Problem 123 - Chemistry: An Atoms-Focused Approach

Question

Colors of Fireworks Barium compounds are a source of the green colors in many fireworks displays. What is the ground-state electron configuration for \(\mathrm{Ba} ?\) b. The lowest-energy excited state of \(\mathrm{Ba}\) has the electron configuration \([\mathrm{Xe}] 5 d^{1} 6 s^{1} .\) What are the possible quantum numbers \(n, \ell,\) and \(m_{e}\) of a \(5 d\) electron?

Step-by-Step Solution

Answer

Answer: The ground-state electron configuration of Barium (Ba) is [Xe] 6s^2. For a 5d electron in the lowest-energy excited state of Ba, the possible quantum numbers are n = 5, l = 2, and m_ℓ = -2, -1, 0, 1, 2.

Step 1: Ground-state electron configuration of Ba

To determine the ground-state electron configuration of Ba, we first look at its atomic number. From the periodic table, we see that Ba has an atomic number of 56, which means it has 56 electrons. We will now fill the electron orbitals following the Aufbau principle by filling the orbitals in the order of increasing energy. The electron configuration will be: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10 4p^6 5s^2 4d^10 5p^6 6s^2 So, the ground-state electron configuration of Ba is: \([\mathrm{Xe}] 6s^2\)

Step 2: Possible quantum numbers for a 5d electron in the lowest-energy excited state

The lowest-energy excited state of Ba has the electron configuration \([\mathrm{Xe}] 5d^1 6s^1\). We want to find the possible quantum numbers (n, l, and \(m_{\ell}\)) for a 5d electron: n: Principal quantum number, which is related to the electron's energy and the electron shell. In this case, since it's a 5d electron, it is in the 5th shell, so n = 5. l: Azimuthal quantum number, which is related to the electron's orbital shape and subshell. For a d orbital, l = 2. \(m_{\ell}\): Magnetic quantum number, which is related to the electron's orbital orientation in space. For a d orbital (l = 2), the possible values of \(m_{\ell}\) range from -l to +l, which are -2, -1, 0, 1, and 2. So, for a 5d electron in the lowest-energy excited state of Ba, the possible quantum numbers are: n = 5, l = 2, \(m_{\ell}\) = -2, -1, 0, 1, 2.

Key Concepts

Quantum Numbers

Quantum numbers describe the unique quantum state of an electron in an atom. They are essential to understanding the arrangement of electrons in atoms and predicting their behavior. There are four types of quantum numbers:

• Principal Quantum Number (n): This number indicates the energy level of an electron within an atom and is denoted by 'n'. Higher 'n' values correspond to higher energy levels and larger orbitals. For instance, when discussing a 5d electron, the principal quantum number is 5.


• Azimuthal Quantum Number (l): Also known as the angular momentum quantum number, 'l' signifies the shape of the electron’s orbital. It can take values from 0 to (n-1). When 'l' is 2, as in the case of a 'd' orbital, it describes the complex multi-lobed shapes characteristic of d orbitals.


• Magnetic Quantum Number (m_l): This number defines the orientation of the electron's orbital in space. It ranges from -l through zero to +l. In a 'd' orbital where l is 2, m_l can be -2, -1, 0, 1, or 2.

Quantum numbers allow us to understand how electrons are distributed in an atom and predict many properties of an element, including its chemical behavior.

Excited States

Electrons in an atom generally exist in the lowest possible energy state, known as the ground state. However, when an atom absorbs energy, its electrons can move to higher energy levels, resulting in an excited state.
The excited state is not stable because electrons tend to seek lower energy levels. They will eventually release energy, often in the form of light, and return to the ground state.
For barium ( Ba ), an excited electron configuration can be expressed as [ Xe ] 5d^{1} 6s^{1} . This indicates that one electron from the ground state 6s^2 has moved to the 5d^1 orbital. This alteration in the configuration due to energy absorption can explain the barium's characteristic green color in fireworks as it returns to the ground state, emitting energy in the process.

Periodic Table

The Periodic Table is an essential tool in chemistry. It organizes all known elements based on their atomic number, electron configuration, and recurring chemical properties. Barium ( Ba ), for instance, is found in Group 2, known as the alkaline earth metals.
The position of an element in the Periodic Table can tell us about its electron configuration. For example, barium's position lets us know it ends with the electron configuration [ Xe ] 6s^2 in its ground state, as it follows the element xenon (Xe), which completes the fifth period.
By examining the Periodic Table, we can predict many physical and chemical properties of elements. This includes their reactivity, strength, and stability, by understanding their electron configurations and occupancies.

Aufbau Principle

The Aufbau Principle is a fundamental concept in chemistry that describes how electrons are added to the atomic orbitals. The term "Aufbau" means "building up" in German, signifying that electrons fill from lower to higher energy levels.
According to the Aufbau Principle, electrons occupy orbitals starting from the lowest energy level and move upwards. For barium, the filling order would ultimately lead to its ground state configuration: [ Xe ] 6s^2 .
The principle simplifies predicting the electron configuration for any element by following the order:

• 1s , 2s , 2p , 3s , 3p , 4s , 3d , 4p , 5s , 4d , 5p , 6s , 4f , 5d , 6p , 7s

Understanding this principle helps forecast which chemistry and interaction each element might exhibit based on its electron arrangement.