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Chapter 7: Problem 63

If the \(n\) quantum number of an atomic orbital is 4, what are the possible values of \(l\) ? If the \(l\) quantum number is 3, what are the possible values of \(m_{l}\) ?

Short Answer

Expert verified
Possible values of \(l\) are 0, 1, 2, 3; possible values of \(m_l\) are -3 to 3.

Step by step solution

01

Understanding Principal Quantum Number

The principal quantum number, denoted as \(n\), defines the energy level or shell of an atom where the electron resides. Given \(n = 4\), it indicates the fourth energy level.
02

Determining Azimuthal Quantum Number Values

The azimuthal quantum number, \(l\), can take values ranging from 0 to \(n-1\). For \(n = 4\), \(l\) can be 0, 1, 2, or 3.
03

Understanding Magnetic Quantum Number

The magnetic quantum number, \(m_l\), describes the orientation of the orbital within a subshell and can take integer values from \(-l\) to \(+l\).
04

Determining \(m_l\) Values for \(l = 3\)

If \(l = 3\), \(m_l\) can take any integer value from \(-3\) to \(+3\), including: -3, -2, -1, 0, +1, +2, +3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Quantum Number
The principal quantum number, represented by the symbol \( n \), is essential in understanding the structure of an atom. It tells us about the energy level or shell that an electron occupies in an atom. Essentially, \( n \) is like the address of the electron within an atom. The value of \( n \) can be any positive integer starting from 1.
  • For example, \( n = 1 \) represents the first energy level, \( n = 2 \) the second, and so forth.
  • The greater the value of \( n \), the higher the energy level and the farther the electron is from the nucleus.
In the exercise, we're given \( n = 4 \), which means the electron is in the fourth energy level, a higher energy state compared to \( n = 1 \) or \( n = 2 \). The principal quantum number is critical as it lays the groundwork for determining other quantum numbers like the azimuthal and magnetic quantum numbers.
Azimuthal Quantum Number
The azimuthal quantum number, denoted by \( l \), gives us insight into the subshell or sublevel that an electron occupies within a particular principal energy level. It is sometimes referred to as the orbital angular momentum quantum number.
  • The values that \( l \) can take range from 0 to \( n-1 \), where \( n \) is the principal quantum number.
  • Each value of \( l \) corresponds to a different type of subshell or orbital, such as \( s \), \( p \), \( d \), and \( f \).
For \( n = 4 \) in the exercise, \( l \) can be 0, 1, 2, or 3. This tells us that the possible subshells available at this energy level are:- \( 0 \): s subshell- \( 1 \): p subshell- \( 2 \): d subshell- \( 3 \): f subshellEach \( l \) value signifies a different shape and spatial distribution of the electron cloud within the atom, thus impacting how electrons interact with each other and the types of bonds they might form.
Magnetic Quantum Number
The magnetic quantum number, represented by \( m_l \), is crucial for describing the specific orientation or direction of an orbital within a subshell. It determines the number of orbitals and their orientation in space.
  • The values of \( m_l \) are integers ranging from \(-l\) to \(+l\).
  • For each value of \( l \), there are \((2l + 1)\) possible values of \( m_l\).
In the exercise, if \( l = 3 \), which corresponds to the \( f \) subshell, \( m_l \) can take any integer value from -3 to +3. These are:- \(-3, -2, -1, 0, +1, +2, +3\)The magnetic quantum number plays a significant role in determining the number of available orbitals that electrons can occupy within a given subshell and affects the magnetic properties of atoms and molecules.

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