Chapter 6, Problem 68GQ

Answer the following questions as a summary quiz on this chapter.

1. (a) The quantum number n describes the ________ of an atomic orbital, and the quantum number ℓ describes its ________.
2. (b) When n = 3, the possible values of ℓ are _________.
3. (c) What type of orbital corresponds to ℓ = 3? ________
4. (d) For a 4d orbital, the value of n is _______, the value of ℓ is ________, and a possible value of m_ℓ is _________.
5. (e) Each of the following drawings represents a type of atomic orbital. Give the letter designation for the orbital, give its value of ℓ, and specify the number of planar nodes.

Letter = ______ ______

ℓ value = ______ ______

Planar nodes = _______ ______

1. (f) An atomic orbital with three planar nodes through the nucleus is a(n) ______ orbital.
2. (g) Which of the following orbitals cannot exist according to modem quantum theory: 2s, 3p, 2d, 3f, 5p, 6p?
3. (h) Which of the following is not a valid set of quantum numbers?

1. (i) What is the maximum number of orbitals that can be associated with each of the following sets of quantum numbers? (One possible answer is ″none. ″)
   1. (i) n = 2 and ℓ =1
   2. (ii) n = 3
   3. (iii) n = 3 and ℓ = 3
   4. (iv) n = 2, ℓ = 1, and m_ℓ = 0

Interpretation Introduction

Interpretation: For an atomic orbital what does the quantum number n and l describes should be given.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a) Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b) Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c) Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d) Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

Answer to Problem 68GQ

n represents energy and the size of an atomic orbital and the shape of an orbital is given by quantum number l

Explanation of Solution

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.

Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number describes the shape of the orbitals.

Therefore, n represents energy and the size of an atomic orbital and the shape of an orbital is given by quantum number l.

Interpretation Introduction

Interpretation: The possible values of l when $\text{n}\text{=}3$ should be determined.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each l value indicates subshell.

Answer to Problem 68GQ

When $\text{n}\text{=}3$, the possible values of l are, $l=0,1,2$

Explanation of Solution

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$.

Therefore, when $\text{n}\text{=}3$, the possible values of l are, $l=0,1,2$

Interpretation Introduction

Interpretation: The type of orbital that corresponds to $l=3$ should be determined.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value of $l$ indicates subshell.

Answer to Problem 68GQ

$l=3$ represents f subshells.

Explanation of Solution

Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number describes the shape of the orbitals.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value indicates subshell. Each value of $l$ indicates subshell.

For $l=3$ represents f subshells.

Interpretation Introduction

Interpretation: The value of n, l and a possible value of $m_l$ should be determined.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The principal quantum number and number of subshells are same in a shell.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value of $l$ indicates subshell.

The values of $m_l$ when the orbital angular quantum number is l are from $-l$ to $+l$.

Answer to Problem 68GQ

The value of n is $4$, value of l is $2$ and a possible value of $m_l$ is $0$ in a $\text{4d}$ orbital.

Explanation of Solution

The principal quantum number and number of subshells are same in a shell. The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. The values of $m_l$ when the orbital angular quantum number is l are from $-l$ to $+l$. Number of $m_l$ in a subshell is $2l+1$ where it indicates number of orbitals in the subshell.

For the orbital $\text{4d}$, the principal quantum number,$n=4$. For d subshell, $l=2$. When $l=2$, the values of $m_l$ are, $m_l=-2,-1,0,1,2$.

Therefore, the value of n is $4$, value of l is $2$ and a possible value of $m_l$ is $0$ in a $\text{4d}$ orbital.

Interpretation Introduction

Interpretation: The appropriate letters, l value and number of planar nodes should be given in the table for the given orbital pictures.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value of $l$ indicates subshell.

Nodal surfaces: the surface which passing through nucleus where there is zero chance of finding electrons.

Answer to Problem 68GQ

  $\begin{array}{ccc}letter& p& d\\ lvalue& 1& 2\\ Planarnodes& 1& 2\end{array}$

Explanation of Solution

The given orbital pictures are,

    

• In a subshell there are $\left(2l+1\right)$ number of orbital. Each value of l indicates subshell and for p subshells, $l=1$. Hence, there is $3$ orbital for p subshells. For p subshells, $l=1$ and has dumbbell shape. One nodal surface is passing through the p subshell and thus has two electron density regions. Therefore, $1$ nodal surfaces for p orbital.
• In a subshell there are $\left(2l+1\right)$ number of orbital. Each value of l indicates subshell and for d subshells, $l=2$. Hence, there is $5$ orbital for d subshells. For d subshells, $l=2$. Two nodal surfaces are passing through the d subshell and thus have four electron density regions. Therefore, $2$ nodal surfaces for d orbital.

  Therefore, from the left the picture represents p and d

  $\begin{array}{ccc}letter& p& d\\ lvalue& 1& 2\\ Planarnodes& 1& 2\end{array}$

Interpretation Introduction

Interpretation: The atomic orbital having three planar nodes through the nucleus should be determined.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

Nodal surfaces: the surface which passing through nucleus where there is zero chance of finding electrons.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value of $l$ indicates subshell.

