Chapter 2, Problem 2.140P

(a)

Interpretation Introduction

Interpretation:

The ratio of the number of neutrons to a number of proton for ${}^{144}\text{Sm}$ is to be determined.

Concept introduction:

Atoms are neutral in nature. They consist of even smaller particles namely, protons, electrons, and neutrons. Protons have a positive charge on them. Electrons have a negative charge. Neutrons have no charge in them. Protons and neutrons make up the nucleus in the atoms whereas the electrons revolve around the nucleus.

The general representation for an atom is given as ${}_Z^A\text{X}$. Here, $A$ is a mass number and $Z$ is the atomic number of an atom.

The superscript in the formula of elements or atoms is the mass number. The mass number is the sum of protons and neutrons in the atom. The subscript in the formula represents the atomic number. The atomic number of an element or atom is the number of protons in that element or atom. The difference between the mass number and the number of protons gives the number of neutrons.

The formula to calculate the number of neutrons is,

  $\text{Total number of neutrons}\left(N\right)=\left[\begin{array}{l}\text{Mass number}\left(A\right)-\\ \text{Total number of protons}\left(Z\right)\end{array}\right]$        (1)

The expression to calculate the ratio of $\left(\frac{N}{Z}\right)$ is as follows:

  $\frac{N}{Z}=\frac{\text{Total number of neutrons}}{\text{Total number of proton}}$        (2)

Answer to Problem 2.140P

The ratio of the number of neutrons to a number of proton for ${}^{144}\text{Sm}$ is $1.3$.

Explanation of Solution

In $\text{S}\text{m}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of ${}^{144}\text{Sm}$ an atom is $62$. Thus, the number of protons in $\text{S}\text{m}$ is $62$.

Substitute $144$ for the mass number and $62$ for the total number of protons in equation (1) to calculate the number of neutrons in $\text{S}\text{m}$.

  $\begin{array}{c}\text{Total number of neutrons}=144-62\\ =82\end{array}$

Substitute $82$ for a total number of neutrons and $62$ for the total number of protons in the equation (2).

  $\begin{array}{c}\frac{N}{Z}=\frac{82}{62}\\ =\mathrm{1.3.}\end{array}$

Conclusion

The value of $\left(\frac{N}{Z}\right)$ for ${}^{144}\text{Sm}$ is $1.3$.

(b)

Interpretation Introduction

Interpretation:

The ratio of the number of neutrons to a number of proton for ${}^{56}\text{Fe}$ is to be determined.

Concept introduction:

Atoms are neutral in nature. They consist of even smaller particles namely, protons, electrons, and neutrons. Protons have a positive charge on them. Electrons have a negative charge. Neutrons have no charge in them. Protons and neutrons make up the nucleus in the atoms whereas the electrons revolve around the nucleus.

The general representation for an atom is given as ${}_Z^A\text{X}$. Here, $A$ is a mass number and $Z$ is the atomic number of an atom.

The superscript in the formula of elements or atoms is the mass number. The mass number is the sum of protons and neutrons in the atom. The subscript in the formula represents the atomic number. The atomic number of an element or atom is the number of protons in that element or atom. The difference between the mass number and the number of protons gives the number of neutrons.

The formula to calculate the number of neutrons is,

  $\text{Total number of neutrons}\left(N\right)=\left[\begin{array}{l}\text{Mass number}\left(A\right)-\\ \text{Total number of protons}\left(Z\right)\end{array}\right]$        (1)

The expression to calculate the ratio of $\left(\frac{N}{Z}\right)$ is as follows:

  $\frac{N}{Z}=\frac{\text{Total number of neutrons}}{\text{Total number of proton}}$        (2)

Answer to Problem 2.140P

The ratio of the number of neutrons to a number of proton for ${}^{56}\text{Fe}$ is $1.2$.

Explanation of Solution

In $\text{F}\text{e}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of $\text{F}\text{e}$ is $\text{26}$. Thus, the number of protons is 26.

Substitute $56$ for the mass number and 26 for the total number of protons in equation (1).

  $\begin{array}{c}\text{Total number of neutrons}=56-26\\ =30\end{array}$

Substitute $30$ for a total number of neutrons and $26$ for a total number of protons in the equation (2).

