Question

If sufficient energy is absorbed by an atom, an electron can be lost by the atom and a positive ion formed. The amount of energy required is called the ionization energy. In the \(\mathrm{H}\) atom, the ionization energy is that required to change the electron from \(n=1\) to \(n=\) infinity. Calculate the ionization energy for the He \(^{+}\) ion. Is the ionization energy of the He \(^{+}\) more or less than that of H? (Bohr's theory applies to He \(^{+}\) because it, like the \(\mathrm{H}\) atom, has a single electron. The electron energy, however, is now given by \(E=-Z^{2} R h c / n^{2},\) where \(Z\) is the atomic number of helium.)

Short answer

The ionization energy for He \(^+\) is more than that for H.

Step-by-step solution

Understand the Formula

The electron energy for a hydrogen-like ion is given by the formula \(E = -Z^2 Rhc / n^2\), where \(Z\) is the atomic number, \(R\) is the Rydberg constant \((1.097 \times 10^7 \text{ m}^{-1})\), \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{ Js})\), and \(c\) is the speed of light \((3 \times 10^8 \text{ m/s})\)." For helium ion (He \(^+\)), \(Z = 2\).

Key concepts

Bohr's Theory

Bohr's Theory is crucial for understanding how atoms and ions behave. It was developed by Niels Bohr in 1913 and revolutionized the way scientists thought about atomic structure. The model proposed that electrons orbit the nucleus in specific, discrete energy levels or shells, much like planets orbiting the sun. Each of these shells corresponds to a quantized energy state, and electrons can move between these levels by absorbing or emitting energy. This theory is particularly useful for explaining the behavior of single-electron systems like hydrogen or, as in this problem, the helium ion. Bohr's model allows us to calculate the required energy to move an electron between orbits or to completely remove it from the influence of the nucleus. In the context of ionization, or the removal of an electron, the model effectively helps determine the ionization energy, as the energy difference between the ground state and a state where the electron is free from the nucleus.

Rydberg Constant

The Rydberg constant is a fundamental constant in atomic physics. It is denoted by the symbol \(R\) and has a value of approximately \(1.097 \times 10^7 \text{ m}^{-1}\). The constant is named after the Swedish physicist Johannes Rydberg. It plays a crucial role in the formulation of the Rydberg formula, which predicts the wavelength of photons emitted from electronic transitions of atoms.In Bohr's model, the Rydberg constant helps calculate the energy levels of an electron in a hydrogen-like system by scaling with the inverse square of the principal quantum number \(n\) and the square of the atomic number \(Z\). This makes it an essential component in determining energies such as ionization energy.

Atomic Number

The atomic number, represented by \(Z\), is a fundamental property of an element. It refers to the number of protons in the nucleus of an atom and defines the identity of the element. In a neutral atom, the atomic number also equals the number of electrons orbiting the nucleus.In the context of Bohr's Theory and calculations involving atomic energy levels, the atomic number is critical for determining the energy states of electrons. For helium, which is the element in question, the atomic number is \(Z = 2\). This means that it typically has two protons and, in its neutral state, two electrons. However, in the helium ion \(He^+\), one of these electrons has been removed, making it analogous to a hydrogen atom in terms of electron configuration.

Helium Ion

A helium ion, symbolized as \(He^+\), is a helium atom that has lost one electron. This process results in a net positive charge, as the helium ion now contains two protons but only one electron. This ion is said to be isoelectronic with hydrogen, meaning it has the same number of electrons as a hydrogen atom, which is one. However, the presence of two protons (as opposed to one in hydrogen) significantly affects the helium ion's energy levels due to its higher atomic number \(Z = 2\). The effect of this higher atomic number is evident when calculating ionization energy. The energy needed to ionize \(He^+\) is much greater than that for hydrogen, due to the stronger electrostatic attraction between the nucleus and the remaining electron. This increased energy requirement highlights how differences in atomic structure influence chemical behavior and the stability of ions.