Question

According to a relationship developed by Niels Bohr, for an atom or ion that has a single electron, the total energy, \(E_{n},\) of an electron in a stable orbit of quantum number \(n\) is \(E_{n}=-\left[Z^{2} / n^{2}\right]\left(2.179 \times 10^{-18} \mathrm{~J}\right)\) where \(Z\) is the atomic number. Calculate the ionization energy for the electron in a ground- state \(\mathrm{He}^{+}\) ion.

Short answer

The ionization energy for the ground-state \( \mathrm{He}^{+} \) ion is \( 8.716 \times 10^{-18} \text{ J} \).

Step-by-step solution

Identify Given Variables

The ion is \( \text{He}^+ \) with atomic number \( Z = 2 \). The electron is in the ground state, which corresponds to quantum number \( n = 1 \).

Substitute Values into Formula

Use Bohr's formula for total energy: \[ E_{n} = -\left[\frac{Z^{2}}{n^{2}}\right]\left(2.179 \times 10^{-18} \text{ J}\right) \] Substituting \( Z = 2 \) and \( n = 1 \): \[ E_{n} = -\left[\frac{2^{2}}{1^{2}}\right]\left(2.179 \times 10^{-18} \text{ J}\right) \]

Calculate Total Energy

Simplify and calculate:- Square \( Z \) and \( n \):\( 2^{2} = 4 \) and \( 1^{2} = 1 \).- Substitute: \( E_{n} = -4 \times 2.179 \times 10^{-18} \text{ J} \)- Calculate \( E_{n} = -8.716 \times 10^{-18} \text{ J} \).

Determine Ionization Energy

Ionization energy is the energy required to remove the electron completely, which is the negative of the total energy. Thus, Ionization Energy = \(-E_{n} = 8.716 \times 10^{-18} \text{ J} \).

Key concepts

Ionization Energy

The concept of ionization energy is crucial in understanding atomic behavior. Ionization energy refers to the amount of energy required to remove an electron from an atom or ion completely. It is a measure of an atom's tendency to retain its electrons.
For the hydrogen-like ion configuration with only one electron, like the \(\text{He}^+\) ion, ionization energy specifically relates to the energy needed to remove that lone electron from its ground state.

• This energy is crucial because it indicates how strongly the nucleus binds the electron.
• Higher ionization energy means a stronger attraction between the nucleus and the electron.

In our example, we calculated the ionization energy using Bohr's model, finding it to be\(8.716 \times 10^{-18} \text{ J}\). This is the energy needed to transition that electron from a bound state to a free, unbound state, missing from the atom's influence, thus effectively ionizing the atom.
The concepts of ionization energy are essential for predicting the reactivity and properties of elements, as elements with low ionization energies tend to lose electrons more readily, making them more reactive.

Atomic Number

The atomic number is a fundamental concept in understanding elements. It represents the number of protons in the nucleus of an atom, and it's denoted by the symbol \(Z\).

• The atomic number determines the identity of the element and its position on the periodic table.
• For example, helium, abbreviated as He, has an atomic number of 2, which means every helium atom has two protons.

The atomic number has profound implications for chemical behavior and element classification.
It is the atomic number that helps in determining the element's electronic configuration and its chemical properties.In the context of the Bohr Model, the atomic number becomes even more critical:

• It directly influences the energy levels of the electrons in an atom or ion.
• In the energy equation \(-\left[Z^{2} / n^{2}\right]\left(2.179 \times 10^{-18} \text{ J}\right)\), the atomic number (\(Z\)) affects the total energy because it signifies how strong the nucleus can hold onto its electrons due to proton count.

Quantum Number

Quantum numbers are a key concept in quantum mechanics, describing the properties of atomic orbitals and electrons within those orbitals.
The principal quantum number, denoted as \(n\), indicates the energy level or shell that an electron inhabits. It is always a positive integer (1, 2, 3,...), where higher numbers correspond to shells farther from the nucleus.

• The principal quantum number determines the size and energy of the orbital.
• In our example of the \(\text{He}^+\) ion, with the electron in the ground state, we refer to the principal quantum number \(n = 1\).

This value of \(n\) is crucial as it dictates the electron's energy and organization around the nucleus.
In Bohr's model, the quantum number is part of the formula that calculates the specific energy levels of electrons. Understanding how quantum numbers work is essential for predicting an electron's behavior in different energy states and for understanding the organization of electrons in atoms, which in turn affects the atom's chemical properties and reactivity.