Question

How many molecules are in \[5{\text{ }}mg\] of aspartame?

Answer 1

Hint: Aspartame is an artificial sweetener which is a non-saccharide sweetener. This is having \[200\] times sweeter than sucrose. The aspartame is substituted over sugar in foods and beverages. Aspartame is a methyl ester of the phenylalanine dipeptide of aspartic acid. In the place of sugar, the aspartame can be used for reducing the calories. Aspartame is having the formula of \[{C_{14}}{H_{18}}{N_2}{O_5}\,\] .

Complete step by step solution:
The \[1\,g\] of molecules will be equal to \[1\] mole.
In equation;
\[1\,g\]of molecule \[ = \,\,1\,\]mole
Since, \[1\] mole contains Avogadro’s number (\[{N_A}\] ) of molecule
So, \[1\,mole\, = \,6.022\, \times \,{10^{23}}\,molecule\]
Therefore, for \[1\] molecule will be;
\[1\,\,molecule\, = \,\dfrac{1}{{6.022\, \times \,{{10}^{23}}}}\,g\,\,molecule\]
So, the answer will be;
\[1\,molecule\, = 1.66\, \times \,{10^{ - 24}}\,g\,molecule\]
\[1\] mole Aspartame contains Avogadro’s number (\[{N_A}\] ) of molecule.
Number of moles is calculated by using below given formula;
\[n\,\, = \,\,\dfrac{{mass}}{{molar\,mass}}\]
where \[n\] is the amount in moles \[(mol)\] , mass in grams \[\;(g)\] , and molar mass in grams per mole \[\;(g/mol)\] .
The given values are;
Mass of aspartame= \[5{\text{ }}mg\]
Molar mass of aspartame; \[{C_{14}}{H_{18}}{N_2}{O_5}\,\] ;
There are \[14\] Carbon atoms and carbon molar mass is \[\left( {12.0{\text{ }}g/mol} \right)\] ,
\[18\] Hydrogens atoms and hydrogen’s molar mass is \[\left( {1.008{\text{ }}g/mol} \right)\]
\[2\] Nitrogen atoms and nitrogen’s molar mass is \[\left( {{\text{14}}{\text{.0 }}g/mol} \right)\]
and \[5\] Oxygen atom and the oxygen’s molar mass is \[\left( {16.0{\text{ }}g/mol} \right){\text{ }}\]
Molar mass of \[{C_{14}}{H_{18}}{N_2}{O_5}\,\] ;
\[\left( {14 \times 12.0{\text{ }}g/mol{\text{ }}C} \right){\text{ }} + {\text{ }}\left( {18 \times 1.00{\text{ }}g/mol{\text{ }}H} \right){\text{ }} + {\text{ }}\left( {2 \times 14.0{\text{ }}g/mol{\text{ N}}} \right){\text{ }} + {\text{ }}\left( {5 \times 16.0{\text{ }}g/mol{\text{ }}O} \right){\text{ }}\] \[ = {\text{ 214 }}g/mol{\text{ }}{C_{14}}{H_{18}}{N_2}{O_5}\,\]
\[n\,\, = \,\,\dfrac{{5\,mg}}{{214\,g/mol}}\]
Since,
\[1\,g = \,{10^3}\,mg\]
Therefore,
\[ = \,\dfrac{{5\, \times \,{{10}^3}\,g}}{{214\,g/mol}}\]
\[ = 1.70\, \times \,{10^{ - 5}}\,mol\]
So, the number of moles \[ = 1.70\, \times \,{10^{ - 5}}\,mol\]
Using the Avogadro’s constant, we need to convert the values into molecules;
\[1.70 \times {10^{ - 5}}\,\,mol\, \times \,\dfrac{{6.022\, \times \,{{10}^{23}}\,\,molecules}}{{1\,mol}}\,\]
\[ = \,\,1\, \times \,{10^{19}}\,\,molecules\] of aspartame.

Note: A mole is a unit measurement for the amount of substance in an international system of units i.e., SI unit. A mole of a particle or a mole of a substance is defined as \[6.02214076 \times {10^{23}}\] of a chemical unit, that can be ions, atoms, molecules, etc. Originally it was defined as the number of atoms in \[12{\text{ }}g\] of carbon-12.