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Chapter 3: Problem 86

Determine the number of orbitals in the \(n=3, n=4\), and \(n=5\) shells.

Short Answer

Expert verified
For \(n=3\): 9 orbitals, \(n=4\): 16 orbitals, \(n=5\): 25 orbitals.

Step by step solution

01

Understand the Principal Quantum Number

The principal quantum number, denoted as \( n \), signifies the main energy level or shell. It can take positive integer values (1, 2, 3, etc.). Each shell at level \( n \) can have orbitals associated with it.
02

Calculate Possible Values of the Azimuthal Quantum Number

The azimuthal quantum number \( l \) describes the shape of the orbital and can have integer values ranging from 0 to \( n-1 \). For example, if \( n=3 \), \( l \) can be 0, 1, or 2.
03

Determine Number of Orbitals per Subshell

Each subshell defined by a specific \( l \) value has orbitals, where the number of orbitals is given by \( 2l + 1 \). For example, if \( l = 0 \), there is 1 orbital; if \( l = 1 \), there are 3 orbitals; and if \( l = 2 \), there are 5 orbitals.
04

Apply Calculation for Each Principal Quantum Number

To find the total orbitals for a shell with quantum number \( n \), sum the orbitals in each of its subshells. This gives the formula \( n^2 \).
05

Apply Formula to \( n=3 \)

For \( n=3 \): the total number of orbitals is \( 3^2 = 9 \).
06

Apply Formula to \( n=4 \)

For \( n=4 \): the total number of orbitals is \( 4^2 = 16 \).
07

Apply Formula to \( n=5 \)

For \( n=5 \): the total number of orbitals is \( 5^2 = 25 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Quantum Number
The principal quantum number, represented by the symbol \( n \), is a fundamental concept in quantum mechanics that signifies the main energy level or shell an electron occupies within an atom. Each value of \( n \) is a positive integer: 1, 2, 3, and so on. As \( n \) increases, the distance of the electron from the nucleus also generally increases, and the electron's energy level becomes higher. Understanding \( n \) is crucial for visualizing the electron configuration in an atom.
  • Each principal quantum number corresponds to a shell.
  • The number of orbitals in any given shell is calculated by \( n^2 \).
  • For example, for \( n = 3 \), there are \( 3^2 = 9 \) orbitals in total.
This simplicity in the calculation reflects the orderly pattern that electrons follow as they fill different energy levels.
Azimuthal Quantum Number
The azimuthal quantum number, denoted by \( l \), describes the shape of an atom's electron cloud or orbitals within a given shell. This quantum number takes on integer values ranging from 0 to \( n-1 \), where \( n \) is the principal quantum number of that specific shell. The value of \( l \) determines the number of subshells within a shell and is essential for understanding the distribution of electrons.
  • Each value of \( l \) corresponds to a different subshell: \( l=0 \) (s-type), \( l=1 \) (p-type), \( l=2 \) (d-type), and so forth.
  • The subshells have distinct shapes and are associated with different electron spatial distributions.
  • For example, for \( n = 3 \), \( l \) can be 0 (s), 1 (p), or 2 (d) indicating three subshells: s, p, and d.
This detail accounts for the complexity of electron arrangements in atoms, influencing chemical properties and bonding behaviors.
Orbitals Calculation
Calculating the number of orbitals within an atom involves a relationship between the quantum numbers. The number of orbitals in a given subshell is determined by the formula \( 2l + 1 \), where \( l \) is the azimuthal quantum number. Each orbital can hold up to 2 electrons. This formula helps in understanding the total orbital structure within a shell: first by evaluating subshell contributions and then by summing up the orbitals.
  • If \( l = 0 \), there is one orbital (s-type, 1 orbital).
  • If \( l = 1 \), there are three orbitals (p-type, 3 orbitals).
  • If \( l = 2 \), there are five orbitals (d-type, 5 orbitals).
When you determine the number of orbitals for a principal quantum number \( n \), calculate for each subshell and sum the results to get the total \( n^2 \). For instance, for \( n = 3 \): the total would be three subshells (s, p, d) with 1 + 3 + 5 = 9 orbitals, aligning with the formula \( n^2 \).

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Most popular questions from this chapter

Draw orbital diagrams for the following atoms. (a) \(\mathrm{Si}\) (b) \(\mathrm{V}\) (c) \(\mathrm{Ga}\) (d) \(\mathrm{Cl}\) (e) \(\mathrm{Na}\)

The most recent estimates give values of about \(10^{-10} \mathrm{~m}\) for the radius of an atom and \(10^{-14} \mathrm{~m}\) for the radius of the nucleus of the atom. Calculate the fraction of the total volume of an atom that is essentially empty space.

Relate to the following data. An atom with an equal number of spin-up and spindown electrons is said to be diamagnetic because the atom is repelled by a magnetic field. In this case we say that all of the electrons are "paired." If this is not the case - if there are one or more unpaired electrons on an atom - the atom is attracted to a magnetic field, and it is said to be paramagnetic. The strength of the attraction is an experimentally measurable quantity known as the magnetic moment. The magnitude of the \(m a g-\) netic moment (measured in magnetons) is related to (but not proportional to) the number of unpaired electrons present. In other words, the larger the number of unpaired electrons, the larger the magnetic moment. Here are some experimental data collected by an investigator of this phenomenon. $$ \begin{array}{llc} {3}{c} {\text { Magnetic Moments of Several Elements }} \\ \hline & & \\ \text { Element } & \text { Type } & \begin{array}{c} \text { Magnetic Moments } \\ \text { (magnetons) } \end{array} \\ \hline \mathrm{H} & \text { Paramagnetic } & 1.7 \\ \mathrm{He} & \text { Diamagnetic } & 0 \\ \mathrm{~B} & \text { Paramagnetic } & 1.7 \\ \mathrm{C} & \text { Paramagnetic } & 2.8 \\ \mathrm{~N} & \text { Paramagnetic } & 3.9 \\ \mathrm{O} & \text { Paramagnetic } & 2.8 \\ \mathrm{Ne} & \text { Diamagnetic } & 0 \\ \hline \end{array} $$ Why is the situation of equal numbers of spin-up and spin-down electrons referred to as all electrons being "paired"?

Which of the following ions has the largest radius? Explain. (a) \(\mathrm{Na}^{+}\) (b) \(\mathrm{Mg}^{2+}\) (c) \(\mathrm{S}^{2-}\) (d) \(\mathrm{Cl}^{-}\) (e) \(\mathrm{Se}^{2-}\)

Sort the following atoms or ions into isoelectronic groups. (a) \(\mathrm{N}^{3-}\) (b) \(\mathrm{Ar}\) (c) \(\mathrm{F}^{-}\) (d) \(\mathrm{Ne}\) (e) \(\mathrm{P}^{3-}\) (f) \(\mathrm{Ca}^{2+}\) (g) \(\mathrm{Al}^{3+}\) (h) \(\mathrm{Si}^{4+}\) (i) \(\mathrm{Na}^{+}\) (j) \(\mathrm{S}^{2-}\) (k) \(\mathrm{Cl}^{-}\) (1) \(\mathrm{O}^{2-}\) \((\mathrm{m}) \mathrm{K}^{+}\) (n) \(\mathrm{Mg}^{2+}\)
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