For gadolinium Gd with atomic number 64, which of the following is the correct outer electronic configuration of the atom?

Introduction / Context:
Electronic configuration describes how electrons are distributed among the orbitals of an atom. For transition and inner transition elements such as the lanthanides, understanding their electronic configurations can be challenging due to the filling of f orbitals and some exceptions to simple rules. Gadolinium, symbol Gd, is a lanthanide with atomic number 64. This question asks you to identify its correct outer electronic configuration by comparing several possible distributions of electrons in the 4f, 5d, and 6s subshells.

Given Data / Assumptions:

• Gadolinium has atomic number 64, so a neutral atom has 64 electrons.
• Its electrons fill orbitals in the general order from lower to higher energy, including 4f, 5d, and 6s orbitals for the outer configuration.
• The options present four combinations of 4f, 5d, and 6s electron counts.
• We assume the inner core up to xenon is the same, and we only compare the outer part beyond xenon.

Concept / Approach:
For lanthanides, the general pattern of filling is [Xe] 4f^n 5d^0 or 5d^1 6s^2 with n increasing from cerium to lutetium. Gadolinium is known to have a particularly stable configuration with a half filled 4f subshell. A half filled f subshell corresponds to 4f^7. To keep 64 electrons in total, we consider that xenon accounts for 54 electrons. That leaves 10 electrons to place. A stable arrangement that gives 4f^7 and keeps 6s^2 filled uses the remaining electron in 5d, giving 4f^7 5d^1 6s^2. This configuration benefits from both a half filled 4f subshell and a partially filled 5d subshell, matching experimental observations for gadolinium.

Step-by-Step Solution:
Step 1: Write the core configuration of xenon, which has 54 electrons, as [Xe]. Step 2: Subtract 54 from the atomic number of gadolinium, 64, to find that 10 electrons remain to be placed in 4f, 5d, and 6s subshells. Step 3: Consider that lanthanides fill the 4f subshell and often have either 5d^0 or 5d^1 along with 6s^2. Step 4: Recognise the special stability of a half filled 4f subshell, which corresponds to 4f^7. Step 5: Allocate 7 electrons to 4f, 2 electrons to 6s, which totals 9, and then place the remaining 1 electron in 5d, giving 4f^7 5d^1 6s^2 as the outer configuration.

Verification / Alternative check:
Check the options against the requirement for 10 electrons beyond xenon. The configuration 4f^7 5d^1 6s^2 adds up to 7 plus 1 plus 2, which equals 10, matching the required number of electrons. It also gives a half filled 4f subshell, known to be particularly stable due to exchange energy and symmetry considerations. Other options either assign too many or too few electrons or give less stable arrangements, such as 4f^8 with 5d^0 or various combinations that do not reflect known ground state data. Reference tables of lanthanide configurations list gadolinium as [Xe] 4f^7 5d^1 6s^2, confirming this choice.

Why Other Options Are Wrong:
The option 4f^3 5d^5 6s^2 still has 10 electrons in the outer shells but does not match the pattern of f subshell filling for lanthanides and does not provide the stable half filled 4f^7 arrangement. The option 4f^8 5d^0 6s^2 has 10 electrons but leads to a more crowded f subshell and lacks the known experimental configuration for gadolinium. The option 4f^4 5d^4 6s^2 again gives a strange distribution that is not supported by spectroscopic data. These alternatives ignore the known stability of a half filled f subshell and do not match reference configurations.

Common Pitfalls:
Students often attempt to apply simple aufbau rules mechanically without considering exceptions in d and f block elements. For lanthanides, it is important to remember that some elements prefer configurations that give half filled or fully filled subshells. Another common mistake is to miscount the number of electrons above xenon and end up with the wrong total in outer shells. To avoid such errors, always subtract the noble gas core electrons, consider the special stability of half filled subshells like 4f^7, and double check that the total number of electrons matches the atomic number.

Final Answer:
For gadolinium with atomic number 64, the correct outer electronic configuration is 4f^7 5d^1 6s^2.