Chapter 12, Problem 88E

(a)

Interpretation Introduction

Interpretation:In mercury number of electrons occupied by atomic orbitals with $n$ equals to 3 should be determined.

Concept introduction:Atomic orbital is mathematical function that depicts wave behavior of electrons in atom. It is used to indicate probability to find electrons in particular region around atomic nucleus. Probability density to find electron in space around sphere located at particular distance from nucleus is determined by radial distribution function.

Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Answer to Problem 88E

Number of electrons occupied by atomic orbitals with $n$ equal to 3 in mercury is 18.

Explanation of Solution

The element $\text{Hg}$ is on sixth period in periodic table and its electronic configuration s as follows:

  $\left[\text{Xe}\right]6s^{2}5d^{10}$

Since the principle quantum number of mercury is 3 that is it is present in third energy level. Type of orbitals present in third level is $s$ , $p$ and $d$ . The relation between value of $l$ and orbitals is as follows:

  $\begin{array}{cccc}\text{Value of }l& 0& 1& 2\\ \text{Orbital}& s& p& d\end{array}$

Total number of orbitals are expressed as $2l+1$ . Therefore, number of orbitals is expressed as follows:

  $\begin{array}{cccc}\text{Orbital}& s& p& d\\ \text{Value of }l& 0& 1& 2\\ 2l+1& 1& 3& 5\end{array}$

Hence, there are total 9 orbitals in third energy level and each orbital has minimum 2 electrons. Therefore, number of electrons in mercury is 18.

(b)

Interpretation Introduction

Interpretation:In mercury number of electrons occupied by $d$ atomic orbitals should be determined.

Concept introduction:Atomic orbital is mathematical function that depicts wave behavior of electrons in atom. It is used to indicate probability to find electrons in particular region around atomic nucleus. Probability density to find electron in space around sphere located at particular distance from nucleus is determined by radial distribution function.

Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Answer to Problem 88E

In mercury, number of electrons occupied by $d$ atomic orbitals is 30.

Explanation of Solution

The element $\text{Hg}$ is on sixth period in periodic table and its electronic configuration s as follows:

  $\left[\text{Xe}\right]6s^{2}5d^{10}$

Since in mercury there are three $d$ atomic orbitals that are $3d$ , $4d$ and $5d$ as seen from electronic configuration. Total number of orbitals is expressed as $2l+1$ . Therefore, number of orbitals in $d$ subshell is 5. Since each orbital occupies 2 electrons; therefore, d orbital contains total 10 electrons. Since in mercury there are 3 d orbitals hence in total 30 electrons are present in d orbital of mercury.

(c)

Interpretation Introduction

Interpretation:In mercury number of electrons occupied by $p_z$ atomic orbitals should be determined.

Concept introduction:Atomic orbital is mathematical function that depicts wave behavior of electrons in atom. It is used to indicate probability to find electrons in particular region around atomic nucleus. Probability density to find electron in space around sphere located at particular distance from nucleus is determined by radial distribution function.

Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Answer to Problem 88E

In mercury number of electrons occupied by $p_z$ atomic orbitals is 8.

Explanation of Solution

The element $\text{Hg}$ is on sixth period in periodic table and its electronic configuration s as follows:

  $\left[\text{Xe}\right]6s^{2}5d^{10}$

Since in mercury there are four $p$ atomic orbitals that are $2p$ , $3p$ , $4p$ and $5p$ as seen from electronic configuration. Total number of orbitals is expressed as $2l+1$ . Therefore, number of orbitals in $p$ subshell is 3 and these are $p_x$ , $p_y$ and $p_z$ . Each orbital contains 2 electrons and since in mercury there are four $p$ atomic orbitals with 4 $p_z$ orbitals, therefore, in total there are 8 electrons in 4 $p_z$ orbitals.

(d)

Interpretation Introduction

Interpretation:In mercury number of electrons that have spin up $\left(m_s=+\frac{1}{2}\right)$ should be determined.

Concept introduction:Atomic orbital is mathematical function that depicts wave behavior of electrons in atom. It is used to indicate probability to find electrons in particular region around atomic nucleus. Probability density to find electron in space around sphere located at particular distance from nucleus is determined by radial distribution function.

Energy, size, shape, and orientation of atomic orbital are determined with help of some numbers. These numbers are called quantum numbers and are obtained from solution of Schrodinger equation of hydrogen atom by application of boundary conditions.

Below mentioned are four quantum numbers.

1. Principal Quantum Number

It is represented by $n$ . This quantum number is related to size and energy of different atomic orbitals. With increase in value of $n$ , size of orbital becomes larger and electron is present farther from atomic nucleus for longer time.

2. Angular Momentum Quantum Number

It is represented by $l$ . It describes angular momentum of electron in any specific atomic orbital. Shape of orbitals is determined by this quantum number. It can have integral values from 0 to $\left(n-1\right)$ .

Designation of orbitals on basis of different $l$ values is as follows:

  $\begin{array}{ccccc}\text{Value of }l& 0& 1& 2& 3\\ \text{Orbital}& s& p& d& f\end{array}$

3. Magnetic Quantum Number

This quantum number is denoted by $m_l$ . It describes orientation of angular momentum associated with atomic orbital. Its value ranges from $-l$ to $+l$ . Total values of magnetic quantum number is governed by $2l+1$

4. Spin Quantum Number

This quantum number is represented by $m_s$ . It indicates direction of electron spin. It can either $+\frac{1}{2}$ or $-\frac{1}{2}$ .

Answer to Problem 88E

In mercury number of electrons that have spin up $\left(m_s=+\frac{1}{2}\right)$ is 40.

Explanation of Solution

The element $\text{Hg}$ is on sixth period in periodic table and its electronic configuration s as follows:

  $\left[\text{Xe}\right]6s^{2}5d^{10}$

Paired electrons have half spin up and half spin down. Since atomic number of mercury is 80 it means it has in total 80 electrons in mercury. Among 80 electrons, half of it that is 40 electrons are spin up and 40 are spin down. Hence, in mercury number of electrons that have spin up $\left(m_s=+\frac{1}{2}\right)$ is 40.