Answer to Problem 68GQ

The atomic orbital having three planar nodes through the nucleus is f orbital

Explanation of Solution

Nodal surfaces is the surface which passing through nucleus where there is zero chance of finding electrons.. For f subshells, $l=3$.

Three nodal surfaces are passing through the f subshell and thus have eight electron density regions.

Therefore, an orbital having $3$ nodal surfaces through the nucleus are f orbital

Interpretation Introduction

Interpretation: The orbitals that cannot exist according to modern quantum theory should be determined from the given orbitals.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each l value indicates subshell

Answer to Problem 68GQ

The orbitals that cannot exist according to modern quantum theory $2d$ and $3f$

Explanation of Solution

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each l value indicates subshell.

When $\text{n}\text{=}2$, the values of l are, $l=\left(n-1\right)=0,1$ since the value of $l$ are from $0$ to $\left(\text{n}-1\right)$. Each $l$ value indicates subshell where $0$ and $1$ represents s and p orbitals. Therefore according to quantum theory $2d$ cannot exist.

When $\text{n}\text{=}3$, the values of l are, $l=\left(n-1\right)=0,1,2$ since the value of $l$ are from $0$ to $\left(\text{n}-1\right)$. Each $l$ value indicates subshell where $0$ and $1$ represents s p, and d orbitals. Therefore according to quantum theory $3f$ cannot exist.

Hence the orbitals that cannot exist are $2d$ and $3f$

Interpretation Introduction

Interpretation: The invalid set of quantum numbers should be determined from the given sets of quantum numbers.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each l value indicates subshell.

The values of $m_l$ when the orbital angular quantum number is l are from $-l$ to $+l$.

Answer to Problem 68GQ

The invalid set is $n=2,l=2,m_l=2,m_s=\text{+1/2}$

Explanation of Solution

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each l value indicates subshell.

Therefore, when $\text{n}\text{=}2$, the values of l are, $l=\left(n-1\right)=0,1$ with the values of $m_l=-1,0,1$. This indicates that the value of $m_l=2$ is not possible. Hence the invalid set is,

  $n=2,l=2,m_l=2,m_s=\text{+1/2}$

Interpretation Introduction

Interpretation: The maximum number of orbitals should be determined for the given set of quantum numbers.

Concept introduction:

Quantum numbers are numbers, which explains the existence and the behavior of electron in an atom.

1. a. Principle quantum number is represented by n and this number describes the energy of the orbital and the size of an atom.
2. b. Angular momentum quantum number (or azimuthal quantum number) is represented by l and this number indicates the shape of the orbitals.
3. c. Magnetic quantum number is represented by ${\text{m}}_{\text{l}}$ and this number indicates the orientation of the orbital.
4. d. Spin quantum number is represented by ${\text{m}}_{\text{s}}$ and this number indicates the spin of the electron.

The principal quantum number and number of subshells are same in a shell.

In a subshell there are $\left(2l+1\right)$ number of orbital.

There are $n^2$ number of orbitals in a shell for the principal quantum number, n.

• Planck’s equation,

    $\begin{array}{c}\text{E}\text{=}\text{hν}\\ \text{where, E}\text{=}\text{energy}\\ \text{h}\text{=}\text{Planck's}\text{constant}\\ \text{ν}=\text{frequency}\end{array}$

The energy increases as the wavelength of the light decrease. Also the energy increases as the frequency of the light increases.

Answer to Problem 68GQ

1. i. $3$ orbitals
2. ii. $9$ orbitals
3. iii. None
4. iv. $1$ orbital

Explanation of Solution

(i)

  $\text{n}\text{=}\text{2,}\text{l}\text{=}\text{1}$

In a subshell there are $\left(2l+1\right)$ number of orbital. Therefore, when $\text{n}\text{=}\text{2,}\text{l}\text{=}\text{1}$, there are $\left(\left(2\times 1\right)+1\right)=3$ orbitals for given sets of quantum numbers

(ii)

  $\text{n}\text{=}3$

In a shell, there are $n^2$ number of orbital. Therefore, when $\text{n}\text{=}3$, there are $9$ orbitals for given sets of quantum numbers.

(ii)

  $\text{n}\text{=}3\text{,}\text{l}\text{=}3$

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value indicates subshell.

Therefore, when $\text{n}\text{=}3$, the values of l are, $l=\left(n-1\right)=0,1,2$ with the values of $m_l=-1,0,1$. This indicates that the value of $l=3$ is not possible for $\text{n}\text{=}3$. So there are no orbitals.

(iv)

    $n=2,l=1,m_l=0$

The values of l when the principal quantum number is n are from $0$ to $\left(\text{n}-1\right)$. Each value indicates subshell. The values of $m_l$ when the orbital angular quantum number is l are from $-l$ to $+l$. In a subshell there are $\left(2l+1\right)$ number of orbital.

Therefore, when $\text{n}\text{=}2$, $l=1$, and $m_l=0$ indicates that there is only $1$ orbital for given sets of quantum.