In Mn, the minimum number of electrons having `m_s = -1/2 ` is _____. 

Answer

To determine the minimum number of electrons in manganese (Mn) that have a spin quantum number \( m_s = -\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the Atomic Number of Manganese
Manganese (Mn) has an atomic number of 25. This means it has 25 electrons.

### Step 2: Write the Electronic Configuration
The electronic configuration of manganese can be determined by filling the orbitals in the order of increasing energy levels:
- 1s²
- 2s²
- 2p⁶
- 3s²
- 3p⁶
- 4s²
- 3d⁵

So, the complete electronic configuration of Mn is:
\[ \text{Mn: } 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5 \]

### Step 3: Count the Electrons in Each Subshell
Now, let's count the number of electrons in each subshell:
- 1s: 2 electrons
- 2s: 2 electrons
- 2p: 6 electrons
- 3s: 2 electrons
- 3p: 6 electrons
- 4s: 2 electrons
- 3d: 5 electrons

### Step 4: Determine the Spin of Electrons
The spin quantum number \( m_s \) can take values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). Electrons in the same orbital must have opposite spins (Pauli Exclusion Principle).

- For the 1s², 2s², 3s², and 4s² subshells:
- Each has 1 electron with \( m_s = +\frac{1}{2} \) and 1 electron with \( m_s = -\frac{1}{2} \) → contributes 1 electron with \( m_s = -\frac{1}{2} \) from each subshell.

Total from these subshells = 4 electrons with \( m_s = -\frac{1}{2} \).

- For the 2p⁶ and 3p⁶ subshells:
- Each has 3 electrons with \( m_s = +\frac{1}{2} \) and 3 electrons with \( m_s = -\frac{1}{2} \) → contributes 3 electrons with \( m_s = -\frac{1}{2} \) from each subshell.

Total from these subshells = 6 electrons with \( m_s = -\frac{1}{2} \).

- For the 3d⁵ subshell:
- According to Hund's rule, each of the 5 electrons will occupy separate orbitals with the same spin first (all \( m_s = +\frac{1}{2} \)). Therefore, there are no electrons with \( m_s = -\frac{1}{2} \) in this subshell.

### Step 5: Calculate the Total
Now, we sum the contributions:
- From 1s, 2s, 3s, and 4s: 4 electrons
- From 2p and 3p: 6 electrons
- From 3d: 0 electrons

Total number of electrons with \( m_s = -\frac{1}{2} \):
\[ 4 + 6 + 0 = 10 \]

### Final Answer
The minimum number of electrons in manganese (Mn) having \( m_s = -\frac{1}{2} \) is **10**.