A possible set of quantum numbers for the last electron added to a gallium atom (Z=31) in its ground state is

Options

A. `{:(n" "l" "m" "m_(s)),(4" "1" "-1" "+1//2):}`
B. `{:(n" "l" "m" "m_(s)),(4" "0" "-1" "-1//2):}`
C. `{:(n" "l" "m" "m_(s)),(3" "2" "+2" "+1//2):}`
D. `{:(n" "l" "m" "m_(s)),(3" "0" "0" "-1//2):}`

Answer

To determine the possible set of quantum numbers for the last electron added to a gallium atom (Z=31) in its ground state, we will follow these steps:

### Step 1: Determine the Electron Configuration
Gallium has an atomic number of 31, which means it has 31 electrons. The electron configuration of gallium can be written as follows:
- 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p¹

### Step 2: Identify the Last Electron
From the electron configuration, we see that the last electron added is in the 4p subshell. This means that the last electron is in the 4p orbital.

### Step 3: Determine the Principal Quantum Number (n)
The principal quantum number (n) corresponds to the energy level of the electron. For the 4p orbital, n = 4.

### Step 4: Determine the Azimuthal Quantum Number (l)
The azimuthal quantum number (l) defines the shape of the orbital:
- For s orbitals, l = 0
- For p orbitals, l = 1
- For d orbitals, l = 2
- For f orbitals, l = 3

Since the last electron is in the p orbital, l = 1.

### Step 5: Determine the Magnetic Quantum Number (m_l)
The magnetic quantum number (m_l) can take on values from -l to +l:
- For l = 1, m_l can be -1, 0, or +1.

Since the last electron is in the 4p orbital, it can have any of these three values.

### Step 6: Determine the Spin Quantum Number (m_s)
The spin quantum number (m_s) indicates the spin of the electron. It can be either +1/2 or -1/2. For the last electron in gallium, we can choose either value, but typically we denote it as +1/2 for the last added electron.

### Conclusion: Possible Set of Quantum Numbers
Thus, the possible set of quantum numbers for the last electron in a gallium atom (Z=31) in its ground state can be expressed as:
- n = 4
- l = 1
- m_l = -1, 0, or +1
- m_s = +1/2 (or -1/2)

### Example of a Possible Set
One example of a complete set of quantum numbers for the last electron could be:
- n = 4
- l = 1
- m_l = 0
- m_s = +1/2