To determine the maximum number of electrons in an atom that can have the quantum numbers \( n = 4 \) and \( m_l = +1 \), we can follow these steps:
### Step 1: Identify the Principal Quantum Number (n)
The principal quantum number \( n \) is given as 4. This indicates that we are dealing with the fourth energy level of an atom.
### Step 2: Determine the Possible Azimuthal Quantum Numbers (l)
The azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). Therefore, for \( n = 4 \):
- Possible values of \( l \) are:
- \( l = 0 \) (s orbital)
- \( l = 1 \) (p orbital)
- \( l = 2 \) (d orbital)
- \( l = 3 \) (f orbital)
### Step 3: Calculate the Magnetic Quantum Number (m_l)
The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \), including zero.
- For \( l = 0 \): \( m_l = 0 \)
- For \( l = 1 \): \( m_l = -1, 0, +1 \)
- For \( l = 2 \): \( m_l = -2, -1, 0, +1, +2 \)
- For \( l = 3 \): \( m_l = -3, -2, -1, 0, +1, +2, +3 \)
### Step 4: Identify the Relevant Orbital for \( m_l = +1 \)
Since we are interested in the case where \( m_l = +1 \):
- This value of \( m_l \) is possible for \( l = 1 \) (p orbital), \( l = 2 \) (d orbital), and \( l = 3 \) (f orbital).
### Step 5: Count the Orbitals with \( m_l = +1 \)
- For \( l = 1 \): There is 1 orbital with \( m_l = +1 \) (the 2p orbital).
- For \( l = 2 \): There is 1 orbital with \( m_l = +1 \) (the 3d orbital).
- For \( l = 3 \): There is 1 orbital with \( m_l = +1 \) (the 4f orbital).
Thus, there are a total of 3 orbitals where \( m_l = +1 \).
### Step 6: Calculate the Maximum Number of Electrons
Each orbital can hold a maximum of 2 electrons (due to the Pauli exclusion principle). Therefore, if we have 3 orbitals with \( m_l = +1 \):
- Maximum number of electrons = Number of orbitals × 2 electrons/orbital
- Maximum number of electrons = \( 3 \times 2 = 6 \)
### Final Answer
The maximum number of electrons in an atom that can have the quantum numbers \( n = 4 \) and \( m_l = +1 \) is **6**.
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