Question

A \(\mathrm{He}^{+}\) ion consists of a nucleus ( containing two protons and two neutrons) and a single electron. Obtain the Bohr radius for this system.

Short answer

The Bohr radius for a helium ion with a single electron (He+) is approximately \(2.64 \times 10^{-11} \mathrm{m}\).

Step-by-step solution

Recall the basic formula for the Bohr radius of hydrogen

For hydrogen, the Bohr radius (denoted as \(a_0\)) is given by the formula: $$a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2}$$ Where \(\epsilon_0\) is the vacuum permittivity, \(\hbar\) is the reduced Planck constant, \(m_e\) is the mass of the electron and \(e\) is the elementary charge.

Modify the formula for the helium ion

The major difference between hydrogen and He+ is the nuclear charge. Since He+ has two protons, the electric force acting on the electron will be doubled. In the Bohr model, this is represented by the following relationship: $$F = \frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r^2}$$ Where \(Z\) is the atomic number (number of protons in the nucleus) and \(r\) is the distance from the electron to the nucleus.

Equate the centripetal force with the Coulomb electric force

In order for the electron to orbit the nucleus in a circular path, the centripetal force acting on it must equal the Coulomb electric force. We have: $$m_e \frac{v^2}{r} = \frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r^2}$$

Use quantization of angular momentum and solve for Bohr radius

According to the Bohr model, the angular momentum of the electron is quantized: $$L = m_e v r = n\hbar$$ Where \(n\) is the principal quantum number. In this exercise, we are asked for the smallest Bohr radius, which corresponds to the ground state (\(n = 1\)). Substitute this angular momentum quantization condition into the equation obtained in Step 3: $$m_e \frac{(1\hbar)^2}{m_e^2 r^2} = \frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r^2}$$ Now solve for the Bohr radius \(r\) (for He+ ion): $$r = \frac{4 \pi \epsilon_0 \hbar^2}{Z m_e e^2}$$

Calculate the Bohr radius for He+ ion

Now we substitute the known values for \(\epsilon_0\), \(\hbar\), \(m_e\) and \(e\), along with \(Z=2\): $$r = \frac{4 \pi (8.85 \times 10^{-12} \, \mathrm{C^2/Nm^2}) (1.054 \times 10^{-34} \, \mathrm{Js})^2}{2 (9.109 \times 10^{-31} \, \mathrm{kg}) (1.602 \times 10^{-19} \mathrm{C})^2}$$ After solving for \(r\), we find the smallest Bohr radius for the He+ ion: $$r \approx 2.64 \times 10^{-11} \mathrm{m}$$

Key concepts

Bohr Model

The Bohr model is a primitive but significant theory in atomic physics introduced by Niels Bohr in 1913. This model depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar to the structure of the Solar System but with attraction provided by electrostatic forces rather than gravity.

According to Bohr's model, electrons move in specific shells or energy levels at a fixed distance from the nucleus, and these orbits correspond to different energy levels. The electrons can jump between levels by emitting or absorbing energy in quantized packets called photons. The model successfully explains the spectral lines of the hydrogen atom. An essential element in this explanation is the quantization of the electron's angular momentum, with the smallest Bohr radius describing the ground state of the atom, where the electron is closest to the nucleus.

In the context of the exercise, to find the Bohr radius for a helium ion (He⁺), the Bohr model provides a straightforward formula adjusted for the helium ion's nuclear charge.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It was developed in the early 20th century as scientists discovered that the classical mechanics by Newton could not explain the peculiar behaviors observed at atomic scales.

One of the core principles of quantum mechanics is the concept of quantization, which means that certain physical quantities, like the energy of an electron in an atom, take on only discrete values. This is analogous to the concept introduced by the Bohr model, where the angular momentum of the electron orbiting a nucleus is quantized. Quantum mechanics also introduces the concept of wave-particle duality, which states that particles can exhibit properties of both particles and waves.

The quantization of angular momentum, as applied in the exercise solution, shows how the principles of quantum mechanics are integral in calculating the electron's permitted orbits around the nucleus and, hence, the Bohr radius of the helium ion.

Helium Ion

A helium ion, specifically a singly ionized helium ion (He⁺), is a helium atom that has lost one of its two electrons. Thus, it consists of a nucleus containing two protons and usually two neutrons, surrounded by a single remaining electron. This ion is of particular interest because it represents a simple system that exhibits the basic principles of quantum mechanics and is effectively a hydrogen-like atom but with a doubly charged nucleus.

The presence of the additional proton in the helium nucleus compared to hydrogen leads to a stronger electrostatic attraction to the electron. This results in a smaller Bohr radius for the He⁺ ion than that of a hydrogen atom. The steps outlined in the original exercise leverage the principles of quantum mechanics to adapt the Bohr radius formula for a helium ion by considering its increased nuclear charge. The exercise demonstrates the direct application of atomic physics concepts by showing how a helium ion’s greater nuclear charge influences the electron’s orbit, as reflected in the Bohr radius.

Atomic Physics

Atomic physics is the field of physics which studies atoms as isolated systems that include electrons and an atomic nucleus. It primarily concerns the arrangement of electrons around the nucleus and the processes by which these arrangements change. This includes studying phenomena like spectra and ionization.

Atomic physics has led to many key developments in science and technology, including lasers, quantum computing, and even the principles behind MRI machines. The theory of atomic structure and spectra is fundamental to the understanding of the physical world.

In the student exercise, we utilize atomic physics to predict the behavior of electrons in a helium ion—an application that goes hand in hand with the foundational knowledge from the Bohr model and quantum mechanics. By understanding the underlying atomic physics, students can make sense of why the formula for the Bohr radius needs to be modified for the helium ion, and how that modification reflects the ion’s unique atomic structure.