Chapter 9: Problem 89
What is the maximum number of electrons that can occupy the \(n=3\) quantum shell?
Short Answer
Expert verified
The maximum number of electrons that can occupy the n=3 quantum shell is 18.
Step by step solution
01
Understanding the quantum number
The principal quantum number, denoted by 'n', determines the size and energy level of the electron shell. For each shell value of 'n', there are 'n' subshells.
02
Determining the subshells
For the third energy level (n=3), there are three possible subshells, which are the s, p, and d subshells. These correspond to angular momentum quantum numbers of 0, 1, and 2, respectively.
03
Calculating the maximum number of electrons
The maximum number of electrons for each type of subshell is given by the formula 2(2l + 1), where 'l' is the angular momentum quantum number. Therefore, for the 's' subshell, l=0, for the 'p' subshell, l=1, and for the 'd' subshell, l=2. Calculate for each and add them together to find the total number of electrons for n=3.
04
Summing up the electrons
2(2(0)+1) = 2 for the s subshell, 2(2(1)+1) = 6 for the p subshell, and 2(2(2)+1) = 10 for the d subshell. Adding these gives 2 + 6 + 10 = 18 electrons.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quantum Shells
In the intriguing world of quantum physics, the principal quantum number, represented by the symbol 'n', plays a vital role in determining the properties of quantum shells. These shells are essentially orbits in which electrons reside around an atom's nucleus. Each shell is associated with a particular energy level, categorized numerically from the innermost shell (n=1) outward.
As the value of 'n' increases, so does the shell's distance from the nucleus, along with its potential energy. The number of available subshells within a given shell is directly proportional to the principal quantum number. A significant aspect of quantum shells is their electron capacity. Using the formula, '2n^2', we can calculate the maximum number of electrons each shell can accommodate. For instance, the n=3 shell has a maximum capacity of 18 electrons, as showcased in the exercise solution. This capacity is essential for understanding the arrangement of electrons in atoms, a concept known as electron configuration.
As the value of 'n' increases, so does the shell's distance from the nucleus, along with its potential energy. The number of available subshells within a given shell is directly proportional to the principal quantum number. A significant aspect of quantum shells is their electron capacity. Using the formula, '2n^2', we can calculate the maximum number of electrons each shell can accommodate. For instance, the n=3 shell has a maximum capacity of 18 electrons, as showcased in the exercise solution. This capacity is essential for understanding the arrangement of electrons in atoms, a concept known as electron configuration.
Electron Configuration
Electron configuration refers to the arrangement of electrons around the nucleus of an atom in atomic orbitals, described by specific rules and notations. It forms a unique 'address' for each electron, specifying its location in terms of energy level (quantum shell), subshell, and its specific orientation and spin within that subshell.
The primary rule to determine the electron configuration of an element is the Pauli exclusion principle, which states that no two electrons can have the same set of four quantum numbers. As a result, building up the structure of electrons starts from the lowest energy levels, a concept known as the Aufbau principle. Hund's rule further specifies that electrons will occupy degenerate orbitals singly before they pair up. This systematic filling of the electron shells and subshells leads to predictable and precise electron configurations for each element on the periodic table.
The primary rule to determine the electron configuration of an element is the Pauli exclusion principle, which states that no two electrons can have the same set of four quantum numbers. As a result, building up the structure of electrons starts from the lowest energy levels, a concept known as the Aufbau principle. Hund's rule further specifies that electrons will occupy degenerate orbitals singly before they pair up. This systematic filling of the electron shells and subshells leads to predictable and precise electron configurations for each element on the periodic table.
Subshells
Within each quantum shell, there is a division into subshells that are differentiated by the angular momentum quantum number 'l'. Subshells are named as s, p, d, and f, correlating to 'l' values of 0, 1, 2, and 3, respectively.
Each subshell type has a characteristic shape and orientation in three-dimensional space, influencing the chemical bonding and reactivity of atoms. The s subshell is spherical, p subshells are dumbbell-shaped, while d and f subshells have more complex shapes. The capacity of each subshell to hold electrons is determined by the equation, '2(2l + 1)'. Thus, as the number of subshells within a quantum shell increases, so does the potential number of electrons that the shell can hold. Understanding subshells is crucial in predicting the chemical properties of atoms and the nature of chemical bonds they can form.
Each subshell type has a characteristic shape and orientation in three-dimensional space, influencing the chemical bonding and reactivity of atoms. The s subshell is spherical, p subshells are dumbbell-shaped, while d and f subshells have more complex shapes. The capacity of each subshell to hold electrons is determined by the equation, '2(2l + 1)'. Thus, as the number of subshells within a quantum shell increases, so does the potential number of electrons that the shell can hold. Understanding subshells is crucial in predicting the chemical properties of atoms and the nature of chemical bonds they can form.
Angular Momentum Quantum Number
The angular momentum quantum number, symbolized by 'l', is one of four quantum numbers used to describe the quantum state of an electron. It determines the shape of the electron cloud and the subshell to which an electron belongs. Values of 'l' range from 0 to n-1 for each principal quantum number 'n'.
Quintessentially, 'l' dictates the number of orbital shapes available within a quantum shell: 0 for s, 1 for p, 2 for d, and so on. The number of orientations each subshell can have is '2l + 1', highlighting the range of possible electron positions. Electrons in different subshells with distinct 'l' values exhibit varying degrees of angular momentum, influencing their energy levels and interaction with other electrons. This quantum number is central to the study of atomic structure and the calculation of electron configurations, as seen in the step-by-step solution of our exercise.
Quintessentially, 'l' dictates the number of orbital shapes available within a quantum shell: 0 for s, 1 for p, 2 for d, and so on. The number of orientations each subshell can have is '2l + 1', highlighting the range of possible electron positions. Electrons in different subshells with distinct 'l' values exhibit varying degrees of angular momentum, influencing their energy levels and interaction with other electrons. This quantum number is central to the study of atomic structure and the calculation of electron configurations, as seen in the step-by-step solution of our exercise.
