Question

Suppose the rules for assigning quantum numbers were as follows $$ \begin{array}{c} \mathrm{n}=1,2,3, \ldots \\ \ell=0,1,2, \ldots, \mathrm{n} \\ \mathrm{m}_{\ell}=0,1,2, \ldots, \ell+1 \\ \mathrm{~m}_{\mathrm{s}}=+\frac{1}{2} \text { or }-\frac{1}{2} \end{array} $$ Prepare a table similar to Table 6.3 based on these rules for \(\mathbf{n}=1\) and \(\mathbf{n}=2\). Give the electron configuration for an tom with eight electrons

Short answer

In summary, we have followed the given rules for assigning quantum numbers to create a table for n=1 and n=2. Then we used these rules to determine the electron configuration for an atom with eight electrons. The resulting electron configuration is written as 1s² (1s+1)² 2s² 1p², showing the distribution of electrons in different orbitals.

Step-by-step solution

Familiarize with the new rules

The rules for assigning quantum numbers in this exercise are different from the usual ones. They are as follows: $$ \begin{gathered} n=1,2,3, \ldots \\ \ell=0,1,2, \ldots, n \\ m_{\ell}=0,1,2, \ldots, \ell+1 \\ m_{s}=+\frac{1}{2} \text { or }-\frac{1}{2} \end{gathered} $$

Create the table for n=1 and n=2

Following the given rules, we can prepare a table for n=1 and n=2 as follows: | n | \(\ell\) | \(m_{\ell}\) | \(m_s\) | |---|-------|----------|-----| | 1 | 0 | 0 | +1/2 | | 1 | 0 | 0 | -1/2 | | 1 | 0 | 1 | +1/2 | | 1 | 0 | 1 | -1/2 | | 1 | 1 | 0 | +1/2 | | 1 | 1 | 0 | -1/2 | | 2 | 0 | 0 | +1/2 | | 2 | 0 | 0 | -1/2 | | 2 | 0 | 1 | +1/2 | | 2 | 0 | 1 | -1/2 | | 2 | 0 | 2 | +1/2 | | 2 | 0 | 2 | -1/2 | | 2 | 1 | 0 | +1/2 | | 2 | 1 | 0 | -1/2 | | 2 | 1 | 1 | +1/2 | | 2 | 1 | 1 | -1/2 | | 2 | 1 | 2 | +1/2 | | 2 | 1 | 2 | -1/2 | | 2 | 2 | 0 | +1/2 | | 2 | 2 | 0 | -1/2 |

Determine the electron configuration for an atom with eight electrons

Using the given rules, electron configuration for an atom with eight electrons is as follows: 1. Two electrons in n=1, \(\ell\)=0, \(m_{\ell}\)=0, (\(m_s\) = +1/2 and -1/2) 2. Two electrons in n=1, \(\ell\)=0, \(m_{\ell}\)=1, (\(m_s\) = +1/2 and -1/2) 3. One electron in n=1, \(\ell\)=1, \(m_{\ell}\)=0, \(m_s\)=+1/2 4. One electron in n=1, \(\ell\)=1, \(m_{\ell}\)=0, \(m_s\)=-1/2 5. One electron in n=2, \(\ell\)=0, \(m_{\ell}\)=0, \(m_s\)=+1/2 6. One electron in n=2, \(\ell\)=0, \(m_{\ell}\)=0, \(m_s\)=-1/2 The electron configuration for an atom with eight electrons can be written as: $$1s^2 (1s+1)^2 2s^2 1p^2$$