To solve the problem of finding the maximum number of electrons in a chromium (Cr) atom that have the quantum numbers \( m = -1 \) and \( s = +\frac{1}{2} \), while ensuring that \( n \neq 2 \), we can follow these steps:
### Step 1: Understand the Quantum Numbers
- The principal quantum number \( n \) indicates the energy level of the electron.
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). For \( m = -1 \), \( l \) must be at least \( 1 \) (since \( m \) can take values from \(-l\) to \( +l\)).
- The spin quantum number \( s \) can be either \( +\frac{1}{2} \) or \(-\frac{1}{2}\).
### Step 2: Identify Possible Values for \( n \)
Since we need \( n \neq 2 \), we can consider \( n = 1, 3, 4, \) etc.
### Step 3: Determine the Values of \( l \) and \( m \)
- For \( n = 1 \): The only value for \( l \) is \( 0 \) (which gives \( m = 0 \)). Not valid since \( m \) must be \(-1\).
- For \( n = 3 \): Possible values for \( l \) are \( 0, 1, 2 \):
- If \( l = 1 \), then \( m \) can be \(-1, 0, +1\). This is valid.
- If \( l = 2 \), then \( m \) can be \(-2, -1, 0, +1, +2\). This is also valid.
- For \( n = 4 \): Possible values for \( l \) are \( 0, 1, 2, 3 \):
- If \( l = 1 \), then \( m \) can be \(-1, 0, +1\). This is valid.
- If \( l = 2 \), then \( m \) can be \(-2, -1, 0, +1, +2\). This is also valid.
- If \( l = 3 \), then \( m \) can be \(-3, -2, -1, 0, +1, +2, +3\). This is also valid.
### Step 4: Count the Electrons
- For \( n = 3 \) and \( l = 1 \) (where \( m = -1 \)):
- The spin \( s = +\frac{1}{2} \) allows for one electron with \( m = -1 \).
- For \( n = 3 \) and \( l = 2 \) (where \( m = -1 \)):
- The spin \( s = +\frac{1}{2} \) allows for one electron with \( m = -1 \).
- For \( n = 4 \) and \( l = 1 \) (where \( m = -1 \)):
- The spin \( s = +\frac{1}{2} \) allows for one electron with \( m = -1 \).
- For \( n = 4 \) and \( l = 2 \) (where \( m = -1 \)):
- The spin \( s = +\frac{1}{2} \) allows for one electron with \( m = -1 \).
- For \( n = 4 \) and \( l = 3 \) (where \( m = -1 \)):
- The spin \( s = +\frac{1}{2} \) allows for one electron with \( m = -1 \).
### Conclusion
Thus, the maximum number of electrons in a chromium atom that have \( m = -1 \) and \( s = +\frac{1}{2} \) while ensuring \( n \neq 2 \) is **5**.
