Problem 34 - Chemistry and Chemical Reactivity

Question

Which of the following represent valid sets of quantum numbers? For a set that is invalid, explain briefly why it is not correct. (a) \(n=3, \ell=3, m_{\ell}=0\) (b) \(n=2, \ell=1, m_{\ell}=0\) (c) \(n=6, \ell=5, m_{\ell}=-1\) (d) \(n=4, \ell=3, m_{\ell}=-4\)

Step-by-Step Solution

Answer

Sets (b) and (c) are valid, while (a) and (d) are not.

Step 1: Understand Quantum Numbers

Quantum numbers are sets of numerical values that describe the quantum state of an electron in an atom. The main quantum numbers are:- Principal quantum number (\(n\)): Determines the energy level, and can be any positive integer (\(n = 1, 2, 3, \ldots\)).- Azimuthal (angular momentum) quantum number (\(\ell\)): Determines the shape of the electron's orbital, and can be any integer from 0 to \(n-1\).- Magnetic quantum number (\(m_{\ell}\)): Describes the orientation of the orbital,and can be an integer value from \(-\ell\) to \(\ell\).Let's evaluate each set of quantum numbers against these rules.

Step 2: Evaluate Set (a)

For(a) \(n=3, \ell=3, m_{\ell}=0\):- \(\ell\) must be less than \(n\). Here, \(\ell = 3\), which is not valid because \(\ell\) should be between 0 and \(n-1 = 2\). Thus, \(\ell = 3\) is invalid for \(n = 3\).

Step 3: Evaluate Set (b)

For(b) \(n=2, \ell=1, m_{\ell}=0\):- \(\ell = 1\) is valid because \(\ell\) is less than \(n\) and can be 0 or 1 for \(n=2\).- \(m_{\ell} = 0\) is valid because it lies between \(-\ell\) and \(\ell\), i.e., -1 to 1.- Therefore, this set is valid.

Step 4: Evaluate Set (c)

For(c) \(n=6, \ell=5, m_{\ell}=-1\):- \(\ell = 5\) is valid because \(\ell\) can be between 0 and \(n-1 = 5\) for \(n=6\).- \(m_{\ell} = -1\) is valid because it must range from \(-\ell\) to \(\ell\), which is from -5 to 5 here.- This is a valid set of quantum numbers.

Step 5: Evaluate Set (d)

For(d) \(n=4, \ell=3, m_{\ell}=-4\):- \(\ell = 3\) is valid since \(\ell\) is less than \(n\) and can range from 0 to 3 for \(n=4\).- However, \(m_{\ell} = -4\) is invalid because \(m_{\ell}\) should range from \(-\ell\) to \(\ell\), which is from -3 to 3 here. Therefore, \(m_{\ell} = -4\) is not valid for \(\ell = 3\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a fundamental aspect of quantum mechanics used to specify the electron's energy level in an atom. It determines the size and energy associated with an electron shell.

• The value of \(n\) can be any positive integer: 1, 2, 3, and so on. This means it represents whole numbers, not fractions or negatives.
• As \(n\) increases, the electron’s energy and the distance from the nucleus both increase. Consequently, the electron is found further from the nucleus in higher energy levels.
• For example, if \(n = 2\), it specifies the second energy level or shell where electrons can be found.

An essential rule for principal quantum numbers is that they must be greater than zero. This quantum number gives us a primary organizational framework for understanding how electrons are arranged in atoms.

Azimuthal Quantum Number

The azimuthal quantum number, or angular momentum quantum number, \(\ell\), describes the shape of an electron's orbital. It plays a critical role in determining the electron's path around the nucleus.

• The value of \(\ell\) depends on the principal quantum number \(n\) and ranges from 0 to \(n-1\).
• Different values of \(\ell\) correspond to different orbital shapes: \(\ell = 0\) corresponds to s orbitals (spherical), \(\ell = 1\) to p orbitals (dumbbell-shaped), \(\ell = 2\) to d orbitals, and \(\ell = 3\) to f orbitals.
• For instance, if \(n = 3\), the possible values for \(\ell\) are 0, 1, and 2.

The azimuthal quantum number gives us deeper insight into the spatial distribution of electron probability, revealing the electron's cloud shape around an atom.

Magnetic Quantum Number

The magnetic quantum number, represented as \(m_{\ell}\), dictates the orientation of an electron’s orbital within a magnetic field. This quantum number helps identify the specific orbital within a given subshell.

• The range of values for \(m_{\ell}\) is determined by \(\ell\) and can range from \(-\ell\) to \(\ell\). This includes zero, offering a total of \(2\ell + 1\) possible orientations for each orbital.
• For example, with \(\ell = 1\), possible \(m_{\ell}\) values are -1, 0, and 1, aligning with three different p orbitals.
• Each \(m_{\ell}\) represents an orbital orientation in three-dimensional space relative to the nucleus.

Understanding \(m_{\ell}\) is crucial when it comes to visualizing how electrons occupy different orbitals and in predicting the behavior of these orbitals in external magnetic fields.