What will be the correct representation of quatum numbers f the last electron entered into Ce?

Options

A. `n=4,l=3,m=-3,s=(-1)/(2)`
B. `n=4,l=3,m=0,s=(-1)/(2)`
C. `n=4,l=2,m=-3,s(+1)/(2)`
D. `n=4,l=3,m=+2,s=(-1)/(2)`

Explanation

To determine the quantum numbers of the last electron entered into cerium (Ce), we need to follow these steps:

### Step 1: Identify the Atomic Number
Cerium has an atomic number of 58. This means it has 58 electrons.

### Step 2: Determine the Electron Configuration
The electron configuration of cerium can be written as:
\[ \text{Ce: } [Xe] 6s^2 4f^1 5d^1 \]
This indicates that cerium has:
- 2 electrons in the 6s subshell
- 1 electron in the 4f subshell
- 1 electron in the 5d subshell

### Step 3: Identify the Last Electron
The last electron that enters cerium is in the 4f subshell. Therefore, we need to focus on the quantum numbers for the 4f electron.

### Step 4: Assign the Principal Quantum Number (n)
The principal quantum number (n) indicates the energy level of the electron. For the 4f subshell:
\[ n = 4 \]

### Step 5: Assign the Azimuthal Quantum Number (l)
The azimuthal quantum number (l) describes the subshell type:
- For s subshell, l = 0
- For p subshell, l = 1
- For d subshell, l = 2
- For f subshell, l = 3

Since we are dealing with the f subshell:
\[ l = 3 \]

### Step 6: Assign the Magnetic Quantum Number (m_l)
The magnetic quantum number (m_l) can take values from -l to +l. For f subshell (l = 3):
\[ m_l = -3, -2, -1, 0, +1, +2, +3 \]
Since there is only one electron in the 4f subshell, we can choose:
\[ m_l = -3 \]

### Step 7: Assign the Spin Quantum Number (m_s)
The spin quantum number (m_s) can be either +1/2 or -1/2. We can assign:
\[ m_s = -\frac{1}{2} \] (for example, but it could also be +1/2 depending on the electron's spin direction)

### Final Representation of Quantum Numbers
Thus, the quantum numbers for the last electron in cerium (4f electron) are:
- \( n = 4 \)
- \( l = 3 \)
- \( m_l = -3 \)
- \( m_s = -\frac{1}{2} \)

### Summary of Quantum Numbers
The correct representation of the quantum numbers for the last electron entered into Ce is:
\[ (n, l, m_l, m_s) = (4, 3, -3, -\frac{1}{2}) \]

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