Question

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

Short answer

Possible sets of quantum numbers for the 4s electrons in Titanium are: 1. n=4, l=0, ml=0, ms=+1/2 2. n=4, l=0, ml=0, ms=-1/2 Possible sets of quantum numbers for the 3d electrons in Titanium are: 1. n=3, l=2, ml=-2, ms=+1/2 2. n=3, l=2, ml=-2, ms=-1/2 3. n=3, l=2, ml=-1, ms=+1/2 4. n=3, l=2, ml=-1, ms=-1/2

Step-by-step solution

Identify the quantum numbers for the 4s electrons

Principal quantum number (n): 4 (since it's the 4s subshell) Angular momentum quantum number (l): 0 (s orbitals have l=0) Magnetic quantum number (ml): 0 (s orbitals have only one value for ml, 0) Spin quantum number (ms): ±1/2 (since there are 2 electrons in the 4s subshell, one will have ms=+1/2 and the other ms=-1/2) The possible set of quantum numbers for the 4s electrons in Titanium are: 1. n=4, l=0, ml=0, ms=+1/2 2. n=4, l=0, ml=0, ms=-1/2

Identify the quantum numbers for the 3d electrons

Principal quantum number (n): 3 (since it's the 3d subshell) Angular momentum quantum number (l): 2 (d orbitals have l=2) Magnetic quantum number (ml): -2, -1, 0, 1, 2 (d orbitals have 5 possible values for ml) Spin quantum number (ms): ±1/2 (since there are 2 electrons in the 3d subshell, one will have ms=+1/2 and the other ms=-1/2) The possible set of quantum numbers for the 3d electrons in Titanium are: 1. n=3, l=2, ml=-2, ms=+1/2 2. n=3, l=2, ml=-2, ms=-1/2 3. n=3, l=2, ml=-1, ms=+1/2 4. n=3, l=2, ml=-1, ms=-1/2 (Note: We only have 2 electrons in the 3d subshell, so we stop listing the quantum numbers at ml=-1. If more electrons were present, we would continue listing quantum numbers with the other possible ml values.)

Key concepts

Principal Quantum Number

The principal quantum number, denoted as "n", is one of the key quantum numbers necessary to describe the arrangement of electrons in an atom. It primarily indicates the size and energy level of the electron's orbit, as electrons orbit the nucleus in specified energy levels. Larger values of "n" indicate electrons are further from the nucleus and usually have higher energy levels.
For example, in the exercise, the number "n" is given for two sets of electrons in titanium:

• For the 4s electrons, "n" is 4, meaning these electrons reside in the fourth shell with a higher energy state compared to those in lower shells.


• For the 3d electrons, "n" is 3, indicating these electrons are located in the third shell, at a moderately high energy level.

Principal quantum numbers play a key role when considering the order in which electron shells fill, known as the Aufbau principle. They also help you to determine the atom's general size.

Angular Momentum Quantum Number

Also known as the azimuthal quantum number, the angular momentum quantum number "l" determines the shape of an electron's orbital. Each type of orbital has a different "l" value:

• s orbitals: l = 0
• p orbitals: l = 1
• d orbitals: l = 2
• f orbitals: l = 3

In the exercise, titanium's 4s electrons have "l = 0," indicating a spherical s orbital. Meanwhile, the 3d electrons of titanium have "l = 2," indicating a d-shaped orbital.
This value is essential for predicting an atom's chemical behavior, as electron interactions often depend on the shape of the orbitals they occupy.

Magnetic Quantum Number

The magnetic quantum number, denoted as "ml," describes the orientation of an electron's orbital in a three-dimensional space around the nucleus. The values of "ml" range from -l to +l, which results in different spatial orientations.
For titanium, the 4s electrons have an "ml" value of 0 since s orbitals only have one possible orientation. Conversely, the 3d electrons can have "ml" values of -2, -1, 0, 1, or 2. These different orientations explain why d orbitals can accommodate multiple electrons in varied positions.
Understanding "ml" is crucial, especially when analyzing magnetic properties and the splitting of spectral lines, which arise due to the spatial distribution of electrons.

Spin Quantum Number

The spin quantum number, symbolized by "ms," describes the intrinsic angular momentum or "spin" of an electron. Electrons have a property akin to spinning on an axis, with possible spin states of +1/2 or -1/2, representing opposite spin directions.
In titanium's case, for both the 4s and 3d electrons, "ms" can be either +1/2 or -1/2, reflecting their paired spins in each orbital.

• Paired electron spins are essential for minimizing electron repulsion within orbitals, balancing the overall spin in an atom.


• This quantum number plays a vital role in defining the magnetic behavior of materials.

Understanding spin quantum numbers can provide insights into the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers.