Question

How many molecules are in $$5mg$$ 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 $$1g$$ of molecules will be equal to $$1$$ mole.
In equation;
$$1g$$of molecule $$=1$$mole
Since, $$1$$ mole contains Avogadro’s number ($$N_A$$ ) of molecule
So, $$1mole=6.022\times {10}^{23}molecule$$
Therefore, for $$1$$ molecule will be;
$$1molecule={\displaystyle \frac{1}{6.022\times {10}^{23}}}gmolecule$$
So, the answer will be;
$$1molecule=1.66\times {10}^{-24}gmolecule$$
$$1$$ mole Aspartame contains Avogadro’s number ($$N_A$$ ) of molecule.
Number of moles is calculated by using below given formula;
$$n={\displaystyle \frac{mass}{molarmass}}$$
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= $$5mg$$
Molar mass of aspartame; $$C_{14}{H}_{18}{N}_2{O}_5$$ ;
There are $$14$$ Carbon atoms and carbon molar mass is $$\left(12.0g/mol\right)$$ ,
$$18$$ Hydrogens atoms and hydrogen’s molar mass is $$\left(1.008g/mol\right)$$
$$2$$ 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 $$\left(16.0g/mol\right)$$
Molar mass of $$C_{14}{H}_{18}{N}_2{O}_5$$ ;
$$\left(14\times 12.0g/molC\right)+\left(18\times 1.00g/molH\right)+\left(2\times 14.0g/mol\text{ N}\right)+\left(5\times 16.0g/molO\right)$$ $$=\text{ 214 }g/mol{C}_{14}{H}_{18}{N}_2{O}_5$$
$$n={\displaystyle \frac{5mg}{214g/mol}}$$
Since,
$$1g={10}^{3}mg$$
Therefore,
$$={\displaystyle \frac{5\times {10}^{3}g}{214g/mol}}$$
$$=1.70\times {10}^{-5}mol$$
So, the number of moles $$=1.70\times {10}^{-5}mol$$
Using the Avogadro’s constant, we need to convert the values into molecules;
$$1.70\times {10}^{-5}mol\times {\displaystyle \frac{6.022\times {10}^{23}molecules}{1mol}}$$
$$=1\times {10}^{19}molecules$$ 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 $$12g$$ of carbon-12.