Chapter 1, Problem 70IP

Using the information in Table 1-1, answer the following questions. In an ion with an unknown charge, the total mass of all the electrons was determined to be 2.55 × 10⁻²⁶ g, while the total mass of its protons was 5.34 × 10⁻²³ g. What is the identity and charge of this ion? What is the symbol and mass number of a neutral atom whose total mass of its electrons is 3.92 × 10⁻²⁶ g, while its neutrons have a mass of 9.35 × 10⁻²³ g?

Interpretation Introduction

Interpretation: Total mass of electrons and total mass of protons in an ion is given. Also, the total mass of electrons and neutrons in a neutral atom is given. By using these values, identity, charge of the ion, symbol and mass number of neutral atom are to be determined.

Concept Introduction: The number of proton in a neutral atom is equal to the atomic number of the atom. The sum of neutrons and protons is equal to the mass number of atom. In a neutral atom, the number of electrons is equal to the number of protons but in case of ion, they are different. The charge (positive or negative) on the ion is the difference of number of proton and number of electrons.

To determine: The identity and charge of the ion and also, the symbol and mass number of the atom.

Answer to Problem 70IP

Answer

The identity of the ion is $\underset{\_}{Ge}$ and the charge on the ion is $\underset{\_}{+4}$. The symbol of the atom is $\underset{\_}{{}_{43}^{99}Tc}$ and the mass number of atom is $\underset{\_}{99}$.

Explanation of Solution

Given

Total mass of all the electrons, $M_e=2.55\times {10}^{-26}\text{g}$

Total mass of all the protons, $M_p=5.34\times {10}^{-23}\text{g}$

The total number of electrons in the ion is calculated using the formula,

$e=\frac{M_e}{m_e}$

Where,

• $e$ is the total number of electrons.
• $M_e$ is the total mass of all the electrons.
• $m_e$ is the mass of one electron $\left(9.10938\times {10}^{-28}\text{g}\right)$.

Substitute the values of $M_e$ and $m_e$ in the above equation.

$\begin{array}{c}e=\frac{M_e}{m_e}\\ =\frac{2.55\times {10}^{-26}}{9.10938\times {10}^{-28}}\\ =28\end{array}$

The total number of protons in the ion is calculated using the formula,

$p=\frac{M_p}{m_p}$

Where,

• $p$ is the total number of protons.
• $M_p$ is the total mass of all the protons.
• $m_p$ is the mass of one proton $\left(1.6726\times {10}^{-24}\text{g}\right)$.

Substitute the values of $M_p$ and $m_p$ in the above equation.

$\begin{array}{c}p=\frac{M_p}{m_p}\\ =\frac{5.34\times {10}^{-23}}{1.6726\times {10}^{-24}}\\ =32\end{array}$

The atomic number of an element is equal to the total number of protons. Thus,

$\begin{array}{c}Z=p\\ =32\end{array}$

In the periodic table, atomic number $32$ corresponds to the elopement Germanium $\left(\text{Ge}\right)$.

The charge on the ion is calculated using the formula,

$\text{Charge}=p-e$

Substitute the values of $p$ and $e$ in the above equation.

$\begin{array}{c}\text{Charge}=p-e\\ =32-28\\ =+4\end{array}$

Thus, the given ion is of $\underset{\_}{Ge}$ and the charge of the ion is $\underset{\_}{+4}$.

Given

Total mass of electrons, $M_e=3.92\times {10}^{-26}\text{g}$

Total mass of neutrons, $M_n=9.35\times {10}^{-23}\text{g}$

The total number of electrons in the atom is calculated using the formula,

$e=\frac{M_e}{m_e}$

Where,

• $e$ is the total number of electrons.
• $M_e$ is the total mass of all the electrons.
• $m_e$ is the mass of one electron $\left(9.10938\times {10}^{-28}\text{g}\right)$.

Substitute the values of $M_e$ and $m_e$ in the above equation.

$\begin{array}{c}e=\frac{M_e}{m_e}\\ =\frac{3.92\times {10}^{-26}}{9.10938\times {10}^{-28}}\\ =43\end{array}$

The total number of neutrons in the atom is calculated using the formula,

$n=\frac{M_n}{m_n}$

Where,

• $n$ is the total number of neutrons.
• $M_n$ is the total mass of all the neutrons.
• $m_n$ is the mass of one neutron $\left(1.67\times {10}^{-24}\text{g}\right)$.

Substitute the values of $M_n$ and $m_n$ in the above equation.

$\begin{array}{c}n=\frac{M_n}{m_n}\\ =\frac{9.35\times {10}^{-23}}{1.67\times {10}^{-24}}\\ =56\end{array}$

For a neutral atom, number of electrons is equal to the number of protons. Thus, the number of protons is $43$.

Number of proton in an atom is equal to the atomic number of the atom, therefore,

$Z=p$

Where,

• $Z$ is the atomic number.
• $p$ is the number of protons.

Substitute the value of $p$ in the above equation,

$Z=43$

In periodic table, atomic number $43$ corresponds to the Technetium $\left(\text{Tc}\right)$.

The mass number is calculated using the formula,

$A=p+n$

Where,

• $A$ is the mass number.
• $p$ is the number of protons.
• $n$ is the number of neutrons.

Substitute the values of $p$ and $n$ in the above equation.

$\begin{array}{c}A=43+56\\ =99\end{array}$

Thus, the symbol of the atom is $\underset{\_}{{}_{43}^{99}Tc}$ and the mass number of the atom is $\underset{\_}{99}$.

Conclusion

The identity and charge of the ion are $\text{Ge}$ and $+4$, respectively.

The symbol and mass number of the atom are  ${}_{43}^{99}\text{Tc}$ and $99$, respectively.