Question

Electrons with the same value of the quantum number \(n\) are said to occupy the same electron shell, \(K, L, M, N,\) or higher. Calculate the maximum allowed number of electrons for the a) \(K\) shell, b) \(L\) shell, and c) \(M\) shell.

Short answer

Answer: The maximum allowed numbers of electrons for the K, L, and M shells are 2 electrons for the K shell, 8 electrons for the L shell, and 18 electrons for the M shell.

Step-by-step solution

Identify the principal quantum number for each shell

The K, L, and M shell corresponds to the principal quantum number values of n = 1, n = 2, and n = 3, respectively.

Apply the formula for maximum number of electrons in a shell

We will use the formula: Maximum number of electrons in a shell = 2n^2 where n is the principal quantum number. a) For the K shell (n = 1): Maximum number of electrons = 2(1)^2 = 2(1) = 2 b) For the L shell (n = 2): Maximum number of electrons = 2(2)^2 = 2(4) = 8 c) For the M shell (n = 3): Maximum number of electrons = 2(3)^2 = 2(9) = 18

Present final answers

The maximum allowed number of electrons for each shell are as follows: a) K shell: 2 electrons b) L shell: 8 electrons c) M shell: 18 electrons

Key concepts

Principal Quantum Number

The principal quantum number, denoted as 'n', is integral to understanding the atomic structure and the arrangement of electrons within an atom. It indicates the relative size and energy of electron shells, as well as their distance from the nucleus. The lowest energy level, closest to the nucleus, is given the principal quantum number 1 (n=1), and the energy levels increase numerically as you move further away.

The principal quantum number also determines the maximum number of electrons that an electron shell can hold. This is calculated using the formula: \(2n^2\), where 'n' is the principal quantum number. For instance, the K shell, which is the innermost shell with n=1, can accommodate up to 2 electrons, as indicated by the formula \(2(1)^2 = 2\).

Understanding this concept is crucial for determining electron configurations and predicting an atom's chemical properties. One way to improve your grasp of this topic is to visualize each electron shell as an 'apartment' where the floor number represents the 'n' value. The higher the floor, the more rooms (places for electrons) are available, following the square of the n value.

Electron Configuration

Electron configuration refers to the distribution of electrons in the atomic orbitals of an element. The configuration is governed by a set of rules, including the Pauli exclusion principle and Hund's rule, and gives insight into an atom’s reactivity, bonding behavior, and magnetic properties. Each element has a unique electron configuration that can be noted using standard notation, which includes the principal quantum number, the type of orbital, and the number of electrons in that orbital.

To achieve a stable state, atoms strive to fill or half-fill their electron shells, following the 'aufbau' principle, which guides the sequence in which orbitals are filled. For example, the configuration for oxygen (with 8 electrons) would be written as 1s² 2s² 2p⁴.

For students, a helpful tip is to utilize the periodic table as a map when writing electron configurations. This map shows how electrons fill up orbitals systematically and why some elements have similar chemical qualities due to their configurations.

Atomic Structure

The atomic structure of an element is a fascinating and complex composition that includes protons, neutrons, and electrons. An atom's nucleus contains protons and neutrons, while electrons orbit the nucleus in various shells and subshells. Each electron shell correlates to a principal quantum number and dictates the atom's electronic architecture.

The understanding of atomic structure is foundational for grasping other chemistry concepts like bonding, periodicity, and molecular geometry. Recognizing how electrons layer in shells and subshells can help map out potential paths for electron transitions, influencing how atoms interact with light and with each other.

For those students finding it difficult to visualize, consider using models or drawings to represent the atomic structure. This can make the concept more tangible, showing the nested layers of electron shells surrounding the nucleus like a set of Russian dolls, each with a maximum capacity determined by the principal quantum number.