Question

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

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

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 \(n\).

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.

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 \(8.854 \times 10^{-12} \text{ C}^2/\text{N} \cdot m^2\).

Electron Velocity at n=1

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

de Broglie Wavelength for n=1

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

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 \(n=1\).\[r_1 = \frac{1^2 \times 6.626^2 \times 10^{-68} \times 8.854 \times 10^{-12}}{\pi \times 9.109 \times 10^{-31} \times 1.602^2 \times 10^{-38}} \approx 5.29 \times 10^{-11} \text{ m}\]\[C_{n=1} = 2\pi \times 5.29 \times 10^{-11} = 3.32 \times 10^{-10} \text{ m}.\]

Electron Velocity at n=4

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

de Broglie Wavelength for n=4

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

Circumference for n=4

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

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.

Key concepts

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.