# Worksheet 10A Solutions

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- 92731

### Solutions

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Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

The Pauli Exclusion Principle states that electrons in a single atom or molecule must have unique quantum numbers. This arises because they are fermions. Recall possible values for quantum numbers:

- Principal quantum number: \(n = 1, 2, 3...\)
- Angular quantum number: \(l = 0, 1, 2...n-1\)
- Magnetic quantum number: \(m_l = -l, -l+1,...0,..., l-1, l\)

### Q1

Consider helium in its ground electronic state. In what orbital will the electrons be found?

What is the value of the principal quantum number, n, for these electrons?

What is the value of the angular quantum number, l, for these electrons?

What is the value of the magnetic quantum number, ml, for these electrons?

What is the value of the spin quantum number, ms, for these electrons?

The total angular momentum in an atom comes from addition of the angular momentum for individual electrons. We do this by adding ml and ms values.

Quantum Number | Electron 1 | Electron 2 | Total | |
---|---|---|---|---|

\(l\) | ||||

\(m_l\) | \(M_l\) | |||

\(m_s\) | \(M_s\) |

We generate \(L\) and \(S\) values from the values of \(M_L\) and \(M_S\).

Using the TOTAL angular momentum, we designate atomic states. We use the same symbolism for atomic states as a atomic orbitals.

L value | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

designation | S | P | D | F | G |

Atomic state:

\[^{2S+1}L_J\]

The value of \(J\) comes from adding orbital, \(L\), and spin, \(S\), angular momentum.

\[J=|L-S|, |L-S+1|,...L+S-1, L-S\]

When an atom has a filled valence shell, we refer to it as a closed shell. For this situation, the quantum state is always the same. What is the atomic state arising from a closed shell configuration?

All electrons in complete, filled shells do not contribute to the atomic state and can be ignored. **We only consider the valence shell and any unfilled shells.**

If we promote one electron in helium to an excited state, what will be the values of the four quantum numbers for the two electrons?

Quantum Number | Electron 1 | Electron 2 | Total | |
---|---|---|---|---|

\(l\) | ||||

\(m_l\) | \(M_l\) | |||

\(m_s\) | \(M_s\) |

What atomic states arise from this configuration?

### Q3

What is the electron configuration of boron in its ground state?

Which electrons contribute to the boron atomic states?

What are the atomic states arising from the boron ground state?

### Q4

When an atom has a shell that is more than half filled, we consider the missing electrons when adding angular momentum rather than adding the angular momentum from the electrons present in the shell.

What is the electron configuration of fluorine in its ground state?

Which electrons contribute to the fluorine atomic states?

What are the atomic states arising from the fluorine ground state?

### Q5: Adding angular momentum from two electrons in an unfilled shell.

What is the electron configuration of carbon in its ground state? Which electrons contribute to the atomic states?

What is the principal quantum number, n, for the electrons in the valence shell?

What are the possible values of the angular quantum number, \(l\), for the electrons in the valence shell?

What are the possible values of the magnetic quantum number, \(m_l\), for the electrons in the valence shell?

What are the possible values of the spin quantum number, \(m_s\), for the electrons in the valence shell?

Remembering that electrons in the atom must have unique quantum numbers, what are the possible combinations of \(m_l\) and \(m_s\)?

Total \(M_L\) | \(m_l\) of Electron 1 | \(m_l\) of Electron 2 | \(m_s\) of Electron 1 | \(m_s\) of Electron 1 | Total \(M_s\) |
---|---|---|---|---|---|

To figure out the atomic states, we need to sum the \(m_l\) and sum the \(m_s\) quantum numbers to get \(M_L\) and \(M_S\), respectively.

\[M_L = \sum_{i=1}^N m_l(i)\]

\[M_S = \sum_{i=1}^N m_s(i)\]

If l = 1, what values are possible for \(m_l\)?

If L = 1, what values are possible for \(M_L\)?

If s = 1, what values are possible for \(m_s\)?

If S = 1, what values are possible for \(M_S\)?

### Q6

To have an atomic state with L=1, we must have the appropriate ML states to comprise the total \(L\) state, that is, when \(L=1\), \(M_L= -1, 0, 1\).

What is the largest value of \(M_L\) in your table for carbon? What value of \(L\) can lead to that value of \(M_L\)?

What is the value of \(M_S\) for this value of \(M_L\)? What value of \(S\) can lead to that value of \(M_S\)?

For this largest value of ML and thus L, what other values of ML must contribute to the state with this value of L?

What atomic states will arise from this combination of L and S?

What \(M_L\) and \(M_S\) values are left?