To solve the problem of finding the sum of the maximum number of electrons having \( m = +1 \) and \( m = -1 \) in Titanium (Ti), we will follow these steps:
### Step 1: Determine the electronic configuration of Titanium
Titanium has an atomic number of 22. Therefore, its electronic configuration is:
\[
\text{Ti: } 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 3d^2 \, 4s^2
\]
### Step 2: Identify the relevant orbitals and their quantum numbers
We need to focus on the orbitals where the magnetic quantum number \( m \) can take the values of \( +1 \) and \( -1 \). The relevant orbitals are:
- **p orbitals** (where \( l = 1 \)): The possible values of \( m \) are \( -1, 0, +1 \).
- **d orbitals** (where \( l = 2 \)): The possible values of \( m \) are \( -2, -1, 0, +1, +2 \).
### Step 3: Count the electrons in the p orbitals
From the electronic configuration:
- The 2p subshell has \( 6 \) electrons, which fill the orbitals as follows:
- \( m = -1 \): 2 electrons
- \( m = 0 \): 2 electrons
- \( m = +1 \): 2 electrons
- The 3p subshell also has \( 6 \) electrons, filled similarly:
- \( m = -1 \): 2 electrons
- \( m = 0 \): 2 electrons
- \( m = +1 \): 2 electrons
Thus, for the p orbitals (2p and 3p):
- Electrons with \( m = +1 \): 2 (from 2p) + 2 (from 3p) = 4 electrons
- Electrons with \( m = -1 \): 2 (from 2p) + 2 (from 3p) = 4 electrons
### Step 4: Count the electrons in the d orbitals
The 3d subshell has \( 2 \) electrons, which can be distributed in the five d orbitals:
- \( m = -2 \): 0 electrons
- \( m = -1 \): 1 electron
- \( m = 0 \): 0 electrons
- \( m = +1 \): 1 electron
- \( m = +2 \): 0 electrons
Thus, for the d orbitals:
- Electrons with \( m = +1 \): 1 electron
- Electrons with \( m = -1 \): 1 electron
### Step 5: Calculate the total number of electrons for \( m = +1 \) and \( m = -1 \)
Now we can sum the electrons:
- Total electrons with \( m = +1 \): 4 (from p) + 1 (from d) = 5 electrons
- Total electrons with \( m = -1 \): 4 (from p) + 1 (from d) = 5 electrons
### Step 6: Find the sum of the maximum number of electrons
Finally, we find the sum of the maximum number of electrons having \( m = +1 \) and \( m = -1 \):
\[
\text{Total} = 5 + 5 = 10
\]
### Final Answer
The sum of the maximum number of electrons having \( +1 \) and \( -1 \) value of \( m \) in Titanium is **10**.
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