Find total no. of orbitals in nickel which have `abs(m)le1` and at least one electron is present, where 'm' is magnetic quantum number.

Answer

To find the total number of orbitals in nickel (Ni) that have \(|m| \leq 1\) and at least one electron present, we can follow these steps:

### Step 1: Determine the electron configuration of Nickel
Nickel has an atomic number of 28, which means it has 28 electrons. The electron configuration for nickel is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^8 \]

### Step 2: Identify the relevant orbitals
We need to find the orbitals where the magnetic quantum number \(m\) satisfies \(|m| \leq 1\). The possible values of \(m\) for different subshells are:
- For \(s\) orbitals (\(l = 0\)): \(m = 0\)
- For \(p\) orbitals (\(l = 1\)): \(m = -1, 0, +1\)
- For \(d\) orbitals (\(l = 2\)): \(m = -2, -1, 0, +1, +2\)

Given that we are interested in \(|m| \leq 1\), we will consider the following orbitals:
- \(2p\) (3 orbitals with \(m = -1, 0, +1\))
- \(3p\) (3 orbitals with \(m = -1, 0, +1\))
- \(3d\) (3 orbitals with \(m = -1, 0, +1\))

### Step 3: Check for electrons in the relevant orbitals
- **2p orbitals**: There are 6 electrons in the 2p subshell, so all three orbitals (\(m = -1, 0, +1\)) have at least one electron.
- **3p orbitals**: There are also 6 electrons in the 3p subshell, so all three orbitals (\(m = -1, 0, +1\)) have at least one electron.
- **3d orbitals**: There are 8 electrons in the 3d subshell. The 3d subshell has 5 orbitals, but we are only interested in those with \(m = -1, 0, +1\). Since there are 8 electrons, at least one electron is present in the orbitals with \(m = -1, 0, +1\).

### Step 4: Count the total number of orbitals
Now, we can count the total number of orbitals that meet the criteria:
- From the 2p subshell: 3 orbitals
- From the 3p subshell: 3 orbitals
- From the 3d subshell: 3 orbitals

Adding these together gives:
\[ 3 \, (\text{from } 2p) + 3 \, (\text{from } 3p) + 3 \, (\text{from } 3d) = 9 \, \text{orbitals} \]

### Final Answer
The total number of orbitals in nickel that have \(|m| \leq 1\) and at least one electron present is **9**.

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