Chapter 5, Problem 23Q

Reference Section 5-2 to find the atomic masses of ¹²C and ¹³C, the relative abundance of ¹²C and ¹³C in natural carbon, and the average mass (in u) of a carbon atom. If you had a sample of natural carbon containing exactly 10,000 atoms, determine the number of ¹²C and ¹³C atoms present. What would be the average mass (in u) and the total mass (in u) of the carbon atoms in this 10,000-atom sample? If you had a sample of natural carbon containing 6.0221 × 10²³ atoms, determine the number of ¹²C and ¹³C atoms present What would be the average mass (in u) and the total mass (in u) of this 6.0221 × 10²³ atom sample? Given that 1 g = 6.0221 × 10²³ u, what is the total mass of I mole of natural carbon in units of grams?

Interpretation Introduction

Interpretation: The atomic masses, relative abundance, the average mass and the number of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms are to be calculated. Also, the total mass of one mole of natural carbon in units of gram is calculated.

Concept introduction: The number of moles is defined as the ratio of mass with the molecular mass of an element. The mass of an element is the amount of the substance present in an element. The mass is calculated by using number of moles in an element.

To determine: The atomic masses, relative abundance, the average mass and the number of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms and the total mass of one mole of natural carbon in units of gram.

Answer to Problem 23Q

The numbers of ${}^{\text{12}}\text{C}$ atoms are $\underset{\_}{9899atomsand5.995\times 10^{23}}$

The numbers of ${}^{\text{13}}\text{C}$ atoms are $\underset{\_}{111atomsand0.0668\times 10^{23}}$.

The average mass of a carbon atom is $\underset{\_}{12.01amu}$.

The mass of one mole of carbon in grams is $\underset{\_}{12.01g}$.

Explanation of Solution

Given

Total number of atoms in a sample of natural carbon is $10,00\text{0 atoms}$.

The composition of ${}^{\text{12}}\text{C}$ in natural carbon is $98.89\%$.

The composition of ${}^{\text{13}}\text{C}$ in natural carbon is $1.11\%$.

The conversion of $\text{1}\text{g}$ into $\text{amu}$ is,

$\text{1 g}=6.0221\times {10}^{\text{23}}\text{ amu}$ of carbon atom

The number of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms is calculated by using the expression,

$\text{Number of atoms}=\text{Percentage composition}\times \text{Total number of atoms}$

For ${}^{\text{12}}\text{C}$,

Substitute the value of percentage composition of ${}^{\text{12}}\text{C}$ and total number of atoms in the above expression,

$\begin{array}{c}{\text{Number of }}^{12}\text{C atoms}={\text{Percentage composition of }}^{\text{12}}\text{C}\times \text{Total number of atoms}\\ =\left(\frac{98.89}{100}\right)\times 10,000\\ =\underset{\_}{9889atoms}\end{array}$

For ${}^{\text{13}}\text{C}$,

Substitute the value of percentage composition of ${}^{\text{13}}\text{C}$ and total number of atoms in the above expression,

$\begin{array}{c}{\text{Number of }}^{13}\text{C atoms}={\text{Percentage composition of }}^{\text{13}}\text{C}\times \text{Total number of atoms}\\ =\left(\frac{1.11}{100}\right)\times 10,000\\ =\underset{\_}{111atoms}\end{array}$

Mass of ${}^{\text{12}}\text{C}$ atom is $1\text{2 amu}$.

Mass of ${}^{\text{13}}\text{C}$ atom is $13.0034\text{ amu}$.

Total mass of carbon atoms is calculated by using mass of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms.

$\begin{array}{c}\text{Total mass of carbon atoms}=\left(988\text{9}\times \text{12 amu}\right)+\left(111\times 13.003\text{4 amu}\right)\\ =120111.377\text{4 amu}\end{array}$

Therefore the average mass of carbon atoms is calculated by using an expression,

$\text{Average mass}=\frac{\text{Total mass}}{\text{Number of atoms}}$

Substitute the value of total mass of carbon atoms and number of atoms in the above expression, the average mass is calculated.

