Question

Using an orbital box diagram and noble gas notation, show the electron configuration of titanium. Give one possible set of four quantum numbers for each of the electrons beyond those of the preceding noble gas.

Short answer

Titanium follows \\[\text{[Ar]} \ 4s^2 \ 3d^2\\] with quantum numbers for 4s electrons: \\(n = 4, \ l = 0, \ m_l = 0, \ m_s = \pm\frac{1}{2}\\) and for 3d: \\(n = 3, \ l = 2, \ m_l = -2, -1, \ m_s = +\frac{1}{2}\\).

Step-by-step solution

Identify the Atomic Number of Titanium

Titanium (Ti) has an atomic number of 22, which means it has 22 electrons when in a neutral state.

Recognize the Preceding Noble Gas

The noble gas preceding titanium is argon, which has an atomic number of 18. Thus, the electron configuration of titanium will begin with the electron configuration of argon.

Write the Noble Gas Notation

The noble gas notation for titanium, incorporating the configuration of argon, is \[\text{[Ar]} \ 4s^2 \ 3d^2\]. This indicates the next four electrons (after argon) fill the 4s and 3d orbitals.

Construct the Orbital Box Diagram

Draw boxes to represent each orbital. Fill the 4s orbital first with two electrons and the 3d orbital with two electrons. Use arrows to denote the spins of these electrons. The 4s orbital is fully paired, whereas the 3d orbital has two unpaired electrons with parallel spins.

Determine Quantum Numbers

The quantum numbers describe each electron. For the two electrons beyond argon in the 4s subshell, a possible set is: 1. First electron in 4s: \(n = 4, \ l = 0, \ m_l = 0, \ m_s = +\frac{1}{2}\)2. Second electron in 4s: \(n = 4, \ l = 0, \ m_l = 0, \ m_s = -\frac{1}{2}\)For the electrons in 3d:1. First electron in 3d: \(n = 3, \ l = 2, \ m_l = -2, \ m_s = +\frac{1}{2}\)2. Second electron in 3d: \(n = 3, \ l = 2, \ m_l = -1, \ m_s = +\frac{1}{2}\)

Key concepts

Orbital Box Diagram

The orbital box diagram is a great visual tool to understand how electrons are distributed among the atomic orbitals of an element. Picture each orbital as a box that can hold a maximum of two electrons. In the orbital box diagram for titanium, which has 22 electrons, we need to first fill up the configurations up to the preceding noble gas, argon. This takes care of the first 18 electrons.
The next step involves adding the four remaining electrons beyond argon. These electrons enter the 4s and 3d orbitals. Each box (orbital) is represented by a line, and electrons are indicated by arrows pointing up or down, representing their spin. The 4s orbital is fully filled with two electrons of opposite spins, while the 3d orbital begins filling with two parallel spin electrons, to reflect Hund's rule.

• 4s: Two boxes (lines) with opposite arrow directions.
• 3d: First two boxes filled with single arrows in the same upward direction.

This intuitive diagram helps in visualizing electron spins and their arrangement in orbitals.

Quantum Numbers

Quantum numbers are essential to describe the specific position and energy of electrons within an atom. There are four quantum numbers:

• Principal Quantum Number (n): Indicates the energy level or shell. For titanium, electrons in the 4s level have a principal quantum number of 4, while those in 3d have 3.
• Azimuthal Quantum Number (l): Represents the subshell or shape of the orbital. Here, the 4s orbital has an azimuthal number of 0, and the 3d has 2.
• Magnetic Quantum Number (m_l): Describes the orientation of the orbital. It can range from –l to +l. For example, in the 3d subshell, values are –2, –1, 0, +1, +2.
• Spin Quantum Number (m_s): Reflects the electron's spin, either +\(\frac{1}{2}\) or –\(\frac{1}{2}\). For instance, in the 4s subshell, one electron can have +\(\frac{1}{2}\) and the other –\(\frac{1}{2}\).

These numbers together provide a unique address for each electron in the atom.

Noble Gas Notation

The noble gas notation is a streamlined way of writing electron configurations by starting from the nearest preceding noble gas. This reduces density in writing electron configurations by condensing the innermost electrons, grouped collectively under the symbol of the noble gas.
For titanium, the noble gas shorthand begins with argon (Ar), as it owns the same electron configuration up to that point. Therefore, the shorthand notation for titanium's electron configuration is \[\text{[Ar]}\, 4s^2\, 3d^2\] This uses argon's complete electron configuration as a starting point and then adds the rest of the electrons. Thus, it makes the representation simple and quick. The remaining electrons are listed explicitly, showing their subshells and occupation.

Titanium

Titanium is a transition metal located in the fourth period and group 4 of the periodic table. Known for its strength and low density, it's used in various applications from aerospace to medical implants. With an atomic number of 22, titanium's electron configuration reveals part of its unique chemical behavior.
Titanium's configuration is notable for its 3d orbital, which contributes to its ability to form various compounds with different oxidation states. The metallic bonding of titanium is attributed to these d-electrons which are less tightly bound and readily engage in interactions with other atoms.
Overall, understanding titanium's electronic structure, including its electron configuration, helps elucidate why it performs so distinctively in chemistry and materials science.