Calculate the minimum and maximum number of electrons which may have magnetic quantum number `m=+1` and spin quantum number `s=+(1)/(2)` in chromium (Cr)
(a)0,1
(b)1,2
(c)4,6
(d)2,3

Options

A. (a) 0,1
B. (b) 1,2
C. (c) 4,6
D. (d) 2,3

Answer

To solve the problem of calculating the minimum and maximum number of electrons with a magnetic quantum number \( m = +1 \) and a spin quantum number \( s = +\frac{1}{2} \) in chromium (Cr), we will follow these steps:

### Step 1: Determine the Electronic Configuration of Chromium
Chromium has an atomic number of 24. Its electronic configuration is:
\[ \text{Cr: } 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^1 \, 3d^5 \]

### Step 2: Identify the Relevant Subshells
We need to identify the subshells that can have the magnetic quantum number \( m = +1 \):
- For the **p subshell** (l = 1), the possible \( m \) values are \( -1, 0, +1 \).
- For the **d subshell** (l = 2), the possible \( m \) values are \( -2, -1, 0, +1, +2 \).

### Step 3: Count Electrons with \( m = +1 \)
Now we will find out how many electrons can have \( m = +1 \) in the electronic configuration of chromium:
- **From the 2p subshell**: The 2p subshell is fully filled with 6 electrons, and it has one electron with \( m = +1 \).
- **From the 3p subshell**: The 3p subshell is also fully filled with 6 electrons, and it has one electron with \( m = +1 \).
- **From the 3d subshell**: The 3d subshell has 5 electrons. The distribution of electrons in the 3d subshell can be represented as follows:
- \( 3d: \uparrow \, \uparrow \, \uparrow \, \uparrow \, \uparrow \) (5 unpaired electrons)
- The electrons can occupy the \( m = +1 \) state.

### Step 4: Determine Maximum and Minimum Electrons
- **Maximum Electrons**:
- One electron from the 2p subshell (with \( m = +1 \)).
- One electron from the 3p subshell (with \( m = +1 \)).
- One electron from the 3d subshell (with \( m = +1 \)).
- Therefore, the maximum number of electrons with \( m = +1 \) and \( s = +\frac{1}{2} \) is **3**.

- **Minimum Electrons**:
- One electron from the 2p subshell (with \( m = +1 \)).
- One electron from the 3p subshell (with \( m = +1 \)).
- We can assume that the 3d electron with \( m = +1 \) has a spin of \( -\frac{1}{2} \), so we do not count it.
- Therefore, the minimum number of electrons with \( m = +1 \) and \( s = +\frac{1}{2} \) is **2**.

### Conclusion
The minimum and maximum number of electrons which may have magnetic quantum number \( m = +1 \) and spin quantum number \( s = +\frac{1}{2} \) in chromium are:
- Minimum: 2
- Maximum: 3

Thus, the correct answer is:
**(d) 2, 3**