$\begin{array}{c}\text{Average mass}=\frac{\text{Total mass}}{\text{Number of atoms}}\\ =\frac{120111.377\text{4 amu}}{10,000}\\ =\underset{\_}{12.01amu}\end{array}$

Hence, the average mass of a carbon atom is $\underset{\_}{12.01amu}$.

When the total number of atoms is $6.0221\times {10}^{\text{23}}\text{ atoms}$.

The number of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms is calculated by using the expression,

$\text{Number of atoms}=\text{Percentage composition}\times \text{Total number of atoms}$.

For ${}^{\text{12}}\text{C}$,

Substitute the value of percentage composition of ${}^{\text{12}}\text{C}$ and total number of atoms in the above expression,

$\begin{array}{c}{\text{Number of }}^{12}\text{C atoms}={\text{Percentage composition of }}^{\text{12}}\text{C}\times \text{Total number of atoms}\\ =\left(\frac{98.89}{100}\right)\times 6.0221\times {10}^{23}\\ =\underset{\_}{5.955\times 10^{23}atoms}\end{array}$

For ${}^{\text{13}}\text{C}$,

Substitute the value of percentage composition of ${}^{\text{13}}\text{C}$ and total number of atoms in the above expression,

$\begin{array}{c}{\text{Number of }}^{13}\text{C atoms}={\text{Percentage composition of }}^{\text{13}}\text{C}\times \text{Total number of atoms}\\ =\left(\frac{1.11}{100}\right)\times 6.0221\times {10}^{23}\\ =\underset{\_}{0.0668\times 10^{23}atoms}\end{array}$

Total mass of carbon atoms is calculated by using mass of ${}^{\text{12}}{\text{C and }}^{\text{13}}\text{C}$ atoms.

$\begin{array}{c}\text{Total mass of carbon atoms}=\left(5.995\times {10}^{23}\times \text{12 amu}\right)+\left(0.0668\times {10}^{23}\times 13.003\text{4 amu}\right)\\ =\left(71.46\times {10}^{23}\right)+\left(0.8686\times {10}^{23}\right)\\ =72.32\times {10}^{23}\text{ amu}\\ =7.232\times {10}^{24}\text{ amu}\end{array}$.

Therefore the average mass of carbon atoms is calculated by using an expression,

$\text{Average mass}=\frac{\text{Total mass}}{\text{Number of atoms}}$

Substitute the value of total mass of carbon atoms and number of atoms in the above expression, the average mass is calculated.

$\begin{array}{c}\text{Average mass}=\frac{\text{Total mass}}{\text{Number of atoms}}\\ =\frac{\text{7}\text{.232}\times {10}^{24}\text{ amu}}{6.022\times {10}^{23}}\\ =\underset{\_}{12.01amu}\end{array}$

Hence, the average mass of a carbon atom is $\underset{\_}{12.01amu}$.

When the mass of carbon is $\text{1 g}=6.0221\times {10}^{\text{23}}\text{ amu}$.

One mole of substance contains $6.0221\times {10}^{\text{23}}\text{ atoms}$.

The mass of $6.0221\times {10}^{\text{23}}\text{ atoms}$ is $7.232\times {10}^{\text{24}}\text{ amu}$ as calculated above.

Therefore, mass of one mole of carbon in grams is calculated by using the given expression,

$\begin{array}{c}\text{Mass of 1mole of carbon}\left(\text{g}\right)=\frac{7.232\times {10}^{24}}{6.0221\times {10}^{23}}\\ =\underset{\_}{12.01g}\end{array}$

Hence, the mass of one mole of carbon in grams is $\underset{\_}{12.01g}$.

Conclusion

The numbers of ${}^{\text{12}}\text{C}$ atoms are $\underset{\_}{9899atomsand5.995\times 10^{23}atoms}$.

The numbers of ${}^{\text{13}}\text{C}$ atoms are $\underset{\_}{111atomsand0.0668\times 10^{23}atoms}$.

The average mass of a carbon atom is $\underset{\_}{12.01amu}$.

The mass of one mole of carbon in grams is $\underset{\_}{12.01g}$