  $\begin{array}{c}\frac{N}{Z}=\frac{30}{26}\\ =\mathrm{1.2.}\end{array}$

Conclusion

The value of $\left(\frac{N}{Z}\right)$ for ${}^{56}\text{Fe}$ is $1.2$.

(c)

Interpretation Introduction

Interpretation:

The ratio of the number of neutrons to a number of proton for ${}^{20}\text{Ne}$ is to be determined.

Concept introduction:

Atoms are neutral in nature. They consist of even smaller particles namely, protons, electrons, and neutrons. Protons have a positive charge on them. Electrons have a negative charge. Neutrons have no charge in them. Protons and neutrons make up the nucleus in the atoms whereas the electrons revolve around the nucleus.

The general representation for an atom is given as ${}_Z^A\text{X}$. Here, $A$ is a mass number and $Z$ is the atomic number of an atom.

The superscript in the formula of elements or atoms is the mass number. The mass number is the sum of protons and neutrons in the atom. The subscript in the formula represents the atomic number. The atomic number of an element or atom is the number of protons in that element or atom. The difference between the mass number and the number of protons gives the number of neutrons.

The formula to calculate the number of neutrons is,

  $\text{Total number of neutrons}\left(N\right)=\left[\begin{array}{l}\text{Mass number}\left(A\right)-\\ \text{Total number of protons}\left(Z\right)\end{array}\right]$        (1)

The expression to calculate the ratio of $\left(\frac{N}{Z}\right)$ is as follows:

  $\frac{N}{Z}=\frac{\text{Total number of neutrons}}{\text{Total number of proton}}$        (2)

Answer to Problem 2.140P

The ratio of the number of neutrons to a number of proton for ${}^{20}\text{Ne}$ is $1.0$.

Explanation of Solution

In $\text{N}\text{e}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of $\text{N}\text{e}$ is 10. Thus, the number of protons is 10.

Substitute $20$ for the mass number and 10 for the total number of protons in equation (1).

  $\begin{array}{c}\text{Total number of neutrons}=20-10\\ =10\end{array}$

Substitute $10$ for a total number of neutrons and $10$ for a total number of protons in the equation (2).

  $\begin{array}{c}\frac{N}{Z}=\frac{10}{10}\\ =1.\end{array}$

Conclusion

The value of $\left(\frac{N}{Z}\right)$ for ${}^{20}\text{Ne}$ is $1.0$.

(d)

Interpretation Introduction

Interpretation:

The ratio of the number of neutrons to a number of proton for ${}^{107}\text{Ag}$ is to be determined.

Concept introduction:

Atoms are neutral in nature. They consist of even smaller particles namely, protons, electrons, and neutrons. Protons have a positive charge on them. Electrons have a negative charge. Neutrons have no charge in them. Protons and neutrons make up the nucleus in the atoms whereas the electrons revolve around the nucleus.

The general representation for an atom is given as ${}_Z^A\text{X}$. Here, $A$ is a mass number and $Z$ is the atomic number of an atom.

The superscript in the formula of elements or atoms is the mass number. The mass number is the sum of protons and neutrons in the atom. The subscript in the formula represents the atomic number. The atomic number of an element or atom is the number of protons in that element or atom. The difference between the mass number and the number of protons gives the number of neutrons.

The formula to calculate the number of neutrons is,

  $\text{Total number of neutrons}\left(N\right)=\left[\begin{array}{l}\text{Mass number}\left(A\right)-\\ \text{Total number of protons}\left(Z\right)\end{array}\right]$        (1)

The expression to calculate the ratio of $\left(\frac{N}{Z}\right)$ is as follows:

  $\frac{N}{Z}=\frac{\text{Total number of neutrons}}{\text{Total number of proton}}$        (2)

Answer to Problem 2.140P

The ratio of the number of neutrons to a number of proton for ${}^{107}\text{Ag}$ is $1.3$.

Explanation of Solution

In $\text{A}\text{g}$ atoms, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of $\text{A}\text{g}$ is 47. Thus, the number of protons is 47.

Substitute $107$ for the mass number and 47 for the total number of protons in equation (1).

  $\begin{array}{c}\text{Total number of neutrons}=107-47\\ =60\end{array}$

Substitute $60$ for a total number of neutrons and $47$ for the total number of protons in the equation (2).

