any energy level \( n \) is related to its de Broglie wavelength, reinforcing the concept that only specific orbits are allowed where the electron's wave nature results in constructive interference. This determines the stable orbits of electrons as seen in reality slightly collapsing into the expected circumference, such as at \( n = 1 \) and \( n = 4 \). The understanding of electron orbits is fundamental for explaining atomic structure and spectra.

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Most popular questions from this chapter

Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers 0.25 \(\mu\)m thick using pulses lasting 12.0 ns from a laser beam of wavelength 193 nm. Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. (a) In what part of the electromagnetic spectrum does this light lie? (b) What is the energy of a single photon? (c) If a 1.50-mW beam is used, how many photons are delivered to the lens in each pulse?

(a) For one-electron ions with nuclear charge Z, what is the speed of the electron in a Bohr-model orbit labeled with \(n\)? Give your answer in terms of \(v_1\), the orbital speed for the \(n\) = 1 Bohr orbit in hydrogen. (b) What is the largest value of Z for which the \(n\) = 1 orbital speed is less than 10\(\%\) of the speed of light in vacuum?

(a) If a photon and an electron each have the same energy of 20.0 eV, find the wavelength of each. (b) If a photon and an electron each have the same wavelength of 250 nm, find the energy of each. (c) You want to study an organic molecule that is about 250 nm long using either a photon or an electron microscope. Approximately what wavelength should you use, and which probe, the electron or the photon, is likely to damage the molecule the least?

A triply ionized beryllium ion, Be\(^{3+}\) (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. (a) What is the ground-level energy of Be\(^{3+}\)? How does this compare to the ground-level energy of the hydrogen atom? (b) What is the ionization energy of Be\(^{3+}\)? How does this compare to the ionization energy of the hydrogen atom? (c) For the hydrogen atom, the wavelength of the photon emitted in the \(n\) = 2 to \(n\) = 1 transition is 122 nm (see Example 39.6). What is the wavelength of the photon emitted when a Be\(^{3+}\) ion undergoes this transition? (d) For a given value of \(n\), how does the radius of an orbit in Be\(^{3+}\) compare to that for hydrogen?

A sample of hydrogen atoms is irradiated with light with wavelength 85.5 nm, and electrons are observed leaving the gas. (a) If each hydrogen atom were initially in its ground level, what would be the maximum kinetic energy in electron volts of these photoelectrons? (b) A few electrons are detected with energies as much as 10.2 eV greater than the maximum kinetic energy calculated in part (a). How can this be?

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