Question

Write down a set of quantum numbers that uniquely defines each of the following atomic orbitals: (a) \(6 s\) (b) each of the five \(4 d\) orbitals.

Short answer

(a) 6s: \(n=6, l=0, m_l=0, m_s=\pm\frac{1}{2}\); (b) 4d: \(n=4, l=2, m_l=-2,-1,0,1,2, m_s=\pm\frac{1}{2}\).

Step-by-step solution

Understanding Quantum Numbers

In atomic physics, four quantum numbers are used to describe the properties of electrons in an atom: the principal quantum number \(n\), the azimuthal (angular momentum) quantum number \(l\), the magnetic quantum number \(m_l\), and the spin quantum number \(m_s\). These numbers together define the state of an electron uniquely.

Interpreting '6s' Orbital

For a '6s' orbital, the principal quantum number \(n\) is 6. Since it's an 's' orbital, the azimuthal quantum number \(l\) is 0. The magnetic quantum number, \(m_l\), can only be 0 as well since \(l = 0\). The electron's spin (\(m_s\)) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Writing Quantum Numbers for 6s Orbital

The quantum numbers for an electron in a '6s' orbital can be written as: \(n = 6, l = 0, m_l = 0, m_s = \pm\frac{1}{2}\), indicating two possible sets due to two potential spin values.

Interpreting '4d' Orbital

For a '4d' orbital, the principal quantum number \(n\) is 4. The azimuthal quantum number \(l\) for a 'd' orbital is 2. The magnetic quantum numbers \(m_l\) range from \(-2\) to \(2\), giving the integer set \(-2, -1, 0, 1, 2\). Each can pair with spins \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).

Writing Quantum Numbers for 4d Orbitals

For each of the five '4d' orbitals, the quantum numbers are: \(\{(n = 4, l = 2, m_l = -2, m_s = \pm\frac{1}{2})\), \((n = 4, l = 2, m_l = -1, m_s = \pm\frac{1}{2})\), \((n = 4, l = 2, m_l = 0, m_s = \pm\frac{1}{2})\), \((n = 4, l = 2, m_l = 1, m_s = \pm\frac{1}{2})\), \((n = 4, l = 2, m_l = 2, m_s = \pm\frac{1}{2})\)\}—resulting in 10 sets due to 5 orbitals each with two spin states.

Key concepts

Atomic Orbitals

Atomic orbitals are regions around an atom's nucleus where electrons are likely to be found. They have distinct shapes and sizes, which are determined by quantum numbers. These quantum numbers provide critical information about an electron's position and energy in an atom.

Each electron in an atom occupies a space defined by these atomic orbitals, described by different letters like 's', 'p', 'd', and 'f'. These letters show the shape of the orbital:

• 's' orbitals are spherical and can hold up to 2 electrons.
• 'p' orbitals are dumbbell-shaped and can hold up to 6 electrons.
• 'd' orbitals are more complex in shape and can accommodate up to 10 electrons.
• 'f' orbitals are even more intricate and can hold up to 14 electrons.

Understanding orbitals helps us to determine the probable location and energy level of an electron in an atom.

Principal Quantum Number

The principal quantum number, symbolized as \( n \), indicates the main energy level occupied by an electron.
It essentially tells us how far an electron is from the nucleus. The larger the value of \( n \), the greater the energy level and the larger the atom.

For example, in a '6s' orbital, the principal quantum number \( n \) is 6. This suggests that the electron is in the sixth energy level, far from the nucleus compared to lower \( n \) values.

• Higher \( n \) values also mean that the orbital is larger, allowing the electron more space to move.
• \( n \) can be any positive integer: 1, 2, 3, and so on.

Knowing the principal quantum number is essential for determining an electron's energy and its possible distance from the nucleus.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), gives us insight into the shape of the orbital an electron occupies.
It is also called the angular momentum quantum number.

The value of \( l \) can range from 0 to \( n-1 \), where \( n \) is the principal quantum number:

• When \( l = 0 \), the orbital is 's', which is spherical.
• When \( l = 1 \), it is 'p', shaped like a dumbbell.
• When \( l = 2 \), the orbital is 'd', with a more complex shape.
• When \( l = 3 \), the orbital is 'f'.

For a '4d' orbital, \( n = 4 \) and \( l = 2 \), reflecting its specific shape and orientation. The azimuthal quantum number is crucial in understanding the orbital's angular momentum and shape.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), describes the orientation of the orbital in space.
It shows how an orbital is aligned in an external magnetic field.

The values \( m_l \) can take are dependent on \( l \):

• For \( l = 0 \), \( m_l \) can only be 0 since there's only one orientation.
• For \( l = 1 \), \( m_l \) can be -1, 0, or +1, corresponding to the three 'p' orbital orientations.
• For \( l = 2 \), \( m_l \) ranges from -2 to +2, indicating five possible 'd' orientations.

Hence, for a '4d' orbital, \( m_l \) ranges from -2 to 2.
This variant value ensures the unique identification of each orbital within a subshell.