  $\begin{array}{c}\frac{N}{Z}=\frac{60}{47}\\ =\mathrm{1.3.}\end{array}$

Conclusion

The value of $\left(\frac{N}{Z}\right)$ for ${}^{107}\text{Ag}$ is $1.3$.

(e)

Interpretation Introduction

Interpretation:

The number of neutrons, proton, and electron in ${}^{234}\text{U}$, ${}^{238}\text{U}$, ${}^{214}\text{Pb}$, ${}^{210}\text{Pb}$ and ${}^{206}\text{Pb}$ is to be determined.

Concept introduction:

Atoms are neutral in nature. They consist of even smaller particles namely, protons, electrons, and neutrons. Protons have a positive charge on them. Electrons have a negative charge. Neutrons have no charge in them. Protons and neutrons make up the nucleus in the atoms whereas the electrons revolve around the nucleus.

The general representation for an atom is given as ${}_Z^A\text{X}$. Here, $A$ is a mass number and $Z$ is the atomic number of an atom.

The superscript in the formula of elements or atoms is the mass number. The mass number is the sum of protons and neutrons in the atom. The subscript in the formula represents the atomic number. The atomic number of an element or atom is the number of protons in that element or atom. The difference between the mass number and the number of protons gives the number of neutrons.

The formula to calculate the number of neutrons is,

  $\text{Total number of neutrons}\left(N\right)=\left[\begin{array}{l}\text{Mass number}\left(A\right)-\\ \text{Total number of protons}\left(Z\right)\end{array}\right]$        (1)

Answer to Problem 2.140P

The number of neutrons, protons, and electrons in ${}^{234}\text{U}$, ${}^{238}\text{U}$, ${}^{214}\text{Pb}$, ${}^{210}\text{Pb}$ and ${}^{206}\text{Pb}$ are as follows:

$\begin{array}{cccc}\text{Isotope}& \text{Neutrons}& \text{Protons}& \text{Electrons}\\ {}^{234}\text{U}& 146& 92& 92\\ {}^{238}\text{U}& 142& 92& 92\\ {}^{214}\text{Pb}& 132& 82& 82\\ {}^{210}\text{Pb}& 128& 82& 82\\ {}^{206}\text{Pb}& 124& 82& 82\end{array}$

Explanation of Solution

In $\text{U}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of $\text{U}$ is $92$. Thus, the number of protons is 92. In neutral atom number of electrons is equal to the number of protons, therefore, the number of electrons is 92.

Substitute $238$ for the mass number and 92 for the total number of protons in equation (1) to calculate the number of neutrons in $\text{U}$.

  $\begin{array}{c}\text{Total number of neutrons}=238-92\\ =146\end{array}$

In $\text{U}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of $\text{U}$ is $92$. Thus, the number of protons is 92 and the number of electrons is also 92.

Substitute 234 for the mass number and 92 for the total number of protons in equation (1) to calculate the number of neutrons in $\text{U}$.

  $\begin{array}{c}\text{Total number of neutrons}=234-92\\ =142\end{array}$

In $\text{P}\text{b}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of this atom is 82. Thus, the number of protons is 82 and the number of electrons is also 82.

Substitute $214$ for the mass number and 82 for the total number of protons in equation (1) to calculate the number of neutrons in $\text{P}\text{b}$.

  $\begin{array}{c}\text{Total number of neutrons}=214-82\\ =132\end{array}$

In $\text{P}\text{b}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of this atom is 82. Thus, the number of protons is 82 and the number of electrons is also 82.

Substitute $210$ for the mass number and 82 for the total number of protons in equation (1) to calculate the number of neutrons in $\text{P}\text{b}$.

  $\begin{array}{c}\text{Total number of neutrons}=210-82\\ =128\end{array}$

In $\text{P}\text{b}$ atom, the number of protons and electrons is the same since the atom is a neutral species. The atomic number is equal to the number of protons. The atomic number of this atom is 82. Thus, a number of protons is 82 and the number of electrons is 82.

Substitute $206$ for the mass number and 82 for the total number of protons in equation (1) to calculate the number of neutrons in $\text{P}\text{b}$.

  $\begin{array}{c}\text{Total number of neutrons}=206-82\\ =124.\end{array}$

Conclusion

Isotopes are the atoms of the same element with the same atomic number but different mass number. In a neutral atom, the number of electrons is equal to the number of protons.