Problem 79 - Chemistry The Central Science

Question

The following do not represent valid ground-state electron configurations for an atom either because they violate the Pauli exclusion principle or because orbitals are not filled in order of increasing energy. Indicate which of these two principles is violated in each example. (a) 1\(s^{2} 2 s^{2} 3 s^{1}\) (b) \([\mathrm{Xe}] 6 s^{2} 5 d^{4}(\mathbf{c})[\mathrm{Ne}] 3 s^{2} 3 d^{5} .\)

Step-by-Step Solution

Answer

In conclusion: - (a) \(1s^2 2s^2 3s^1\) is a valid ground-state electron configuration and does not violate any principle. - (b) \([Xe] 6s^2 5d^4\) is not a valid ground-state electron configuration and violates the Aufbau principle. - (c) \([Ne] 3s^2 3d^5\) is not a valid ground-state electron configuration and violates the Aufbau principle.

Step 1: Example (a) - 1s² 2s² 3s¹

For this electron configuration, the order of orbitals is 1s, 2s, 3s. Comparing this to the Aufbau principle's order of orbitals given above, we can see that the orbitals are filled in order of increasing energy, so this configuration does not violate the Aufbau principle. Additionally, each of the orbitals has a maximum of 2 electrons, which adheres to the Pauli Exclusion Principle. Therefore, this is a valid ground-state electron configuration and does not violate either principle.

Step 2: Example (b) - [Xe] 6s² 5d⁴

For this electron configuration, we start with [Xe] which represents the electron configuration of xenon. Xenon has an electron configuration of 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶. The new electron configuration would then be 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 5d⁴. From the configuration, we can see that the 6s orbital is filled before the 5d orbitals. Comparing this to the Aufbau principle's order of orbitals mentioned above, this electron configuration does not follow the order of increasing energy, violating the Aufbau principle. Therefore, this electron configuration is not valid and violates the Aufbau principle.

Step 3: Example (c) - [Ne] 3s² 3d⁵

For this electron configuration, we start with [Ne] which represents the electron configuration of neon. Neon has an electron configuration of 1s² 2s² 2p⁶. The new electron configuration would then be 1s² 2s² 2p⁶ 3s² 3d⁵. It is important to note that the Aufbau principle is followed until the 3p orbital. The 3p orbital is not filled before filling the 3d orbital, which violates the Aufbau principle. Therefore, this electron configuration is not valid and violates the Aufbau principle. In conclusion: - (a) 1s² 2s² 3s¹ is a valid ground-state electron configuration and does not violate any principle. - (b) [Xe] 6s² 5d⁴ is not a valid ground-state electron configuration and violates the Aufbau principle. - (c) [Ne] 3s² 3d⁵ is not a valid ground-state electron configuration and violates the Aufbau principle.

Key Concepts

Aufbau Principle

Imagine you're building a pyramid; you wouldn't start from the top, right? Similarly, the Aufbau principle guides us in constructing the electron configuration of atoms. It states that electrons fill atomic orbitals from the lowest to the highest energy levels.

Electron orbitals are like homes for electrons, and each home has a certain amount of energy. The Aufbau principle ensures that electrons fill up the 'homes' starting with the lowest energy level before moving to higher levels. This orderly filling is crucial for understanding why certain electron configurations are more stable than others. For example, in exercise (b), the electrons should fill the 4d orbitals before the 5d orbitals, but in the given configuration, this order was not followed, leading to a violation of the Aufbau principle.

Pauli Exclusion Principle

The Pauli exclusion principle is akin to a strict rule at a party: no two guests can wear the exact same outfit. In the world of electrons, this means that no two electrons in an atom can have the same four quantum numbers. Each electron is characterized by these quantum numbers, which describe properties such as energy, shape, and orientation of their orbitals, and their spin direction.

In a given orbital, only two electrons with opposite spins (represented by up and down arrows) can exist. If we take example (a), each orbital mentioned contains a maximum of two electrons, each with opposite spins, complying with the Pauli exclusion principle.

Ground-State Electron Configurations

Ground-state electron configurations are like the default seating arrangement for electrons in an atom when it is at its most relaxed or most stable state. It's when the electrons occupy the lowest possible energy levels.

When we write an electron configuration, we're listing the addresses of all the electrons according to the rules set by the Aufbau and Pauli exclusion principles. A proper ground-state electron configuration must follow these rules, filling the lowest energy orbitals first and ensuring electrons don't break the 'no matching quantum numbers' rule. In the exercise, while (a) was correct, (b) and (c) failed to meet these criteria, indicating incorrect ground-state electron configurations.

Orbital Energy Levels

Orbital energy levels are like the different floors of a building, where the ground floor is the least energy-intensive to reach (think no stair climbing!). Electrons tend to 'live' on the lower floors unless an external force 'pushes' them to a higher one.

Each type of orbital (s, p, d, and f) corresponds to a certain energy level, and this level can be affected by the electron configuration of the elements. As you move up the periodic table, orbitals of the same type increase in energy. For example, 3d is higher in energy than 2d, but lower than 4d. Therefore, in a well-ordered electron configuration, such as the valid one in example (a), electrons fill the lower energy 3s orbital before they would fill the higher energy 3d orbital.