Question

Calculate the energy needed to change a single ionized helium atom into a double ionized helium atom (that is, change it from \(\mathrm{He}^{+}\) into \(\mathrm{He}^{2+}\) ). Compare it to the energy needed to ionize the hydrogen atom. Assume that both atoms are in their fundamental state.

Short answer

Answer: The energy required to change a helium atom from single to double ionization is approximately 5 times greater than the energy required to ionize a hydrogen atom.

Step-by-step solution

Calculate the energy required for the first ionization of helium

To find the energy required for the first ionization of helium, we need to calculate the energy that goes into overcoming the electrostatic attraction between the electrons and the nucleus. The energy for ionization is given by: \(n_1\) = first Bohr orbit, \(Z\) = atomic number (for helium, \(Z = 2\)), and \(E_0\) = ionization energy of hydrogen = 13.6 eV \(E_1 = -\frac{Z^2 * E_0}{n_1^2} = -\frac{(2)^2 * 13.6 \ \text{eV}}{(1)^2} = -54.4 \ \text{eV}\) The energy required for the first ionization of helium is \(54.4\) eV.

Calculate the energy required for the second ionization of helium

For the second ionization of helium, we need to calculate the energy needed to overcome the electrostatic attraction between the only remaining electron and the nucleus. Since, after the first ionization, the helium ion \(\text{He}^+\) has a positive charge, the remaining electron moves to the first Bohr orbit. Then, using the same formula as before, but with the atomic number being reduced by 1 (\(Z = 1\) in this case): \(n_2\) = first Bohr orbit and \(Z\) = atomic number \(E_2 = -\frac{Z^2 * E_0}{n_2^2} = -\frac{(1)^2 * 13.6 \ \text{eV}}{(1)^2} = -13.6 \ \text{eV}\) The energy required for the second ionization of helium is \(13.6\) eV.

Calculate the total energy required to change the helium atom

To calculate the total energy required to fully ionize the helium atom, we will add the two energies obtained in previous steps: \(E_{total} = E_1 + E_2 = -54.4 \ \text{eV} + (-13.6 \ \text{eV}) = -68.0 \ \text{eV}\) The total energy required to change the helium atom from \(\mathrm{He}^{+}\) to \(\mathrm{He}^{2+}\) is \(68.0\) eV.

Compare the energy values

Now that we have the energies required for the ionization processes of both helium and hydrogen, we can make a comparison: - The energy required for ionizing hydrogen atom in its fundamental state is \(13.6\) eV. - The energy required for changing a single ionized helium atom into a double ionized helium atom is \(68.0\) eV, which includes both the first ionization energy (\(54.4\) eV) and the second ionization energy (\(13.6\) eV). Therefore, the energy required to change a helium atom from single to double ionization is significantly greater than the energy required to ionize a hydrogen atom, by a factor of \(\frac{68.0}{13.6} \approx 5\).

Key concepts

Helium Ionization

The process of helium ionization involves removing electrons from a helium atom, transitioning it from a neutral state to a charged ion. Helium, with two electrons in its natural state, can undergo two ionizations. The first ionization removes one electron, forming a helium ion with a single positive charge, ( Nor Div Nor Nor �He+�). The second ionization strips away the final electron, resulting in a double positive charge, �He2+�. This process is energetically demanding because it involves overcoming the electrostatic attraction between the positively charged nucleus and the negatively charged electrons. In atomic physics, the ionization energy is a crucial concept, as it indicates how strongly an electron is bound to an atom and the effort required to remove it.

In the context of the textbook exercise, we calculated the energy required for both ionizations. The first requires significantly more energy because when helium is neutral, two electrons are being held by the nucleus; after the first ionization, only one electron remains, making it easier to remove.

Bohr Model

The Bohr model is a fundamental concept in atomic physics that describes the structure of atoms, particularly hydrogen-like atoms. According to this model, electrons revolve around the nucleus in discrete orbitals without radiating energy, which contradicts classical electromagnetism. The energy levels are quantized, meaning electrons can only inhabit specific orbits with fixed energies.

Quantum Leaps in the Bohr Model

When an electron transitions between these orbits, it must absorb or emit energy equal to the difference between the energy levels. This model is a cornerstone for understanding atomic behavior, especially for simple, single-electron systems like hydrogen or ionized helium, which are akin to the hydrogen-like models.

In the provided exercise solution, we applied the Bohr formula to determine the ionization energies of helium. This formula reflects the energy of an electron in a particular orbit, laying the groundwork for calculating the ionization energy needed to free the electron from its atomic bond.

Electrostatic Attraction

At the heart of ionization processes lies the principle of electrostatic attraction. This fundamental force governs the interaction between charged particles—positive protons in the nucleus attract negative electrons, holding them within the atom. The strength of this attraction depends on the distance between the particles and the magnitude of their charges, described by Coulomb's law.

Coulomb’s Law and Ionization

Coulomb's law quantitatively expresses the electrostatic force between two charges. As this force is inversely proportional to the square of the distance between the charges, it becomes significantly stronger when the electron is close to the nucleus, such as in smaller atoms like helium. Thus, a higher ionization energy is needed to overcome this force in such atoms, as seen when comparing helium's ionizing energy to hydrogen's.

Atomic Physics

The field of atomic physics is the study of atoms as isolated systems, including their electron configurations and the way electrons transition between energy levels. This discipline is fundamental for understanding the properties of elements on the atomic scale and plays a crucial role in the development of various technologies and scientific theories.

It extends beyond the structural details provided by the Bohr model to include principles of quantum mechanics and the behavior of electrons under the influence of external fields. In educational exercises like the one we discussed, atomic physics concepts allow students to understand the energy dynamics within atoms and how environmental changes, such as the addition of energy, can lead to significant structural changes like ionization.