Question

Following the steps used in the text for the hydrogen atom, apply the Bohr model of the atom to derive an expression for a) the radius of the \(n\) th orbit, b) the speed of the electron in the \(n\) th orbit, and c) the energy levels in a hydrogen-like ionized atom with charge number \(Z\) that has lost all of its electrons except for one. Compare the results with the corresponding ones for the hydrogen atom.

Short answer

Question: Using the Bohr model for hydrogen-like ionized atoms, derive expressions for the radius, speed, and energy levels of the nth orbit and compare the results to the hydrogen atom. Answer: For hydrogen-like ionized atom with charge number Z, the expressions derived for radius, speed and energy levels of the nth orbit are: a) Radius, \(a_n\) = \(\frac{n^2\hbar^2}{m_ee^2Z}4\pi \epsilon_0\) b) Speed, \(v_n\) = \(\frac{Ze^2}{n\hbar}\frac{1}{4\pi\epsilon_0}\) c) Energy level, \(E_n\) = -\(\frac{m_ee^4Z^2}{2n^2\hbar^2(4\pi\epsilon_0)^2}\) When comparing these expressions to the hydrogen atom (with Z = 1), they are indeed consistent with the expressions derived for hydrogen atoms, confirming the validity of the applied Bohr model.

Step-by-step solution

Recall the Bohr Model Assumptions

In the Bohr model, there are three main assumptions: 1) Electrons revolve around the nucleus in stable orbits with a fixed radius (a), 2) Only specific angular momenta (mvr) are allowed, and 3) An electron won't radiate electromagnetic energy while in a stable orbit.

Write the Equations for Centripetal Force and Coulomb Force

For a hydrogen-like ionized atom with charge number \(Z\), the centripetal force is given by: \(F_c = \frac{m_ev^2}{a_n}\) and the Coulomb force is given by: \(F_e = \frac{e^2Z}{4\pi \epsilon_0 a_n^2}\).

Equate Centripetal Force and Coulomb Force for nth Orbit

Since the electron in the circular orbit is in equilibrium, the centripetal force is balanced by the Coulomb force: \(\frac{m_ev^2}{a_n} = \frac{e^2Z}{4\pi \epsilon_0 a_n^2}\). a)

Derive the Expression for the Radius of nth Orbit

We can rearrange the equation to find \(a_n\), the radius of the \(n\)th orbit: \(a_n = \frac{n^2\hbar^2}{m_ee^2Z}4\pi \epsilon_0\), where \(\hbar = \frac{h}{2\pi}\), \(e\) is the elementary charge, \(m_e\) is the mass of the electron, and \(n\) is the principal quantum number. b)

Derive the Expression for the Speed of the Electron in the nth Orbit

To find the speed, \(v_n\), of the electron in the \(n\)th orbit, we'll use the quantization of angular momentum: \(m_ev_na_n = n\hbar\), where \(\hbar = \frac{h}{2\pi}\). Rearrange this equation to find the expression for \(v_n\): \(v_n = \frac{n\hbar}{m_ea_n}\), and substitute the previously derived expression for \(a_n\): \(v_n = \frac{Ze^2}{n\hbar}\frac{1}{4\pi\epsilon_0}\). c)

Derive the Expression for the Energy Levels of nth Orbit

The total energy, \(E_n\), of the electron in the \(n\)th orbit has two parts: kinetic energy (\(\frac{1}{2}m_ev_n^2\)) and potential energy (\(-\frac{e^2Z}{4\pi\epsilon_0a_n}\)). Combining the kinetic and potential energies, we get: \(E_n = \frac{1}{2}m_ev_n^2 - \frac{e^2Z}{4\pi\epsilon_0a_n}\). Substitute the expressions for \(a_n\) and \(v_n\) derived in the previous steps: \(E_n = -\frac{m_ee^4Z^2}{2n^2\hbar^2(4\pi\epsilon_0)^2}\).

Compare the Results to the Hydrogen Atom

When comparing the results to the hydrogen atom (with \(Z = 1\)), the expressions for the radius, speed, and energy levels are all consistent with the ones derived for a hydrogen atom. This confirms that our derivation is correct, and it shows the ability to apply the Bohr model of the atom to hydrogen-like ionized atoms.

Key concepts

Radius of the nth Orbit

In the Bohr model of the atom, the radius of the nth orbit is crucial for understanding the structure of atoms. According to Bohr's theory, which blends classical mechanics with early quantum theory, the electron orbits the nucleus in discrete paths without radiating energy.

Using the balance of forces, the radius for the nth orbit in a hydrogen-like ion can be derived. The expression for the radius, denoted as \( a_n \), is \( a_n = \frac{n^2\hbar^2}{m_ee^2Z}4\pi \epsilon_0 \), where \( \hbar \) is the reduced Planck's constant, \( n \) is the principal quantum number, \( m_e \) the mass of the electron, \( e \) the elementary charge, and \( Z \) the atomic number representing the number of protons in the nucleus. This formula demonstrates that as we move to higher orbits (increase \( n \)), the radius grows quadratically, reflecting larger orbits for electrons with greater energy.

Speed of the Electron in the nth Orbit

The speed of the electron in the nth orbit is another important aspect of the Bohr model. Bohr postulated that the angular momentum of the electron is quantized, meaning only certain values are allowed. This is expressed by the relationship \( m_ev_na_n = n\hbar \).

To find the speed, \( v_n \), we rearrange this to yield \( v_n = \frac{n\hbar}{m_ea_n} \). By substituting in our previously found expression for \( a_n \), the speed of the electron in the nth orbit for a hydrogen-like ion becomes \( v_n = \frac{Ze^2}{n\hbar}\frac{1}{4\pi\epsilon_0} \), showing that the speed decreases as we move to orbits with higher values of \( n \). It's intriguing to note that the speed directly depends on the nuclear charge \( Z \), meaning that electrons move faster in ions with a higher atomic number.

Energy Levels of Hydrogen-like Ions

Energy levels in atoms describe where electrons reside in relation to the nucleus. Bohr's model introduces distinct energy levels for electrons, depending on their orbit. For a hydrogen-like ion, we can calculate each energy level using the total energy equation, comprising both kinetic and potential energy components.

The total energy, \( E_n \), of the nth orbit is given by the sum of the kinetic energy (\( \frac{1}{2}m_ev_n^2 \)) and the potential energy (\( -\frac{e^2Z}{4\pi\epsilon_0a_n} \)). This results in \( E_n = -\frac{m_ee^4Z^2}{2n^2\hbar^2(4\pi\epsilon_0)^2} \). From here, it is clear that the energy levels become more negative (indicating a lower energy state) as we increase the nuclear charge \( Z \), and less negative (higher energy) as the principal quantum number \( n \) increases, reflecting the energy required to move an electron to a higher orbit.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides an understanding of the physical properties of nature at the scale of atoms and subatomic particles. It is in this framework that Bohr's model of the atom is often discussed.

The Bohr model was a significant step towards developing quantum mechanics by introducing quantized orbits for electrons. However, the model has its limitations and was later superseded by more advanced quantum theories that incorporate wave-particle duality and the probabilistic nature of quantum phenomena. In the context of the atoms and ions, quantum mechanics offers a more comprehensive and detailed approach to predict and explain the behavior of electrons, such as the Heisenberg Uncertainty Principle and Schrödinger's wave equation, which go beyond what Bohr's model could explain. Such principles form the core of modern atomic theory and provide a more accurate and complete description of atomic behavior.