In the ground state, an element has 13 electrons in its M-shell. The element is_____.

The correct option is Chromium.

To understand why this is correct, we can analyze the electronic shell structure of atoms step-by-step:

1. Identify the Shells and Quantum Numbers:
The principal quantum number, represented by n, defines the electron shells of an atom:
• K-shell corresponds to n = 1
• L-shell corresponds to n = 2
• M-shell corresponds to n = 3
• N-shell corresponds to n = 4

2. Determine the Orbitals in the M-shell:
For the M-shell (n = 3), the available subshells are 3s, 3p, and 3d. Therefore, the total number of electrons in the M-shell is the sum of the electrons occupying these three subshells:

$$\text{Electrons in M-shell}=\left(\text{electrons in 3s}\right)+\left(\text{electrons in 3p}\right)+\left(\text{electrons in 3d}\right)$$

3. Calculate the Required 3d Electrons:
In transition metals, the inner 3s and 3p subshells are completely filled before electrons occupy the 3d and 4s levels. A completely filled s-subshell holds 2 electrons, and a completely filled p-subshell holds 6 electrons.
Substituting these values along with the given 13 M-shell electrons:

$$2+6+\text{electrons in 3d}=13$$

$$\text{electrons in 3d}=13-8=5$$

4. Analyze the Ground State Configuration of Chromium:
Chromium has an atomic number of Z = 24. Based on the standard Aufbau principle, we might expect its electron configuration to be:

$$1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{4}4s^2$$

However, half-filled d-orbitals (3d⁵) are exceptionally stable due to their symmetrical distribution and high exchange energy. Consequently, one electron from the 4s subshell is promoted to the 3d subshell. Thus, the actual ground-state electronic configuration of Chromium is:

$$1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{5}4s^1$$

5. Sum the M-shell Electrons:
Grouping the electrons by their principal quantum number n = 3 (the M-shell):
• 3s subshell: 2 electrons
• 3p subshell: 6 electrons
• 3d subshell: 5 electrons

$$\text{Total M-shell electrons}=2+6+5=13$$

This matches the requirement of 13 electrons in the M-shell, confirming that the element is Chromium.