Given
(a) `n=5,m_(i)=+1` (b) `n=2,l=1,m_(i)=-1,m_(s)=-1//2`
The maximum number of electron(s) in an atom that can have the quanturm numbers as given in (a) and (b) are respectively:

Options

A. 8 and 1
B. 25 and 1
C. 2 and 4
D. 4 and 1

Answer

To solve the question regarding the maximum number of electrons that can have the specified quantum numbers, we will analyze each case step by step.

### Step 1: Analyze Case (a) where n=5 and m_i=+1

1. **Identify the values of l**:
- For a principal quantum number \( n = 5 \), the possible values of the azimuthal quantum number \( l \) can be \( 0, 1, 2, 3, 4 \).

2. **Determine the magnetic quantum number (m_i)**:
- The magnetic quantum number \( m_i \) can take values from \( -l \) to \( +l \).
- For \( l = 0 \): \( m_i = 0 \) (1 value)
- For \( l = 1 \): \( m_i = -1, 0, +1 \) (3 values)
- For \( l = 2 \): \( m_i = -2, -1, 0, +1, +2 \) (5 values)
- For \( l = 3 \): \( m_i = -3, -2, -1, 0, +1, +2, +3 \) (7 values)
- For \( l = 4 \): \( m_i = -4, -3, -2, -1, 0, +1, +2, +3, +4 \) (9 values)

3. **Count the valid m_i values**:
- Since \( m_i = +1 \) is valid for \( l = 1, 2, 3, 4 \), we have:
- \( l = 1 \): 1 valid \( m_i \)
- \( l = 2 \): 1 valid \( m_i \)
- \( l = 3 \): 1 valid \( m_i \)
- \( l = 4 \): 1 valid \( m_i \)
- Total valid \( m_i \) values = 4.

4. **Determine the spin quantum number (m_s)**:
- Each \( m_i \) can have 2 possible spin states: \( +\frac{1}{2} \) and \( -\frac{1}{2} \).
- Therefore, the total number of electrons for \( n=5 \) and \( m_i=+1 \) is:
\[ \text{Total electrons} = \text{Number of valid } m_i \times 2 = 4 \times 2 = 8. \]

### Step 2: Analyze Case (b) where n=2, l=1, m_i=-1, m_s=-1/2

1. **Identify the values of m_i**:
- For \( n = 2 \) and \( l = 1 \), the possible values of \( m_i \) are \( -1, 0, +1 \).

2. **Check the given m_i**:
- The given \( m_i = -1 \) is valid.

3. **Determine the spin quantum number**:
- The given spin quantum number \( m_s = -\frac{1}{2} \) indicates that we are looking for the specific electron with this spin state.

4. **Count the number of electrons**:
- Since we are given a specific \( m_i \) and \( m_s \), there can only be one electron that fits these quantum numbers:
\[ \text{Total electrons} = 1. \]

### Final Answer:
- The maximum number of electrons for (a) is **8** and for (b) is **1**.