Chapter 6: Problem 161

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge \(q=+Z e\). Derive the formulas for the energy \(E_{n}\) of the electron in \(n\) th orbit and the orbital radius \(r_{n}\).

Short Answer

Expert verified

The energy levels and orbital radii for a hydrogen-like ion with charge q = Z*e can be derived using the Bohr model, Coulomb force, and Newton's second law. The final formulas are: Energy levels: \[E_n = -\frac{Z^2e^4m_e}{32\pi^2\epsilon_0^2\hbar^2n^2}\] Orbital radii: \[r_n = \frac{n\hbar^2}{2\pi Ze^2\epsilon_0 m_e}\]

Step by step solution

Write down the Bohr model formula

The Bohr model provides a formula for the quantization of angular momentum in a hydrogen-like ion with charge q = Z*e. According to the Bohr model, the angular momentum L of the electron in orbit n is: \[L_n = n\hbar\] Where n is the orbit number (positive integer), and \(\hbar\) is the reduced Planck constant, given as \(\hbar = \frac{h}{2\pi}\), where h is the Planck constant.

Write down the electron energy formula

The energy of the electron in orbit n is given by the sum of its kinetic energy (K) and potential energy (U). Since the electron moves in a circular orbit around the nucleus, we can write: \[E_n = K_n + U_n\] Where E_n, K_n, and U_n represent the total energy, kinetic energy, and potential energy of the electron in the n-th orbit, respectively.

Write down the Coulomb force formula

The attractive force between the electron and the nucleus can be described using the Coulomb force formula: \[F = \frac{Ze^2}{4\pi\epsilon_0 r^2}\] Where F is the force, Z is the atomic number, e is the fundamental charge, \(\epsilon_0\) is the vacuum permittivity, and r is the distance between the electron and the nucleus.

Use Newton's second law

In the n-th orbit, the electron is in a circular motion due to the Coulomb force, so we can use Newton's second law (F = ma) to write the centripetal force equation: \[F = m_ev^2/r_n\] Where \(m_e\) is the electron mass, and \(v\) is the speed of the electron in the n-th orbit.

Derive expressions for electron speed and radius

Use the Bohr model formula and Newton's second law to eliminate the force from the momentum and speed equations, and obtain expressions for the electron speed and radius in the n-th orbit: \[v_n = \frac{Ze^2}{4\pi\epsilon_0 n\hbar}\] \[r_n = \frac{n\hbar^2}{2\pi Ze^2\epsilon_0 m_e}\]

Derive the formula for energy in the n-th orbit

Substitute the derived expressions for the speed and radius into the energy equation (E_n = K_n + U_n). Then simplify the equation to obtain the general formula for the energy levels of a hydrogen-like ion: \[E_n = -\frac{Z^2e^4m_e}{32\pi^2\epsilon_0^2\hbar^2n^2}\]

Write the final formulas for the energy levels and orbital radii

Now we have derived the general formulas for the energy levels and the orbital radii of a hydrogen-like ion with charge q = Z*e: Energy levels: \[E_n = -\frac{Z^2e^4m_e}{32\pi^2\epsilon_0^2\hbar^2n^2}\] Orbital radii: \[r_n = \frac{n\hbar^2}{2\pi Ze^2\epsilon_0 m_e}\]

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Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge \(q=+Z e\). Derive the formulas for the energy \(E_{n}\) of the electron in \(n\) th orbit and the orbital radius \(r_{n}\).

Short Answer

Step by step solution

Write down the Bohr model formula

Write down the electron energy formula

Write down the Coulomb force formula

Use Newton's second law

Derive expressions for electron speed and radius

Derive the formula for energy in the n-th orbit

Write the final formulas for the energy levels and orbital radii

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