Question

Summarize the rules for the allowable values of the four quantum numbers of an electron in an atom.

Short answer

The four quantum numbers are , u\textit\(l\), u\textit\(m_l\), u\textit\(m_s\) with specific sets of possible values for each.

Step-by-step solution

Principal Quantum Number (\textit\(n\))

The principal quantum number, denoted as \textit\(n\), indicates the energy level of the electron in an atom and its most probable distance from the nucleus. It can take any positive integer value: \textit\(n = 1, 2, 3, 4, \ldots\)

Angular Momentum Quantum Number (\textit\(l\))

The angular momentum quantum number, denoted as \textit\(l\), describes the shape of the orbital. It can take integer values from 0 up to \textit\(n-1\): \textit\(l = 0, 1, 2, ..., (n-1)\)

Magnetic Quantum Number (\textit\(m_l\))

The magnetic quantum number, denoted as \textit\(m_l\), specifies the orientation of the orbital in space. It can take integer values ranging from \textit\(-l\) to \textit\(+l\): \textit\(m_l = -l, -(l-1), ..., -1, 0, 1, ..., (l-1), +l\)

Spin Quantum Number (\textit\(m_s\))

The spin quantum number, denoted as \textit\(m_s\), describes the intrinsic spin of the electron within its orbital. It can take one of two possible values: \textit\(m_s = -\frac{1}{2}\) or \textit\(+\frac{1}{2}\)

Key concepts

Principal Quantum Number

The principal quantum number, symbolized as \( n \), is one of the four quantum numbers that describe the state of an electron in an atom. It tells us about the energy level the electron resides in and how far it is from the nucleus. The principal quantum number can be any positive integer: \( n = 1, 2, 3, 4, \text{and so on} \). The larger the value of \( n \), the higher the energy level and the farther the electron is from the nucleus. For example, if \( n = 1 \), the electron is in the first energy level, closest to the nucleus.

Angular Momentum Quantum Number

The angular momentum quantum number is represented by the symbol \( l \). This quantum number is essential for describing the shape of the electron's orbital. The value of \( l \) depends on the principal quantum number \( n \). It can take any integer value from 0 to \( n-1 \). For instance, if \( n = 3 \), then \( l \) can be 0, 1, or 2. Different values of \( l \) correspond to different shapes of orbitals:

• \( l = 0 \): s - orbital (spherical shape)
• \( l = 1 \): p - orbital (dumbbell shape)
• \( l = 2 \): d - orbital (cloverleaf shape)
• \( l = 3 \): f - orbital (complex shapes)

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), gives us information about the orientation of the orbital in three-dimensional space. The value of \( m_l \) depends on the angular momentum quantum number \( l \). It can range from \( -l \) to \( +l \), including zero. This means \( m_l \) can be any integer in this range. For example, if \( l = 2 \), \( m_l \) can be -2, -1, 0, 1, or 2. These different values correspond to different orientations of the orbital in space.
Understanding the magnetic quantum number is crucial for predicting how electrons interact within magnetic fields.

Spin Quantum Number

The spin quantum number, represented by \( m_s \), describes the intrinsic spin of the electron. This intrinsic property can be thought of as the electron spinning on its axis, which gives it a magnetic moment. The spin quantum number can only have one of two possible values: \( m_s = -\frac{1}{2} \) or \( m_s = +\frac{1}{2} \). These two values represent the two possible spin states of an electron:

• \( m_s = +\frac{1}{2} \): Spin-up state
• \( m_s = -\frac{1}{2} \): Spin-down state

Electrons in the same orbital must have opposite spins due to the Pauli Exclusion Principle, which states that no two electrons can have the same set of quantum numbers.