Question

A $1.00{\text{ carat}}$ pure diamond has a mass of $0.2{\text{ grams}}$. How many carbon atoms are there within this diamond?

Answer 1

Hint:We are given the mass of a substance i.e. diamond. To solve this first calculate the number of moles of diamond in $0.2{\text{ grams}}$ of diamond. We know that ${\text{1 mol}}$ of any substance contains $6.022 \times {10^{23}}$ atoms. $6.022 \times {10^{23}}$ is Avogadro's number. Then calculate the number of atoms of carbon in $0.2{\text{ grams}}$ of diamond.

Complete answer:
We are given the mass of a substance i.e. diamond. We know that diamond is an allotrope of carbon. Thus, diamond contains $0.2{\text{ grams}}$ of carbon. The number of moles of any substance is the ratio of mass (in grams) to the molar mass (in gram per mol).
Thus, the equation to calculate the number of moles is as follows:
${\text{Number of moles}}\left( {{\text{mol}}} \right) = \dfrac{{{\text{Mass}}\left( {\text{g}} \right)}}{{{\text{Molar mass}}\left( {{\text{g/mol}}} \right)}}$
Substitute $0.2{\text{ grams}}$ for the mass, $12.01{\text{ g/mol}}$ for the molar mass. Thus,
Number of moles of diamond $ = \dfrac{{0.2{\text{ g}}}}{{12.01{\text{ g/mol}}}}$
Number of moles of diamond $ = 0.01665{\text{ mol}}$
Thus, moles of diamond i.e. carbon in $0.2{\text{ grams}}$ of diamond are $0.01665{\text{ mol}}$.
We know that the amount of substance having exactly the same number of atoms as are present in $12{\text{ g}}$ of ${{\text{C}}^{{\text{12}}}}$ is known as mole. Moles is the ratio of the mass of substance in g to the molar mass of the substance in \[{\text{g/mol}}\]. ${\text{1 mol}}$ of any substance contains $6.022 \times {10^{23}}$ atoms. $6.022 \times {10^{23}}$ is Avogadro's number.
We know that ${\text{1 mol}}$ of any substance contains $6.022 \times {10^{23}}$ atoms. We have $0.01665{\text{ mol}}$ of carbon. Thus,
Number of carbon atoms $ = 0.01665{\text{ mol}} \times \dfrac{{6.022 \times {{10}^{23}}{\text{ atoms}}}}{{1{\text{ mol}}}}$
Number of carbon atoms $ = 0.100 \times {10^{23}}{\text{ atoms}}$
Number of carbon atoms $ = {10^{22}}{\text{ atoms}}$
Thus, $0.2{\text{ grams}}$ of diamond contains ${10^{22}}{\text{ atoms}}$ atoms of carbon.

Note:Amount of substance having exactly the same number of atoms as are present in 12 grams of ${{\text{C}}^{{\text{12}}}}$ is known as mole. The number of molecules in one mole of a compound is Avogadro’s number. The number $6.022 \times {10^{23}}$ is known as Avogadro’s number.