Chapter 2, Problem 2.41P

(a)

Interpretation Introduction

Interpretation:

Whether ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ have the same number of protons, electrons or neutrons 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 2.41P

${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ have equal numbers of protons and electrons. The number of neutrons is different in both. The given pair of isotopes have the same $Z$ value.

Explanation of Solution

In atoms, the number of protons and electrons is the same since atom is a neutral species. Hence, there are $8$ protons and $8$ electrons in ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ each.

The formula for calculating the number of neutrons in ${}_8^{16}\text{O}$ is,

$\text{Total number of neutrons}=\text{Mass number}-\text{Total number of protons}$ (1)

Substitute $16$ for the mass number and $8$ for the total number of protons in equation (1) to calculate the number of neutrons in ${}_8^{16}\text{O}$ .

$\begin{array}{c}\text{Total number of neutrons}=16-8\\ =8\end{array}$

Substitute $17$ for the mass number and $8$ for the total number of protons in equation (1) to calculate the number of neutrons in ${}_8^{17}\text{O}$ .

$\begin{array}{c}\text{Total number of neutrons}=17-8\\ =9\end{array}$

The number of neutrons in ${}_8^{16}\text{O}$ is $8$ and the number of neutrons in ${}_8^{17}\text{O}$ is $9$.

The atomic number of isotopes ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ are the same. Hence ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ have the same $Z$ value.

Conclusion

The number of protons and electrons in ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ are the same whereas the number of neutrons is different. Hence ${}_8^{16}\text{O}$ and ${}_8^{17}\text{O}$ have the same $Z$ value.

(b)

Interpretation Introduction

Interpretation:

Whether ${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ have the same number of protons, electrons or neutrons 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 2.41P

${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ have different numbers of protons and electrons. The number of neutrons is the same. The pair have the same $N$ value.

Explanation of Solution

In an atom, the number of protons and electrons is the same since atom is a neutral species. Hence, there are $18$ protons and $18$ electrons in ${}_{18}^{40}\text{Ar}$.

Substitute $40$ for the mass number and $18$ for the total number of protons in equation (1) to calculate the number of neutrons in ${}_{18}^{40}\text{Ar}$.

$\begin{array}{c}\text{Total number of neutrons}=40-18\\ =22\end{array}$

In an atom, the number of protons and electrons is the same since atom is a neutral species. Hence, there are $19$ protons and electrons in ${}_{19}^{41}\text{K}$.

Substitute $41$ for the mass number and $19$ for the total number of protons in equation (1) to calculate the number of neutrons in ${}_{19}^{41}\text{K}$.

$\begin{array}{c}\text{Total number of neutrons}=41-19\\ =22\end{array}$

The number of neutrons in ${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ is $22$ for each.

The number of neutrons in ${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ are the same. Hence ${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ have the same $N$ value.

Conclusion

${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ have different numbers of protons and electrons. The number of neutrons is the same. Hence ${}_{18}^{40}\text{Ar}$ and ${}_{19}^{41}\text{K}$ have the same $N$ value.

(c)

Interpretation Introduction

Interpretation:

Whether ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$ have the same number of protons, electrons or neutrons 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 2.41P

The numbers of protons, neutrons, and electrons are different in ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$. The given pair has the same $A$ value.

Explanation of Solution

In an atom, the number of protons and electrons is the same since atom is a neutral species. Hence, there are $27$ protons and $27$ electrons in ${}_{27}^{60}\text{Co}$.

Substitute $60$ for the mass number and $27$ for the total number of protons in equation (5) to calculate the number of neutrons in ${}_{27}^{60}\text{Co}$.

$\begin{array}{c}\text{Total number of neutrons}=60-27\\ =33\end{array}$

In an atom, the number of protons and electrons is the same since atom is a neutral species. Hence, there are $28$ protons and $28$ electrons in ${}_{28}^{60}\text{Ni}$.

Substitute $60$ for the mass number and $28$ for the total number of protons in equation (6) to calculate the number of neutrons in ${}_{28}^{60}\text{Ni}$.

$\begin{array}{c}\text{Total number of neutrons}=60-28\\ =32\end{array}$

The numbers of neutrons in ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$ are $33$ and $32$ respectively.

The mass number ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$ are the same. Hence ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$ have the same $A$ value.

Conclusion

The number of protons, electrons, and neutrons are different in ${}_{27}^{60}\text{Co}$ and ${}_{28}^{60}\text{Ni}$. The mass number is the same in both elements. Hence they have the same $A$ value.