Question

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

Answer 1

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.