23. a. Find the specific electron configuration of an element which has the following four quantum numbers.

To determine the specific electron configuration of the element with the given quantum numbers, we need to interpret each of the quantum number values:

1. Principal quantum number (n): This indicates the energy level of the electron. In this case, \( n = 5 \).
2. Azimuthal quantum number (l): This represents the subshell. The value \( l = 2 \) corresponds to the d subshell.
3. Magnetic quantum number (ml): This specifies the particular orbital within the subshell. The value \( ml = -1 \) refers to one of the d orbitals (specifically, one of the d orbitals that can hold two electrons).
4. Spin quantum number (ms): A value of \( ms = -\frac{1}{2} \) indicates that the electron has a spin down.

Based on this information, the specific electron configuration will involve an electron in the 5d subshell.

a. Specific Electron Configuration:

The electron configuration of an element with an electron in the \( 5d \) subshell, where the other electrons are filled according to the Aufbau principle, would look like this:

For a complete electron configuration up to \( 5d \), you would have:

• The filling of lower energy orbitals occurs first:
  • \( 1s^2 2s^2 2p^6 \) (total 10 electrons)
  • \( 3s^2 3p^6 \) (total 18 electrons)
  • \( 4s^2 3d^{10} \) (total 28 electrons)
  • \( 4p^6 \) (total 36 electrons)
  • Then, we start filling \( 5s^2 \), which gives us 2 more electrons, bringing the total to 38.
  • Next, the \( 4d^{10} \) subshell is filled, adding 10 more electrons for a total of 48.
  • Finally, we fill the \( 5d \) subshell, where we will have 1 electron represented by the quantum numbers given.

Thus, the relevant portion of the electron configuration is \( 5d^1 \).

So concentrating on the specific electron configuration considering the first and partially filled subshell, we have:

Electron Configuration: \[ \text{[Kr]}, 5s^2 , 4d^{10} , 5d^1 \]

b. Name of the Element:

The element in question here can be identified by its atomic number.

• The configuration \( [Kr], 5s^2 , 4d^{10} , 5d^1 \) corresponds to the element Zinc (Zn), which has atomic number 30, along with completing the filling of \( 4d \) orbitals and starting \( 5d \) orbitals.

However, the element that specifically has the \( 5d^1 \) is Zirconium - which has atomic number 40.

Final answer:

Specific Electron Configuration: \[ \text{[Kr]}, 5s^2 , 4d^{10} , 5d^1 \]

Name of the Element: Zirconium (Zr)

Certainly! Here’s a concise version:

a. Specific Electron Configuration:

The given quantum numbers correspond to an electron in the 5d subshell. The complete electron configuration up to that point is:
\[ \text{[Kr]} , 5s^2 , 4d^{10} , 5d^1 \]

b. Name of the Element:

The element with this configuration is Zirconium (Zr), which has atomic number 40.