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Question: A \(1.00{\text{ carat}}\) pure diamond has a mass of \(0.2{\text{ grams}}\). How many carbon atoms a...

A 1.00 carat1.00{\text{ carat}} pure diamond has a mass of 0.2 grams0.2{\text{ grams}}. How many carbon atoms are there within this diamond?

Explanation

Solution

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 grams0.2{\text{ grams}} of diamond. We know that 1 mol{\text{1 mol}} of any substance contains 6.022×10236.022 \times {10^{23}} atoms. 6.022×10236.022 \times {10^{23}} is Avogadro's number. Then calculate the number of atoms of carbon in 0.2 grams0.2{\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 grams0.2{\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:
Number of moles(mol)=Mass(g)Molar mass(g/mol){\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 grams0.2{\text{ grams}} for the mass, 12.01 g/mol12.01{\text{ g/mol}} for the molar mass. Thus,
Number of moles of diamond =0.2 g12.01 g/mol = \dfrac{{0.2{\text{ g}}}}{{12.01{\text{ g/mol}}}}
Number of moles of diamond =0.01665 mol = 0.01665{\text{ mol}}
Thus, moles of diamond i.e. carbon in 0.2 grams0.2{\text{ grams}} of diamond are 0.01665 mol0.01665{\text{ mol}}.
We know that the amount of substance having exactly the same number of atoms as are present in 12 g12{\text{ g}} of C12{{\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 g/mol{\text{g/mol}}. 1 mol{\text{1 mol}} of any substance contains 6.022×10236.022 \times {10^{23}} atoms. 6.022×10236.022 \times {10^{23}} is Avogadro's number.
We know that 1 mol{\text{1 mol}} of any substance contains 6.022×10236.022 \times {10^{23}} atoms. We have 0.01665 mol0.01665{\text{ mol}} of carbon. Thus,
Number of carbon atoms =0.01665 mol×6.022×1023 atoms1 mol = 0.01665{\text{ mol}} \times \dfrac{{6.022 \times {{10}^{23}}{\text{ atoms}}}}{{1{\text{ mol}}}}
Number of carbon atoms =0.100×1023 atoms = 0.100 \times {10^{23}}{\text{ atoms}}
Number of carbon atoms =1022 atoms = {10^{22}}{\text{ atoms}}
Thus, 0.2 grams0.2{\text{ grams}} of diamond contains 1022 atoms{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 C12{{\text{C}}^{{\text{12}}}} is known as mole. The number of molecules in one mole of a compound is Avogadro’s number. The number 6.022×10236.022 \times {10^{23}} is known as Avogadro’s number.