Chapter 3, Problem 3.58E

(a)

Interpretation Introduction

Interpretation:

Total mass of electrons present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ has to be determined.

Concept Introduction:

Atom of any element contains three subatomic particles as electron, proton, neutron. Protons of atom are equal to atomic number of element. For neutral atom, electrons are equal to total proton. Neutrons in atom are equal to difference between mass number and number of protons. Mass of electron in $9.11\times {10}^{-31}\text{ kg}$ and mass of proton in $1.67\times {10}^{-27}\text{ kg}$.

Answer to Problem 3.58E

Mass of electrons present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ is $1.13\text{ g}$.

Explanation of Solution

Ceramic ${\text{BaTiO}}_{\text{3}}$ contains 1 $\text{Ba}$ atom that has atomic number 56, 1 $\text{Ti}$ atom that has atomic number 22, and 3 $\text{O}$ atoms that have atomic number 8. Since molecule is neutral thus total electrons are equal to sum of atomic number of total atoms. Thus, total number of electron in 1 molecule of ${\text{BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Total electrons}=1\left(56\right)+1\left(22\right)+3\left(8\right)\\ =102\text{ electrons}\end{array}$

Number of moles present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Mole}=\frac{\text{Given mass}}{\text{Molar mass}}\\ =\left(\frac{4.72\text{ kg}}{233.19\text{ g/mol}}\right)\left(\frac{{10}^3\text{ g}}{1\text{ kg}}\right)\\ =20.24\text{ mol}\end{array}$

Expression to determine number of molecule of ${\text{BaTiO}}_{\text{3}}$ present in $20.24\text{ mol}$ is as follows:

  $\text{Number of molecule}=\left(\text{Moles}\right)\left(6.022\times {10}^{23}\text{ molecules}\cdot {\text{mol}}^{-1}\right)$        (1)

Substitute $20.24\text{ mol}$ for moles in equation (1).

  $\begin{array}{c}\text{Number of molecule}=\left(20.24\text{ mol}\right)\left(6.022\times {10}^{23}\text{ molecules}\cdot {\text{mol}}^{-1}\right)\\ =1.22\times {10}^{25}\text{ molecules}\end{array}$

Since 1 molecule of ${\text{BaTiO}}_{\text{3}}$ contains 102 electrons, thus electrons contained by $1.22\times {10}^{25}\text{ molecules}$ of ${\text{BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Total electrons}=\left(102\text{ electrons}\right)\left(1.22\times {10}^{25}\text{ molecules}\right)\\ =1.24\times {10}^{27}\text{ electrons}\end{array}$

Total mass of $1.24\times {10}^{27}\text{ electrons}$ can be calculated as follows:

  $\begin{array}{c}\text{Total mass of electrons}=\left(1.24\times {10}^{27}\text{ electrons}\right)\left(9.11\times {10}^{-31}\text{ kg}\cdot {\text{electron}}^{-1}\right)\\ =\left(0.00113\text{ kg}\right)\left(\frac{{10}^3\text{ g}}{1\text{ kg}}\right)\\ =1.13\text{ g}\end{array}$

Hence mass of $1.24\times {10}^{27}\text{ electrons}$ is $1.13\text{ g}$.

(b)

Interpretation Introduction

Interpretation:

Total mass of protons present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ has to be determined.

Concept Introduction:

Refer to part (a).

Answer to Problem 3.58E

Mass of protons present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ is $2.07\text{ kg}$.

Explanation of Solution

Ceramic ${\text{BaTiO}}_{\text{3}}$ contains 1 $\text{Ba}$ atom that has atomic number 56, 1 $\text{Ti}$ atom that has atomic number 22, and 3 $\text{O}$ atoms that have atomic number 8. Thus total number of protons in 1 molecule of ${\text{BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Total protons}=1\left(56\right)+1\left(22\right)+3\left(8\right)\\ =102\text{ protons}\end{array}$

Number of moles present in $\text{4}{\text{.72 kg BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Mole}=\frac{\text{Given mass}}{\text{Molar mass}}\\ =\left(\frac{4.72\text{ kg}}{233.19\text{ g/mol}}\right)\left(\frac{{10}^3\text{ g}}{1\text{ kg}}\right)\\ =20.24\text{ mol}\end{array}$

Expression to determine number of molecule of ${\text{BaTiO}}_{\text{3}}$ present in $20.24\text{ mol}$ is as follows:

  $\text{Number of molecule}=\left(\text{Moles}\right)\left(6.022\times {10}^{23}\text{ molecules}\cdot {\text{mol}}^{-1}\right)$        (1)

Substitute $20.24\text{ mol}$ for moles in equation (1).

  $\begin{array}{c}\text{Number of molecule}=\left(20.24\text{ mol}\right)\left(6.022\times {10}^{23}\text{ molecules}\cdot {\text{mol}}^{-1}\right)\\ =1.22\times {10}^{25}\text{ molecules}\end{array}$

Since 1 molecule of ${\text{BaTiO}}_{\text{3}}$ contains 102 protons, thus protons contained by $1.22\times {10}^{25}\text{ molecules}$ of ${\text{BaTiO}}_{\text{3}}$ can be calculated as follows:

  $\begin{array}{c}\text{Total protons}=\left(102\text{ protons}\right)\left(1.22\times {10}^{25}\text{ molecules}\right)\\ =1.24\times {10}^{27}\text{ protons}\end{array}$

Total mass of $1.24\times {10}^{27}\text{ protons}$ can be calculated as follows:

  $\begin{array}{c}\text{Total mass of electrons}=\left(1.24\times {10}^{27}\text{ electrons}\right)\left(1.67\times {10}^{-27}\text{ kg}\cdot {\text{electron}}^{-1}\right)\\ =2.07\text{ kg}\end{array}$

Hence mass of $1.24\times {10}^{27}\text{ protons}$ is $2.07\text{ kg}$.