To find the number of electrons in a Krypton (Kr) atom that satisfy the condition \( n \cdot l + m = 3 \), we will follow these steps:
### Step 1: Understand the Quantum Numbers
The quantum numbers for an electron in an atom are:
- Principal quantum number (\( n \)): Indicates the energy level.
- Azimuthal quantum number (\( l \)): Indicates the shape of the orbital.
- Magnetic quantum number (\( m \)): Indicates the orientation of the orbital.
### Step 2: Determine the Electron Configuration of Krypton
Krypton has an atomic number of 36, which means it has 36 electrons. The electron configuration of Krypton is:
\[
1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6
\]
### Step 3: Identify the Values of \( n \), \( l \), and \( m \)
We need to find combinations of \( n \), \( l \), and \( m \) such that:
\[
n \cdot l + m = 3
\]
### Step 4: Calculate Possible Combinations
Let’s evaluate possible values of \( n \), \( l \), and \( m \):
1. **For \( n = 1 \)**:
- \( l = 0 \) (only \( s \) orbital)
- \( m = 0 \)
- Calculation: \( 1 \cdot 0 + 0 = 0 \) (not valid)
2. **For \( n = 2 \)**:
- \( l = 0 \) (s orbital)
- \( m = 0 \)
- Calculation: \( 2 \cdot 0 + 0 = 0 \) (not valid)
- \( l = 1 \) (p orbital)
- Possible values for \( m \): -1, 0, +1
- Calculation for \( m = 1 \): \( 2 \cdot 1 + 1 = 3 \) (valid)
- Calculation for \( m = 0 \): \( 2 \cdot 1 + 0 = 2 \) (not valid)
- Calculation for \( m = -1 \): \( 2 \cdot 1 - 1 = 1 \) (not valid)
3. **For \( n = 3 \)**:
- \( l = 0 \) (s orbital)
- \( m = 0 \)
- Calculation: \( 3 \cdot 0 + 0 = 0 \) (not valid)
- \( l = 1 \) (p orbital)
- Possible values for \( m \): -1, 0, +1
- Calculation for \( m = 1 \): \( 3 \cdot 1 + 1 = 4 \) (not valid)
- Calculation for \( m = 0 \): \( 3 \cdot 1 + 0 = 3 \) (valid)
- Calculation for \( m = -1 \): \( 3 \cdot 1 - 1 = 2 \) (not valid)
- \( l = 2 \) (d orbital)
- Possible values for \( m \): -2, -1, 0, +1, +2
- Calculation for \( m = 0 \): \( 3 \cdot 2 + 0 = 6 \) (not valid)
- Calculation for \( m = 1 \): \( 3 \cdot 2 + 1 = 7 \) (not valid)
- Calculation for \( m = -1 \): \( 3 \cdot 2 - 1 = 5 \) (not valid)
- Calculation for \( m = 2 \): \( 3 \cdot 2 + 2 = 8 \) (not valid)
- Calculation for \( m = -2 \): \( 3 \cdot 2 - 2 = 4 \) (not valid)
4. **For \( n = 4 \)**:
- \( l = 0 \) (s orbital)
- \( m = 0 \)
- Calculation: \( 4 \cdot 0 + 0 = 0 \) (not valid)
- \( l = 1 \) (p orbital)
- Possible values for \( m \): -1, 0, +1
- Calculation for \( m = 1 \): \( 4 \cdot 1 + 1 = 5 \) (not valid)
- Calculation for \( m = 0 \): \( 4 \cdot 1 + 0 = 4 \) (not valid)
- Calculation for \( m = -1 \): \( 4 \cdot 1 - 1 = 3 \) (valid)
- \( l = 2 \) (d orbital)
- Possible values for \( m \): -2, -1, 0, +1, +2
- Calculation for \( m = 0 \): \( 4 \cdot 2 + 0 = 8 \) (not valid)
- Calculation for \( m = 1 \): \( 4 \cdot 2 + 1 = 9 \) (not valid)
- Calculation for \( m = -1 \): \( 4 \cdot 2 - 1 = 7 \) (not valid)
- Calculation for \( m = 2 \): \( 4 \cdot 2 + 2 = 10 \) (not valid)
- Calculation for \( m = -2 \): \( 4 \cdot 2 - 2 = 6 \) (not valid)
### Step 5: Count the Valid Electrons
From the calculations, we found valid combinations:
- \( (n=2, l=1, m=1) \)
- \( (n=3, l=1, m=0) \)
- \( (n=4, l=1, m=-1) \)
Each of these combinations corresponds to one electron. Therefore, the total number of electrons having \( n \cdot l + m = 3 \) is **3**.
### Final Answer
The number of electrons having \( n \cdot l + m = 3 \) for the Kr atom is **3**.
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