Problem 4 - Fundamentals of Physics

Question

How many electron states are there in the following shells: (a) \(n=4,(\) b) \(n=1,(\mathrm{c}) n=3,(\) d \() n=2 ?\)

Step-by-Step Solution

Answer

(a) 32 states, (b) 2 states, (c) 18 states, (d) 8 states.

Step 1: Understanding Electron Shells and Quantum Numbers

An electron shell is defined by the principal quantum number \(n\). The number of electron states in a shell depends on \(n\) because this defines the number of subshells, which are characterized by the angular momentum quantum number \(l\). The possible range for \(l\) is from 0 to \(n-1\). For each \(l\), there are \(2(2l+1)\) possible states, including spin degeneracy.

Step 2: Calculate Electron States for \(n = 4\)

For \(n = 4\), the possible \(l\) values are 0, 1, 2, and 3. Calculate the number of states for each \(l\):- \(l = 0\): \(2(2 \times 0 + 1) = 2\)- \(l = 1\): \(2(2 \times 1 + 1) = 6\)- \(l = 2\): \(2(2 \times 2 + 1) = 10\)- \(l = 3\): \(2(2 \times 3 + 1) = 14\)Add these together: \(2 + 6 + 10 + 14 = 32\). There are 32 electron states for \(n = 4\).

Step 3: Calculate Electron States for \(n = 1\)

For \(n = 1\), the only possible \(l\) value is 0:- \(l = 0\): \(2(2 \times 0 + 1) = 2\)Thus, there are 2 electron states for \(n = 1\).

Step 4: Calculate Electron States for \(n = 3\)

For \(n = 3\), the possible \(l\) values are 0, 1, and 2. Calculate the number of states for each \(l\):- \(l = 0\): \(2(2 \times 0 + 1) = 2\)- \(l = 1\): \(2(2 \times 1 + 1) = 6\)- \(l = 2\): \(2(2 \times 2 + 1) = 10\)Add these together: \(2 + 6 + 10 = 18\). There are 18 electron states for \(n = 3\).

Step 5: Calculate Electron States for \(n = 2\)

For \(n = 2\), the possible \(l\) values are 0 and 1. Calculate the number of states for each \(l\):- \(l = 0\): \(2(2 \times 0 + 1) = 2\)- \(l = 1\): \(2(2 \times 1 + 1) = 6\)Add these together: \(2 + 6 = 8\). There are 8 electron states for \(n = 2\).

Key Concepts

Quantum Numbers

Quantum numbers are like the precise address where you can find an electron within an atom. Each electron in an atom has a unique set of four quantum numbers, which helps us describe its energy and location. The four quantum numbers are:

• Principal Quantum Number ( ): Determines the electron's energy level or shell. Higher values mean the electron is further from the nucleus.
• Angular Momentum Quantum Number ( ): Indicates the shape of the electron's orbital. It can have integer values from 0 to ( -1).
• Magnetic Quantum Number ( ): Describes the orientation of the orbital in space relative to the nucleus.
• Spin Quantum Number ( ): Expresses the direction of the electron's spin, which can be either +1/2 or -1/2.

The primary quantum number, , dictates the number of subshells for an electron shell, and each subshell is linked to a particular angular momentum quantum number, which we explore further below.

Electron Shells

Electron shells are the layers of electrons orbiting the nucleus of an atom. The shells are defined by the principal quantum number, . Here's what you need to know about electron shells:

• Each shell can contain a set number of electrons and is composed of one or more subshells.
• The number of subshells within a shell equals the principal quantum number ().
• Each subshell correlates to a different angular momentum quantum number (l).
• In each shell, the maximum number of electrons is given by the formula \(2n^2\).

As increases, the shells expand, allowing more electrons and thus increasing the atom's complexity. Understanding electron shells is essential to grasp how electrons are arranged and how they affect chemical properties.

Angular Momentum Quantum Number

The angular momentum quantum number ( ) is key to understanding electron subshells. While the principal quantum number states the shell, l details the subshells:

• The values of l range from 0 to −1.
• A specific l correlates to a certain shape of the electron orbital.
• For example, =0 represents a spherical 's' orbital, =1 denotes a 'p' orbital, =2 corresponds to a 'd' orbital, and =3 signifies an 'f' orbital.

The variance in subshells leads to diverse energy states. The electron's angular momentum ( l) also affects how many orbitals are in a subshell. For instance, =1 ('p' subshells) can have three different orientations, hence, three orbitals. Each orbital can hold two electrons because of the magnetic and spin quantum numbers, culminating in 2(2l+1) states for each l. With more subshells, the complexity of electron arrangement increases, influencing an atom's interactions and bonding behavior.