Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3 s , (b) 4 p , (c) 3 d .

Question

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Answer 1

Textbook Question

Chapter 7, Problem 7.58QP

Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3s, (b) 4p, (c) 3d.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

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Answer 2

Textbook Question

Chapter 7, Problem 7.58QP

Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3s, (b) 4p, (c) 3d.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

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Answer 3

Textbook Question

Chapter 7, Problem 7.58QP

Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3s, (b) 4p, (c) 3d.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

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Answer 4

Textbook Question

Chapter 7, Problem 7.58QP

Give the values of the four quantum numbers of an electron in the following orbitals: (a) 3s, (b) 4p, (c) 3d.

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

Answer 8

(a)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (a) 3s

Get the values of the quantum numbers ‘n’, ‘l’ in (a)

Answer 13

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½) (a).

Answer 14

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3s is 3. The letter in orbital designation gives the value of ‘s’. For subshell ‘s’, the possible value of ‘l’ is 0.

Find the values of ‘m_l’ and ‘m_s’

For l = 0, m_l values are from ‒ l, ..., 0, ..., +l. So, 0 is resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3s orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3s is (3, 0, 0, ±½).

Answer 15

(b)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (b) 4p

Get the values of the quantum numbers ‘n’, ‘l’ in (b)

Answer 20

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 4p is (4, 1, (‒1, 0, +1), ±½) (b).

Answer 21

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 4p is 4. The letter in orbital designation gives the value of ‘l’. For subshell ‘p’, the possible value of ‘l’ is 1.

Find the values of ‘m_l’ and ‘m_s’

For l = 1, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒1, 0 and +1 are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 2p orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with 4p is (4, 1, (‒1, 0, +1), ±½).

Answer 22

(c)

Expert Solution

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Interpretation Introduction

Interpretation:

The values of the quantum numbers associated with the given orbitals should be identified using the concept of quantum numbers.

Concept Introduction:

Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n)

The principal quantum number (n) assigns the size of the orbital and specifies the energy of an electron. If the value of n is larger, then the average distance of an electron in the orbital from the nucleus will be greater. Therefore the size of the orbital is large. The principal quantum numbers have the integral values of 1, 2, 3 and so forth and it corresponds to the quantum number in Bohr’s model of the hydrogen atom. If all orbitals have the same value of ‘n’, they are said to be in the same shell (level). The total number of orbitals for a given n value is n². As the value of ‘n’ increases, the energy of the electron also increases.

Angular Momentum Quantum Number (l)

The angular momentum quantum number (l) explains the shape of the atomic orbital. The values of l are integers which depend on the value of the principal quantum number, n. For a given value of n, the possible values of l range are from 0 to n − 1. If n = 1, there is only one possible value of l (l=0). If n = 2, there are two values of l: 0 and 1. If n = 3, there are three values of l: 0, 1, and 2. The value of l is selected by the letters s, p, d, and f. If l = 0, we have an s orbital; if l = 1, we have a p orbital; if l = 2, we have a d orbital and finally if l = 3, we have a f orbital. A collection of orbitals with the same value of n is called a shell. One or more orbitals with the same n and l values are referred to a subshell (sublevel). The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) explains the orientation of the orbital in space. The value of m_l depends on the value of l in a subshell. This number divides the subshell into individual orbitals which hold the electrons. For a certain value of l, there are (2l + 1) integral values of m_l which is explained as follows:

m_l = ‒ l, ..., 0, ..., +l

If l = 0, there is only one possible value of m_l: 0.

If l = 1, then there are three values of m_l: −1, 0, and +1.

If l = 2, there are five values of m_l, namely, −2, −1, 0, +1, and +2.

If l = 3, there are seven values of m_l, namely, −3, −2, −1, 0, +1, +2, and +3, and so on.

The number of m_l values indicates the number of orbitals in a subshell with a particular l value. Therefore, each m_l value refers to a different orbital.

Electron Spin Quantum Number (m_s)

It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron spins in the clockwise direction. Another electron spins in the anticlockwise direction. But, no two electrons should have the same spin quantum number.

To find: Get the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital (c) 3d

Get the values of the quantum numbers ‘n’, ‘l’ in (c)

Answer 27

Answer to Problem 7.58QP

The values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½) (c).

Answer 28

Explanation of Solution

The integer at the beginning of the orbital designation is n. Hence, n in the given orbital, 3d is 3. The letter in orbital designation gives the value of ‘l’. For subshell ‘d’, the possible value of ‘l’ is 2.

Find the values of ‘m_l’ and ‘m_s’

For l = 2, m_l values are from ‒ l, ..., 0, ..., +l. So, ‒2, ‒1, 0, +1 and +2 values are resulted. There are two possible ways to represent m_s values. They are +½ and ‒½. One electron is given in upward direction. Another electron is given in the downward direction. But, no two electrons should have the same spin quantum number. Hence the values of ‘m_s’ for 3d orbital is ±½. Therefore, the values of the quantum numbers (n, l, m_l, m_s) associated with the given orbital 3d is (3, 2, (‒2, ‒1, 0, +1, +2), ±½).