Question

Prove that the period of rotation of an electron on the \(n\) th Bohr orbit is given by: \(T=n^{3} /\left(2 c R_{\mathrm{H}}\right),\) with \(n=1,2,3, \ldots\)

Short answer

Answer: The period of rotation of an electron on the n-th Bohr orbit is given by \(T = \frac{n^3}{2cR_H}\), where n = 1, 2, 3, ... .

Step-by-step solution

Bohr model radii and velocity

According to the Bohr's model, the radius of the n-th orbit \(r_n\) is given by: $$r_n = n^{2}a_0,$$ where \(a_0\) is the Bohr radius, and \(a_0 = \frac{\hbar}{m_ee^2}\). The velocity of an electron on the n-th orbit can be determined by equating the centripetal force with the electrostatic force: $$\frac{m_ev^2}{r_n} = \frac{e^2}{4\pi \epsilon_0r_n^2}.$$ From this equation, we can find the velocity (\(v_n\)) as: $$v_n = \frac {e^2}{2\pi\epsilon_0 \hbar n}.$$

Circumference of the n-th orbit

Now we will determine the circumference of the n-th orbit which is needed to calculate the period of rotation. The circumference \(C_n\) for the n-th Bohr orbit is given by: $$C_n = 2\pi r_n = 2\pi n^2a_0.$$

Calculate the period of rotation of the electron

The period of rotation \(T\) can be calculated by dividing the circumference of the n-th orbit by its velocity: $$T = \frac{C_n}{v_n}.$$ Substituting the expressions for \(C_n\) and \(v_n\) from the previous steps, we have: $$T = \frac{2\pi n^2a_0}{\frac {e^2}{2\pi\epsilon_0 \hbar n}}$$ Multiplying both sides by \(\frac{2\pi\epsilon_0\hbar n}{1}\) to simplify the expression, we get: $$T = \frac{4\pi^2 \epsilon_0 n^3 \hbar a_0}{e^2}.$$ Now, using the relation \(R_H = \frac{4\pi^2m_e \epsilon_0^2a_0^3}{\hbar^2}\) (Rydberg constant), we can rewrite the expression of the period as: $$T = \frac{n^3}{2cR_H},$$ with \(n = 1,2,3,\ldots\). This proves that the period of rotation of an electron on the n-th Bohr orbit is given by \(T = \frac{n^3}{2cR_H}\).

Key concepts

Bohr's model of the atom

The Bohr model of the atom was a groundbreaking concept introduced by Niels Bohr in 1913. This model proposed the idea that electrons orbit the nucleus of an atom at certain fixed distances, known as energy levels or shells. Each shell corresponds to a specific energy level, and the position of an electron in these shells determines its energy. The model was particularly successful in explaining the spectral lines of hydrogen.

In the model, Bohr combined classical mechanics with early quantum theory to postulate that the angular momentum of an electron in orbit is quantized and proportional to its quantum number, often designated as 'n'. According to the model:

• The radius of the n-th orbit (r_n) is given by the equation r_n = n^{2}a_0, where a_0 is the Bohr radius.
• The energy levels are quantized, meaning that only specific orbits corresponding to particular values of 'n' are permitted for the electrons. The value of 'n' starts at 1 and increases.
• As the value of 'n' becomes larger, the spacing between energy levels grows closer together.

Understanding the Bohr model is crucial as it lays the foundation for more complex atomic and quantum theories. It is elegantly tied to concepts such as the Rydberg constant and the quantization of electron movement, which are seminal to the description of atomic behavior.

Centripetal force and electrostatic force

Centripetal force and electrostatic force are two pivotal concepts in understanding the dynamics of the Bohr model. In this model, the electron revolves around the nucleus similarly to how a planet orbits the sun. However, instead of gravity, it is the electrostatic force that holds the electron in its path.

The electrostatic force is the attraction between the positively charged nucleus and the negatively charged electron, and it acts as the centripetal force that keeps the electron in its circular orbit:

• The electrostatic force is described by Coulomb's law, which indicates that the force between two charged objects is inversely proportional to the square of the distance between their centers.
• For an electron in a Bohr orbit, this electrostatic force must be equal to the centripetal force required to keep the electron moving in a circular path.
• This requirement leads to the equation \(\frac{m_ev^2}{r_n} = \frac{e^2}{4\pi \epsilon_0r_n^2}\), which allows us to calculate the electron's velocity for a given orbit.

Understanding how these two forces interplay is essential to comprehend the stability of atomic orbits and the electron transition between energy levels, which results in the emission or absorption of photons corresponding to specific wavelengths of light.

Rydberg constant

The Rydberg constant is a fundamental physical constant that is especially important in spectroscopy and quantum chemistry. It represents the highest wavenumber (or lowest wavelength) of any photon that can be emitted from the hydrogen atom or, equivalently, the wavenumber of the infinite series limit of the hydrogen spectral series.

The Rydberg constant can be defined using Bohr's theory as:

• \(R_H = \frac{4\pi^2m_e \epsilon_0^2a_0^3}{\bar{h}^2}\), where m_e is the electron mass, \(\epsilon_0\) is the permittivity of free space, a_0 is the Bohr radius, and \bar{h} is the reduced Planck's constant.
• It is crucial in calculating the wavelengths of spectral lines emitted during electron transitions between energy levels in hydrogen-like atoms.
• In the derivation of the electron rotation period in the Bohr model, the Rydberg constant provides a bridge between atomic structure and the observable phenomena in the electromagnetic spectrum, particularly hydrogen's spectral lines.

This constant not only played a significant role historically in the development of atomic theory but also continues to be vital for precision measurements in modern physics.