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: [Ar] 4s^2 3d^2. Quantum numbers: (3, 2, -2, +1/2) and (3, 2, -1, +1/2).

Step-by-step solution

Identify the Preceding Noble Gas

Titanium is located in the 4th period and the d-block of the periodic table. The preceding noble gas is Argon, with the atomic number 18.

Determine the Atomic Number of Titanium

Titanium has an atomic number of 22, which means it has 22 electrons in its neutral state.

Write the Electron Configuration Using Orbital Box Diagram

After Argon (18 electrons), the following electrons are placed: - 19th and 20th electrons go into the 4s orbital. - 21st and 22nd electrons go into the 3d orbital. The orbital box representation for Titanium looks like this: - [Ar] - 4s: ↑↓ - 3d: ↑ ↑

Express the Electron Configuration in Noble Gas Notation

The noble gas notation for Titanium's electron configuration is:\[ ext{[Ar]} ext{ } 4s^2 ext{ } 3d^2 \]

Assign Quantum Numbers for the 3d Electrons

The quantum numbers for any electron are given as (n, l, m_l, m_s). For the first 3d electron: - n = 3, l = 2, m_l could be -2, -1, 0, +1, or +2; select m_l = -2 - m_s = +1/2 (spin-up) For the second 3d electron: - n = 3, l = 2, m_l could be -2, -1, 0, +1, or +2; select m_l = -1 - m_s = +1/2 (spin-up) Thus, a possible set of quantum numbers beyond Argon would be (3, 2, -2, +1/2) and (3, 2, -1, +1/2).

Key concepts

Orbital Box Diagram

An orbital box diagram is a visual way of representing the electron configurations of an atom. It shows the distribution of electrons in different atomic orbitals, represented as boxes or squares. Each box corresponds to a single orbital that can hold up to two electrons. Electrons within these boxes are shown as arrows, pointing up or down to indicate their spin states.
Electrons fill orbitals following Hund's Rule and the Aufbau principle. The Aufbau principle tells us electrons fill the lowest energy orbitals first, while Hund's Rule states that electrons will fill empty orbitals before pairing up.
For titanium, which has an atomic number of 22, the orbital box diagram starts after the preceding noble gas, argon. The 19th and 20th electrons fill the 4s orbital, and the 21st and 22nd go into the 3d orbital. This configuration reflects the electron arrangement:

• 4s: a box with two paired arrows (↑↓), indicating a filled orbital
• 3d: two boxes each with a single arrow (↑), showing electrons in separate orbitals first

Noble Gas Notation

Noble gas notation is a shorthand way to express an element's electron configuration using the electron configuration of the previous noble gas. This simplifies complex configurations by focusing only on the outer electrons following the noble gas core.
The noble gases are chemically inert due to their full valence shells, making them useful as reference points.
For example, titanium's electron configuration in noble gas notation is \[ \text{[Ar]} \ 4s^2 \ 3d^2 \], where [Ar] represents the 18 electrons of argon.
Using noble gas notation helps in recognizing patterns in electron configurations quickly, especially for elements with many electrons. It reduces the complexity of writing long chains of electron arrangements.

Quantum Numbers

Quantum numbers are a set of four numbers that provide a complete and unique description of an electron's position and movement in an atom. Each electron in an atom is defined by:

• Principal quantum number (n): Indicates the main energy level or shell. It can take positive integer values like 1, 2, 3, etc. For titanium's 3d electrons, n=3.
• Angular momentum quantum number (l): Determines the shape of the orbital. For d orbitals, l=2.
• Magnetic quantum number (m_l): Specifies the orientation of the orbital in space. It ranges from -l to +l. For instance, the values for titanium's 3d electrons could be -2 and -1.
• Spin quantum number (m_s): Represents the electron spin direction, either +1/2 or -1/2. For the 3d electrons in titanium, m_s is +1/2.

Understanding these numbers is crucial in quantum mechanics and provides insights into an atom's chemical properties and bonding behavior.

Transition Metals

Transition metals are elements found in the middle of the periodic table, specifically in the d-block. These metals are characterized by their ability to form various oxidation states and complex ions, thanks to the partially filled d orbitals.
Transition metals, like titanium, often have properties such as high melting points, good conductivity, and the ability to form colored compounds. Their electron configurations include the progressive filling of the 3d, 4d, and 5d orbitals, which affects their chemical and physical behavior.

• This group of metals plays a critical role in industries and technology, forming the backbone of many alloys and catalysts.
• The variable oxidation states allow them to participate in a variety of chemical reactions, which is crucial for metallurgical processes and biological functions in some cases.
• Their unique electron configurations are what give them their extraordinary properties, such as conducting electricity and heat efficiently.