Problem 79 - Chemistry: An Atoms-Focused Approach

Question

What are the possible values of quantum number \(\ell\) when \(n=4 ?\)

Step-by-Step Solution

Answer

Answer: The possible values of \(\ell\) when \(n=4\) are 0, 1, 2, and 3.

Step 1: Understanding quantum numbers

In this problem, we're dealing with two quantum numbers: the principal quantum number (\(n\)) and the angular momentum quantum number (\(\ell\)). The principal quantum number \(n\) represents the energy level of the electron in an atom and can be any positive integer. The angular momentum quantum number \(\ell\) describes the shape of the electron's orbital and can have integral values ranging from \(0\) to \(n-1\).

Step 2: Set up the equation for the range of \(\ell\)

We are given the principal quantum number \(n=4\). The range of possible values of the angular momentum quantum number \(\ell\) can be determined using the rule that \(\ell\) ranges from \(0\) to \(n-1\). So, we can write this relationship as: 0 ≤ \(\ell\) ≤ \(n-1\).

Step 3: Find the possible values of \(\ell\)

Now we will substitute the given value of \(n\) into the equation to find the range of possible values of \(\ell\). For \(n = 4\), we have: 0 ≤ \(\ell\) ≤ \(4-1\). Therefore, 0 ≤ \(\ell\) ≤ 3. This means the possible values of \(\ell\) when \(n=4\) are: \(\ell = 0, 1, 2, 3\).