Question

How many subshells and electrons are associated with \(n=4\) ? (a) 32,64 (b) 16,32 (c) 4,16 (d) 8,16

Short answer

For n=4, there are 4 subshells and 32 electrons.

Step-by-step solution

Identify Subshells

For a given principal quantum number n, the number of subshells is equal to n. Therefore, for n = 4, there are 4 subshells.

Calculate Number of Electrons

The total number of electrons in a shell can be determined by the formula 2n^2. Plug n = 4 into the formula to calculate the number of electrons.

Perform the Calculation

Calculating, we get 2(4)^2 = 2(16) = 32 electrons. Thus, there are 32 electrons that can be associated with n = 4.

Key concepts

Understanding Subshells

In the context of atomic physics, subshells are categories within an electron shell which are differentiated by their angular momentum quantum number, or azimuthal quantum number (). During the exercise, it was mentioned that for a principal quantum number (), the number of subshells is equal to that number. So, with = 4, we'd have four subshells. These subshells are designated as s, p, d, and f in ascending order of energy.

The 's' subshell can hold up to 2 electrons, 'p' can hold 6, 'd' can accommodate 10, and 'f' subshell can fill with 14 electrons. To know how many electrons a principal energy level can hold, we sum up the capacities of all the subshells within that level. Corrections for higher levels can be introduced where subshells like 'g' (with 18 electrons) exist, but for = 4, 'g' is not relevant.

In summary, knowing the number of subshells provides valuable information on the potential complexity of electron arrangements within an atom and forms the groundwork for further understanding electron configurations and quantum chemistry.

Decoding Electronic Configuration

Electronic configuration offers a roadmap for the location of electrons within the various shells and subshells of an atom. To tackle the problem at hand: How many electrons can fit within the principal quantum number = 4? We learned from the step-by-step solution that calculating using the formula 2^2 gives us 32 electrons.

Electronic configuration uses the principles of the Pauli exclusion principle and Hund’s rule to stack electrons methodically into the available subshells starting from the lowest energy level onward. The 's' subshell gets filled first, followed by 'p', then 'd', and lastly 'f'. From a learning perspective, visualizing or even drawing out the filling of these subshells can significantly aid in understanding.

Furthermore, knowing these configurations is critical for predicting chemical properties and behaviors of elements, as elements with similar electron configurations tend to exhibit similar chemical properties. This is why electronic configuration is a central concept in chemistry and physics.

Quantum Mechanics Fundamentals

Quantum mechanics is the underlying theory that explains the behavior of electrons in atoms. It shapes our understanding of subshells and electronic configuration. It deals with probabilities rather than certainties, which means that we describe electron placement in terms of 'probability densities' rather than fixed orbits, as was previously thought in classical physics.

The principal quantum number, , originates from quantum mechanics. It describes not only the electron's energy level but also its distance from the nucleus. The greater the value of , the greater the energy and the farther the electron is likely to be from the nucleus. Other quantum numbers such as the azimuthal (related to subshells), magnetic, and spin quantum numbers add additional detail to our model of how electrons are arrayed around the nucleus.

A solid understanding of quantum mechanics allows students to better understand the architecture of atoms and the cause-and-effect relationships of atomic interactions in chemistry and materials science. With its principles at play in subatomic particles, quantum mechanics serves as a foundation for modern scientific explorations and technologies.