Question

In the ground state of cadmium, Cd, a. how many electrons have \(\ell=2\) as one of their quantum numbers? b. how many electrons have \(n=4\) as one of their quantum numbers? c. how many electrons have \(m_{\ell}=-1\) as one of their quantum numbers? d. how many electrons hav \(m_{s}=-\frac{1}{2}\) as one of their quantum numbers?

Short answer

a. 18 electrons have \(\ell=2\) as one of their quantum numbers. b. 18 electrons have \(n=4\) as one of their quantum numbers. c. 4 electrons have \(m_\ell=-1\) as one of their quantum numbers. d. 24 electrons have \(m_s=-\frac{1}{2}\) as one of their quantum numbers.

Step-by-step solution

1. Determine electron configuration of Cadmium

Using Aufbau's principle, the electron configuration of Cd can be represented as follows: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10}\] This represents the distribution of electrons in various orbitals for the ground state of cadmium. Now we will analyze each part of the exercise: a. Finding the number of electrons with \(\ell=2\)

2. Identify \(\ell=2\) orbitals in the electron configuration

In the electron configuration, orbitals with \(\ell=2\) are denoted by p orbitals. We need to find all the p orbitals in the electron configuration.

3. Count the electrons in \(\ell=2\) orbitals

In cadmium's ground state electron configuration, 2p, 3p and 4p are the orbitals with \(\ell=2\). We will count the electrons in these orbitals: \(2p^6 \Rightarrow 6\) \: 3p^6 \Rightarrow 6\) \: 4p^6 \Rightarrow 6\) So, there are a total of \(6+6+6=18\) electrons with \(\ell=2\) as one of their quantum numbers. b. Finding the number of electrons with \(n=4\)

4. Identify \(n=4\) orbitals in the electron configuration

In the electron configuration, the orbitals with the principal quantum number \(n=4\) are 4s, 4p, and 4d.

5. Count the electrons in \(n=4\) orbitals

We will count the electrons in these orbitals: \(4s^2 \Rightarrow 2\) \: 4p^6 \Rightarrow 6\) \: 4d^{10} \Rightarrow 10\) So, there are a total of \(2+6+10 = 18\) electrons with \(n=4\) as one of their quantum numbers. c. Finding the number of electrons with \(m_\ell=-1\)

6. Identify orbitals with corresponding \(m_\ell\) values

In the electron configuration, we identify the orbitals and corresponding \(m_\ell\) values as follows: \[\begin{array}{c|c} Orbital & \: \: m_{\ell} \: \: \\ \hline s & 0 \\ p & -1,0,1 \\ d & -2,-1,0,1,2 \\ f & -3,-2,-1,0,1,2,3 \end{array}\] Using this table, we identify orbitals with \(m_\ell=-1\) as 2p (\(2nd\: in\: p\)), 3p (\(2nd\: in\: p\)), 4p (\(2nd\: in\: p\)), and 3d (\(2nd\: in\: d\)).

7. Count the electrons in orbitals with \(m_\ell=-1\)

Now we will count the electrons in these orbitals with \(m_\ell=-1\): \[\begin{array}{c|c} Orbital & Number\: of\: electrons \\ \hline 2p & 1 \\ 3p & 1 \\ 4p & 1 \\ 3d & 1 \end{array}\] So, there are a total of \(1+1+1+1=4\) electrons with \(m_\ell=-1\) as one of their quantum numbers. d. Finding the number of electrons with \(m_s=-\frac{1}{2}\)

8. Calculate the electrons with \(m_s=-\frac{1}{2}\)

All electrons have either \(m_s=\frac{1}{2}\) or \(m_s=-\frac{1}{2}\). In each orbital, half of the electrons will have \(m_s=-\frac{1}{2}\). Cadmium has a total of 48 electrons. Therefore, half of these electrons will have \(m_s = -\frac{1}{2}\). So, there are \(48 / 2 = 24\) electrons with \(m_s=-\frac{1}{2}\) as one of their quantum numbers. In conclusion: a. 18 electrons have \(\ell=2\) as one of their quantum numbers. b. 18 electrons have \(n=4\) as one of their quantum numbers. c. 4 electrons have \(m_\ell=-1\) as one of their quantum numbers. d. 24 electrons have \(m_s=-\frac{1}{2}\) as one of their quantum numbers.

Key concepts

Quantum Numbers

Quantum numbers serve as unique identifiers for electrons within an atom. They describe the properties and locations of electrons:

• The principal quantum number ( ext{n} ) indicates the electron's shell or energy level, with higher numbers representing greater distances from the nucleus.
• The azimuthal quantum number ( ext{l} ) defines the electron's subshell or orbital shape—such as s, p, d, or f. Typically, l = 0 represents the s orbital, l = 1 the p orbital, l = 2 the d orbital, and so on.
• The magnetic quantum number ( m_ ext{l} ) specifies the spatial orientation of the orbital, with values ranging from - l to + l
• The spin quantum number ( m_s ) describes the electron's intrinsic spin, being either +1/2 or -1/2.

Understanding quantum numbers allows us to map out where electrons reside in atoms, helping to explain chemical behavior and bonding characteristics. By using these four numbers, we can predict many aspects of an atom's electron configuration.

Aufbau Principle

The Aufbau Principle is a guideline for writing the electron configuration of atoms. According to this principle, electrons fill atomic orbitals in order of increasing energy levels. It helps predict how electrons in an atom will be distributed among the various orbitals:

• Electrons first occupy the lowest energy orbital available, typically starting with the 1s orbital.
• Orbitals are filled in order of increasing energy, moving up to 2s, 2p, 3s, and so on.
• The order of filling is not strictly sequential numerically owing to energy variations between s, p, d, and f orbitals.

For example, in cadmium, electrons fill up to the 5s and 4d orbitals according to this principle. Understanding the Aufbau Principle is crucial for determining how electrons are arranged, which in turn influences an element's chemical properties and reactivity.

Cadmium

Cadmium (Cd) is a chemical element with the atomic number 48. Its electron configuration in the ground state is [1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10}]. Cadmium sits in the transition metals category of the periodic table, commonly encountered with atomic orbitals in higher energy levels:

• In its ground state, cadmium’s electrons are completely filled up to the 3d orbitals.
• This element is characterized by having a full complement of 48 electrons, evenly split between positive and negative spins.
• Cadmium is often used in electroplating and battery manufacturing due to its corrosion resistance and excellent electrical properties.

Cadmium's electron configuration and properties are fundamental in understanding its place in chemical reactions and material science.

Ground State

The ground state of an atom refers to its lowest energy configuration. When electrons occupy their minimal energy levels, the atom is said to be in its ground state. Here's why understanding ground state is vital:

• In the ground state, electrons are arranged in stable configurations according to the Aufbau Principle and Hund's Rule.
• When atoms or molecules absorb energy, electrons can jump to a higher orbit, leading to an "excited state." The return to the ground state often results in energy release, observable in the form of light.'
• The electron configuration determined in the ground state of an element is primarily used for studying the element's chemical properties and potential interactions.

Studying the ground state conditions of elements, such as cadmium, facilitates deeper insights into their behaviors in various chemical contexts.