Question

Following the steps outlined in our treatment of 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 of charge number \(Z\) that has lost all of its electrons except for one electron. Compare the results with the corresponding ones for the hydrogen atom.

Short answer

Radius: \(r_n = \frac{n^2\frac{h^2}{4\pi^2}}{kZq^2m_e}\) Speed: \(v = \frac{2\pi kZq^2}{nh}\) Energy Levels: \(E_n = -\frac{k^2Z^2q^4m_e}{2n^2h^2}\)

Step-by-step solution

Part a: Deriving the radius of the nth orbit

1. We know centripetal force is given by \(F_c = \frac{m_ev^2}{r_n}\), where \(m_e\) is the mass of the electron, \(v\) is the speed of the electron, and \(r_n\) is the radius of the nth orbit. 2. The Coulomb force between the nucleus and the electron is given by \(F_e = \frac{kZq^2}{r_n^2}\), where \(k\) is the Coulomb's constant, \(Z\) is the charge number, and \(q\) is the charge of the electron. 3. Since the centripetal force is equal to the Coulomb force, we equate the two expressions: \(\frac{m_ev^2}{r_n} = \frac{kZq^2}{r_n^2}\) 4. We will use the quantization of angular momentum, which states that the angular momentum of the electron is an integer multiple of \(\frac{h}{2\pi}\): \(m_evr_n = n\frac{h}{2\pi}\), where \(n\) is the orbit number. 5. Solving for \(v\), we get \(v = \frac{n\frac{h}{2\pi}}{m_er_n}\) 6. Substituting this back into the equated centripetal and Coulomb forces, we get \(\frac{m_e(\frac{n\frac{h}{2\pi}}{m_er_n})^2}{r_n} = \frac{kZq^2}{r_n^2}\) 7. Simplify and solve for \(r_n\): \(r_n = \frac{n^2\frac{h^2}{4\pi^2}}{kZq^2m_e}\)

Part b: Deriving the speed of the electron in the nth orbit

1. Recall the quantization of angular momentum: \(m_evr_n = n\frac{h}{2\pi}\) 2. Now that we have the expression for \(r_n\) from part a, substitute this expression into the quantization of angular momentum formula: \(m_ev\frac{n^2\frac{h^2}{4\pi^2}}{kZq^2m_e} = n\frac{h}{2\pi}\) 3. Simplify and solve for \(v\): \(v = \frac{2\pi kZq^2}{nh}\)

Part c: Deriving the energy levels in a hydrogen-like ionized atom of charge number Z

1. We know that the total energy of an atom in the Bohr model is given by the sum of the kinetic and potential energies, which are \(\frac{1}{2}m_ev^2\) and \(-\frac{kZq^2}{r_n}\), respectively. 2. Substitute the expression for \(v\) from part b and the expression for \(r_n\) from part a into the kinetic and potential energy expressions: \(\frac{1}{2}m_e(\frac{2\pi kZq^2}{nh})^2\) and \(-\frac{kZq^2}{\frac{n^2\frac{h^2}{4\pi^2}}{kZq^2m_e}}\) 3. Simplify and add the two energy expressions to get the total energy: \(E_n = -\frac{k^2Z^2q^4m_e}{2n^2h^2}\) Now, let's compare the results to the hydrogen atom with Z=1: For hydrogen (Z=1): 1. \(r_n = \frac{n^2\frac{h^2}{4\pi^2}}{kq^2m_e}\) (this the same as the Bohr radius formula) 2. \(v = \frac{2\pi kq^2}{nh}\) (electron speed in the hydrogen atom) 3. \(E_n = -\frac{k^2q^4m_e}{2n^2h^2}\) (energy levels for hydrogen atom) We can see that the derived expressions for the radius, speed, and energy levels for the hydrogen-like ionized atom of charge number Z can be reduced to the corresponding expressions for the hydrogen atom when setting Z=1.

Key concepts

Quantization of Angular Momentum

The Bohr model introduces a revolutionary concept that the angular momentum of an electron orbiting the nucleus is quantized. This means the angular momentum can only have certain discrete values. The formula describing this phenomenon is given by

\[\begin{equation}L = n\frac{h}{2\pi},\end{equation}\]where L represents the angular momentum, n is an integer known as the principal quantum number, and h stands for Planck's constant.

By quantizing the angular momentum, Niels Bohr explained the stability of the electron's orbit and the discrete energy levels observed in atoms. In practice, quantization restricts the electron to orbit at particular distances from the nucleus, leading to defined energy levels within the atom.

Coulomb Force

The Coulomb force is the electrostatic force of attraction or repulsion between two charged particles. In the context of the Bohr model, it refers to the attractive force between the positively charged nucleus and the negatively charged electron. This force is described by Coulomb's law:

\[\begin{equation}F = \frac{kZq^2}{r^2},\end{equation}\]where F is the force magnitude, k is Coulomb's constant, Z is the nuclear charge number, q represents the charge on an electron, and r is the distance between the charges. The Coulomb force plays a critical role in determining the electron's orbit around the nucleus in the Bohr model.

Hydrogen-like Ionized Atoms

Hydrogen-like ionized atoms, or ions, are atoms that have only one electron and a nucleus with a charge Z. This is similar to the hydrogen atom, which naturally has only one electron, but the nucleus of a hydrogen-like ion holds more than one proton, increasing its positive charge. The Bohr model can be applied to these ions to determine their electron's possible energy levels and orbits. Bohr's formulas are scalable with the charge number Z, allowing comparisons with the basic hydrogen atom by simply setting Z to 1. This scaling shows that as Z increases, the electron's orbits become smaller and the energy levels more negative, indicating a tighter bond with the nucleus.

Energy Levels of Atoms

Bohr's model predicts that electrons reside in certain energy levels, or shells, each corresponding to a unique distance from the nucleus. The energy levels are quantized, meaning the electron can only contain certain amounts of energy and not values in between. These levels are represented by:

\[\begin{equation}E_n = -\frac{k^2Z^2q^4m_e}{2n^2h^2},\end{equation}\]where E_n denotes the energy of the electron in the nth energy level, and the other symbols have their usual meanings. As long as the electron remains in a given level, it does not radiate energy. However, when transitioning between levels, it either absorbs or emits energy in the form of light. The energy differences between these levels generate the atomic spectral lines characteristic of each element.

Atomic Orbit Radius

In Bohr's model, the radius of the electron's orbit around the nucleus increases with the principal quantum number n. This corresponds to an increase in energy level and is expressed by the formula:

\[\begin{equation}r_n = \frac{n^2\frac{h^2}{4\pi^2}}{kZq^2m_e},\end{equation}\]where r_n is the radius of the nth orbit. For hydrogen-like atoms, this radius scales with the square of n and inversely with the charge number Z, indicating that higher energy electrons are found further away from the nucleus. Additionally, the electron's orbit gets smaller in more highly charged nuclei, as seen by the presence of Z in the denominator. The smallest possible orbit, corresponding to the ground state of hydrogen (with n=1 and Z=1), is known as the Bohr radius and is a fundamental constant in atomic physics.