Problem 74 - Chemistry An Atoms First Approach

Question

Which of the following sets of quantum numbers are not allowed? For each incorrect set, state why it is incorrect. a. \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\) b. \(n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}\) c. \(n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\) e. \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\) f. \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

The sets of quantum numbers that are not allowed are: a. Incorrect due to \(\ell = 3\) (should be between 0 and 2) d. Incorrect due to \(m_s = -1\) (should be \(-\frac{1}{2}\) or \(\frac{1}{2}\)) e. Incorrect due to \(\ell = -4\) (should be between 0 and 4) f. Incorrect due to \(m_\ell = 2\) (should be between \(-1\) and \(1\))

Step 1: Check set (a)

For set (a), we have \(n = 3\), \(\ell = 3\), \(m_\ell = 0\), and \(m_s = -\frac{1}{2}\). According to rule number 2, \(\ell\) should be between 0 and \(n-1\). In this case, \(\ell\) should be between 0 and 2. As \(\ell = 3\), this set of quantum numbers is not allowed. The value for \(\ell\) exceeds the maximum allowed value for this \(n\) value.

Step 2: Check set (b)

For set (b), we have \(n = 4\), \(\ell = 3\), \(m_\ell = 2\), and \(m_s = -\frac{1}{2}\). For this \(n\) value, \(\ell\) can take values from 0 to 3. As \(\ell = 3\) falls within this range, it is allowed. Also, \(m_\ell\) falls within its allowed range of \(-\ell\) to \(\ell\). And the value for \(m_s\) is permitted. So, this set of quantum numbers is allowed.

Step 3: Check set (c)

For set (c), we have \(n = 4\), \(\ell = 1\), \(m_\ell = 1\), and \(m_s = +\frac{1}{2}\). All the quantum numbers in this set are within their allowed ranges. Therefore, this set of quantum numbers is allowed.

Step 4: Check set (d)

For set (d), we have \(n = 2\), \(\ell = 1\), \(m_\ell = -1\), and \(m_s = -1\). The first three quantum numbers are within their allowed ranges. However, \(m_s\) can only take values \(-1/2\) or \(1/2\), and \(m_s = -1\) is not allowed. Therefore, this set of quantum numbers is not allowed due to the incorrect value of \(m_s\).

Step 5: Check set (e)

For set (e), we have \(n = 5\), \(\ell = -4\), \(m_\ell = 2\), and \(m_s = +\frac{1}{2}\). According to rule number 2, the value of \(\ell\) should be between 0 and \(n-1\). The value of \(\ell\) is negative and not allowed. Therefore, this set of quantum numbers is not allowed due to the incorrect value of \(\ell\).

Step 6: Check set (f)

For set (f), we have \(n = 3\), \(\ell = 1\), \(m_\ell = 2\), and \(m_s = -\frac{1}{2}\). For this \(\ell\) value, \(m_\ell\) can take values from \(-1\) to \(1\). The value of \(m_\ell\) given in the set is 2, which is not allowed. Therefore, this set of quantum numbers is not allowed due to the incorrect value of \(m_\ell\). To sum up, the sets of quantum numbers that are not allowed are: a. Incorrect due to \(\ell = 3\) (should be between 0 and 2) d. Incorrect due to \(m_s = -1\) (should be \(-1/2\) or \(1/2\)) e. Incorrect due to \(\ell = -4\) (should be between 0 and 4) f. Incorrect due to \(m_\ell = 2\) (should be between \(-1\) and \(1\))