How many electron in atom may have the following quantum number ? A `n = 4, m_(s) = -(1)/(2)` b `n = 3,l = 0`

Explanation

To solve the question of how many electrons in an atom may have the given quantum numbers, we will break it down into two parts as per the question.

### Part (a): For `n = 4` and `m_s = -1/2`

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n = 4 \) indicates that we are looking at the fourth energy level of an atom.

2. **Determine Possible Values of Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n = 4 \), \( l \) can be \( 0, 1, 2, \) or \( 3 \).
- Corresponding orbitals:
- \( l = 0 \) corresponds to the s orbital (2 electrons)
- \( l = 1 \) corresponds to the p orbital (6 electrons)
- \( l = 2 \) corresponds to the d orbital (10 electrons)
- \( l = 3 \) corresponds to the f orbital (14 electrons)

3. **Calculate Total Electrons**:
- The total number of electrons that can be accommodated in each of these orbitals is:
- s: 2 electrons
- p: 6 electrons
- d: 10 electrons
- f: 14 electrons
- Thus, the total number of electrons is:
\[ 2 + 6 + 10 + 14 = 32 \text{ electrons} \]

4. **Consider the Spin Quantum Number (m_s)**:
- The spin quantum number \( m_s = -\frac{1}{2} \) indicates that we are only considering electrons with this specific spin.
- Each orbital can hold two electrons with opposite spins. Thus, for each type of orbital, we can only count half of the total electrons:
- s: 1 electron (since it can have -1/2 and +1/2)
- p: 3 electrons (only counting the -1/2 spin)
- d: 5 electrons
- f: 7 electrons
- Therefore, the total number of electrons with \( m_s = -\frac{1}{2} \) is:
\[ 1 + 3 + 5 + 7 = 16 \text{ electrons} \]

### Part (b): For `n = 3` and `l = 0`

1. **Identify the Principal Quantum Number (n)**:
- Here, \( n = 3 \) indicates we are looking at the third energy level.

2. **Determine the Azimuthal Quantum Number (l)**:
- Given \( l = 0 \), this corresponds to the s orbital.

3. **Calculate Total Electrons**:
- The s orbital can hold a maximum of 2 electrons.

4. **Consider the Spin Quantum Number (m_s)**:
- Since no specific spin quantum number is given for this part, we can assume that both spins are possible. Therefore, we can have:
- 2 electrons (one with \( m_s = -\frac{1}{2} \) and one with \( m_s = +\frac{1}{2} \)).

### Final Answers:
- For part (a): **16 electrons** can have the quantum numbers \( n = 4 \) and \( m_s = -\frac{1}{2} \).
- For part (b): **2 electrons** can have the quantum numbers \( n = 3 \) and \( l = 0 \).