Question

A hydrogen-like ion is an ion containing only one electron. The energies of the electron in a hydrogen-like ion are given by $$ E_{n}=-\left(2.18 \times 10^{-18} \mathrm{~J}\right) Z^{2}\left(\frac{1}{n^{2}}\right) $$ where \(n\) is the principal quantum number and \(Z\) is the atomic number of the element. Calculate the ionization energy (in \(\mathrm{kJ} / \mathrm{mol}\) ) of the \(\mathrm{He}^{+}\) ion.

Short answer

The ionization energy of \( \mathrm{He}^{+} \) is \( 5252.4 \, \text{kJ/mol} \).

Step-by-step solution

Identify Variables

For the ionization energy calculation of the \( \text{He}^+ \) ion, note that \( Z = 2 \) since Helium has an atomic number of 2 and \( n = 1 \) for the ground state, since ionization energy relates to removing an electron from the lowest energy level.

Use the Ionization Energy Formula

The ionization energy of an electron in a hydrogen-like ion is given by \( E_{n} = - (2.18 \times 10^{-18} \, \text{J}) \cdot Z^2 \cdot \left( \frac{1}{n^2} \right) \). Substitute \( Z = 2 \) and \( n = 1 \). This gives:\[E_{1} = - (2.18 \times 10^{-18} \, \text{J}) \cdot 2^2 \cdot \left( \frac{1}{1^2} \right) = - 8.72 \times 10^{-18} \, \text{J}\]

Convert Energy to Per Mole

Ionization energy must be expressed in \( \text{kJ/mol} \). First, convert \( -8.72 \times 10^{-18} \text{J} \) to kJ by dividing by \( 10^3 \): \[-8.72 \times 10^{-18} \, \text{J} = -8.72 \times 10^{-21} \, \text{kJ}\]Next, multiply this by Avogadro's number \( 6.022 \times 10^{23} \) to find the energy per mole: \[-8.72 \times 10^{-21} \, \text{kJ} \cdot 6.022 \times 10^{23} = -5252.4 \, \text{kJ/mol}\]

Adjust Sign for Ionization Energy

The ionization energy is always a positive value, as it represents the energy required to remove an electron. Thus, the ionization energy is \( 5252.4 \, \text{kJ/mol} \).

Key concepts

Hydrogen-like Ion

A hydrogen-like ion is a fascinating concept in chemistry. It refers to any ion that has only one electron, similar to a hydrogen atom. Despite the name, it doesn't have to involve hydrogen exclusively; it can belong to any atom that has lost all its electrons except one. This concept is significant because:

• The electronic structure is much simpler since there's only one electron to consider.
• Understanding the behavior of these ions helps in the study of more complex atoms.

In the exercise above, the example of \(\mathrm{He}^+\) shows how we use this model to calculate ionization energies.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is an essential term in quantum mechanics. It indicates the energy level or shell of an electron in an atom. The larger the value of \(n\), the higher the energy level and the farther the electron is from the nucleus. Here are some important points:

• It is a positive integer (1, 2, 3, etc.).
• For \(n = 1\), electrons are in the ground state, closest to the nucleus.
• Each increase in \(n\) represents a higher energy level.

In calculations like the ionization energy of \(\mathrm{He}^+\), knowing \(n\) helps us find the energy needed to move the electron from its current state.

Atomic Number

The atomic number, symbolized as \(Z\), is a fundamental property of elements that defines the number of protons in the nucleus of an atom. In chemistry, it also indicates the total number of electrons in a neutral atom. Knowing the atomic number helps us understand several aspects:

• It determines the element type. For instance, helium has \(Z = 2\).
• It affects the chemistry of an element, as it helps define the electronic structure.
• It plays a crucial role in ionization calculations for hydrogen-like ions.

In the example, \(\mathrm{He}^+\) uses atomic number 2 to calculate ionization energy.

Avogadro's Number

Avogadro's number, \(6.022 \times 10^{23}\), is a constant that represents the number of atoms, ions, or molecules in one mole of substance. This number makes it possible to convert between microscopic and macroscopic scales in chemistry:

• It bridges the gap between atoms/molecules and observable quantities like moles.
• Avogadro's number helps in converting calculated ionization energies from energy per ion to energy per mole.
• It's used across chemistry for various unit conversions.

In our calculation, multiplying by Avogadro's number transformed the energy from \(\mathrm{kJ}\) per ion to \(\mathrm{kJ/mol}\), which is standard for expressing ionization energies.