Question

Give a possible set of values of the four quantum numbers for the \(4 s\) and \(3 d\) electrons in titanium.

Short answer

The possible sets of quantum numbers for 4s electrons in titanium are: - \((n, l, m_l, m_s) = (4, 0, 0, +\frac{1}{2})\) - \((n, l, m_l, m_s) = (4, 0, 0, -\frac{1}{2})\) For 3d electrons, we have four possible sets of quantum numbers (ignoring the cases of 3 other m_l values): - \((n, l, m_l, m_s) = (3, 2, m_{l1}, +\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l1}, -\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, +\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, -\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\)

Step-by-step solution

Determine the electron configuration of titanium

Titanium has an atomic number of 22, and its electron configuration can be written as: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2\] Now we can analyze the 4s and 3d electrons:

Quantum numbers for 4s electrons

1. Principal quantum number (n): For the 4s orbital, the principal quantum number is \(n = 4\). 2. Angular momentum quantum number (l): For s orbitals, the angular momentum quantum number is \(l = 0\). 3. Magnetic quantum number (m_l): For l = 0, there is only one possible value of the magnetic quantum number, \(m_l = 0\). 4. Electron Spin quantum number (m_s): There are two possibilities for the electron spin quantum number: either \(m_s = +\frac{1}{2}\) or \(m_s = -\frac{1}{2}\). Therefore, the two possible sets of quantum numbers for 4s electrons in titanium are: - \((n, l, m_l, m_s) = (4, 0, 0, +\frac{1}{2})\) - \((n, l, m_l, m_s) = (4, 0, 0, -\frac{1}{2})\)

Quantum numbers for 3d electrons

1. Principal quantum number (n): For the 3d orbital, the principal quantum number is \(n = 3\). 2. Angular momentum quantum number (l): For d orbitals, the angular momentum quantum number is \(l = 2\). 3. Magnetic quantum number (m_l): For l = 2, we have five possible values of the magnetic quantum number: \(-2, -1, 0, 1, 2\). However, the titanium atom has only two 3d electrons; therefore, m_l can have any two of the five possible values (we don't know which two values those are). 4. Electron Spin quantum number (m_s): There are two possibilities for the electron spin quantum number: either \(m_s = +\frac{1}{2}\) or \(m_s = -\frac{1}{2}\). So, for the 3d electrons, we have four possible sets of quantum numbers (ignoring the cases of 3 other m_l values): - \((n, l, m_l, m_s) = (3, 2, m_{l1}, +\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l1}, -\frac{1}{2})\), where \(m_{l1}\) is one of \(-2, -1, 0, 1, 2\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, +\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\) - \((n, l, m_l, m_s) = (3, 2, m_{l2}, -\frac{1}{2})\), where \(m_{l2}\) is one of \(-2, -1, 0, 1, 2\) and \(m_{l2} \neq m_{l1}\)

Key concepts

Electron Configuration

Electron configuration is a method used to distribute electrons among the various orbitals of an atom. It provides a detailed portrayal of an atom's electron arrangement, revealing how the electrons are organized around the nucleus. The configuration is vital for understanding many chemical properties of elements.

• Electrons fill orbitals in a specific order, following the Aufbau principle, which states that electrons occupy the lowest energy orbitals first. This forms the basis for constructing electron configurations.
• Pauli’s exclusion principle permits only two electrons per orbital, with opposite spins. And Hund's rule advises that electrons will occupy degenerate orbitals (orbitals with the same energy) singly first before any orbital gets a second electron.

For titanium, with an atomic number of 22, its electron configuration is presented as:\[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2\].
This distribution provides titanium with distinct properties and a stepping stone for predicting its reactivity and bonding characteristics.

Electron Orbitals

Electron orbitals are regions around the nucleus where the probability of finding an electron is high. Each orbital can hold a maximum of two electrons.

• s Orbitals: These orbitals are spherical in shape and are present in every principal energy level, with only one orbital existing per level.
• p Orbitals: Shaped like dumbbells, these orbitals appear starting from the second principal energy level. Each level has three p orbitals (px, py, pz) which can hold a total of six electrons.
• d Orbitals: More intricate in shape, these orbitals start from the third principal energy level. There are five d orbitals in a given level, capable of accommodating a total of ten electrons.

The knowledge of electron orbitals aids in visualizing electron arrangements and predicting how atoms will interact chemically with one another.

Titanium Electron Structure

The titanium electron structure is pivotal in cementing its position in the periodic table and understanding its chemical behavior. With an atomic number of 22, titanium’s electron configuration is \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2\].
The 4s electrons (2 of them) are taken into play before the 3d orbitals start filling. The reason these 4s electrons fill before the 3d is due to their lower energy level. This arrangement sheds light on several characteristics of titanium:

• Chemical Stability: Titanium's limited reactivity in the 3d orbitals contributes to its strength and resistance to corrosion.
• Behavior in Compounds: Its typical oxidation states (like +2, +3, and +4) are reliant on its ability to lose these outer electrons.

Understanding titanium's electron structure not only enriches the appreciation of its role in alloys and industrial applications but also its significant contribution to modern technology.