Problem 104 - Chemistry

Question

$$ \begin{array}{lccccc} \lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\ \hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\ & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} \end{array} $$ (a) What is the lowest possible value of the principal quantum number \((n)\) when the angular momentum quantum number \((\ell)\) is \(1 ?\) (b) What are the possible values of the angular momentum quantum number ( \(\ell\) ) when the magnetic quantum number \(\left(m_{\ell}\right)\) is 0 , given than \(n \leq 4 ?\)

Step-by-Step Solution

Answer

The lowest possible value for the principal quantum number when the angular momentum quantum number is \(1\) is \(2\). The possible values for the angular momentum quantum number when the magnetic quantum number is \(0\) and the principal quantum number is less than or equal to \(4\) are \(0, 1, 2, 3\).

Step 1: Understand Quantum Number Relationships

Understand quantum number relationships. The principal quantum number, n, can be any positive integer. The angular momentum quantum number, l, is dependent on n and can take on any integer value from 0 to n-1. The magnetic quantum number, ml, is related to l and can take integer values from -l to +l, including 0.

Step 2: Find the quantum number n For part (a)

Notice that for l=1, according to the relationship between n and l, the smallest possible value for the principal quantum number n that allows l to be 1 would be n=2, as n has to be at least one unit larger than l.

Step 3: Identify the Possible Values of l For part (b)

The magnetic quantum number ml is 0, and we know n is less than or equal to 4. Using the relationship between ml and l, there are four possible scenarios for l, which are l=0, l=1, l=2, and l=3. These are the possible values of angular momentum when ml is 0, and n is less than or equal to 4.

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized by \( n \), is a fundamental part of the quantum mechanical model of the atom. It defines the energy level or shell where an electron resides. Unlike other quantum numbers, \( n \) can only take positive integer values (1, 2, 3, etc.).

Think of \( n \) as the size or distance of an electron's path around the nucleus.

• Higher \( n \) values mean the electron is further from the nucleus and has higher energy.
• The larger the \( n \), the greater the number of available subshells (types of orbitals).

In our specific example, when \( \ell = 1 \) (where \( \ell \) represents the angular momentum quantum number, more on that later), the smallest \( n \) can be is \( 2 \), since \( \ell \) must be less than \( n \).

This relation implies that \( n \) being greater than \( \ell \) ensures there is an available subshell for electrons to occupy.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \( \ell \), is related to the shape of the electron's orbital. It determines the subshell that the electron belongs to, and each subshell corresponds to a different orbital shape.

The value of \( \ell \) ranges from 0 to \( n-1 \). Here are a few details about \( \ell \):

• \( \ell = 0 \) corresponds to an "s" orbital, which is spherical.
• \( \ell = 1 \) corresponds to a "p" orbital, which has a dumbbell shape.
• \( \ell = 2 \) corresponds to a "d" orbital, with a more complex shape.
• \( \ell = 3 \) relates to an "f" orbital with an even more complicated geometry.

For this particular exercise, given a magnetic quantum number \( m_{\ell} \) of 0, possible \( \ell \) values include 0, 1, 2, and 3 when \( n \leq 4 \). The broader range of \( \ell \) signifies the different shapes the electron's path could take as it moves around the nucleus.

Magnetic Quantum Number

The magnetic quantum number, symbolized by \( m_{\ell} \), indicates the orientation of the electron’s orbital in space relative to the other orbitals. It plays a crucial role in defining how the subshell is arranged and how electrons might pair up within the atom.

In terms of value, \( m_{\ell} \) spans from \(-\ell\) to \( +\ell \), including zero. Consequently, each \( \ell \) value implies multiple possible \( m_{\ell} \) values which represent different orientations for a given shape of the orbital.

• For \( \ell = 0 \), \( m_{\ell} \) is 0, meaning just one orientation.
• For \( \ell = 1 \), \( m_{\ell} \) can be -1, 0, or +1.
• For \( \ell = 2 \), \( m_{\ell} \) ranges from -2 to +2.

In our exercise, for \( m_{\ell} = 0 \), the electron orientation is positioned without angular direction bias. By limiting the value of \( n \) to 4, our focus narrows to electron orbitals with relatively low energy levels, boasting several potential orientations.