Question

The active ingredient in some antiperspirants is aluminum chlorohydrate, \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl. Analysis of a \(2.000-\mathrm{g}\) sample of antiperspirant yields \(0.334 \mathrm{~g}\) of aluminum. What percent (by mass) of aluminum chlorohydrate is present in the antiperspirant? (Assume that there are no other compounds containing aluminum in the antiperspirant.)

Short answer

Answer: The mass percentage of aluminum chlorohydrate in the antiperspirant is approximately 65.1%.

Step-by-step solution

Calculate moles of aluminum in the sample

We are given the mass of aluminum in the sample, which is \(0.334 \mathrm{~g}\). To find the moles of aluminum, we need to use its molar mass, which is approximately \(26.98 \mathrm{~g/mol}\). The formula to calculate the moles is given by: Moles of Aluminum = \(\frac{Mass ~of ~Aluminum}{Molar ~Mass ~of ~Aluminum}\) Moles of Aluminum = \(\frac{0.334 \mathrm{~g}}{26.98 \mathrm{~g/mol}}\) Moles of Aluminum = \(0.01238 ~mol\)

Calculate moles of aluminum chlorohydrate

Since there are two moles of aluminum in one mole of aluminum chlorohydrate, \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl, we can find the moles of aluminum chlorohydrate using the following formula: Moles of Aluminum chlorohydrate = \(\frac{Moles ~of ~Aluminum}{2}\) Moles of Aluminum chlorohydrate = \(\frac{0.01238 ~mol}{2}\) Moles of Aluminum chlorohydrate = \(0.00619 ~mol\)

Calculate the mass of aluminum chlorohydrate using its molecular formula

Now we'll calculate the molar mass of aluminum chlorohydrate, which can be found from its molecular formula, \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl. Molar mass of \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl = \(2(Molar ~Mass ~of ~Aluminum + 5(Molar ~Mass ~of ~Oxygen + Molar ~Mass ~of ~Hydrogen)+ Molar ~Mass ~of ~Chlorine)\) Molar mass of \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl = \(2(26.98 \mathrm{~g/mol} + 5(16.00 \mathrm{~g/mol} + 1.008 \mathrm{~g/mol}) + 35.45 \mathrm{~g/mol})\) Molar mass of \(\mathrm{Al}_{2}(\mathrm{OH})_{5}\) Cl = \(210.38 \mathrm{~g/mol}\) Now we can find the mass of aluminum chlorohydrate in the sample: Mass of Aluminum chlorohydrate = Moles of Aluminum chlorohydrate × Molar mass of Aluminum chlorohydrate Mass of Aluminum chlorohydrate = \(0.00619 ~mol \times 210.38 \mathrm{~g/mol}\) Mass of Aluminum chlorohydrate = \(1.302 \mathrm{~g}\)

Calculate the mass percentage of aluminum chlorohydrate in the antiperspirant

The mass percentage is given by the following formula: Mass percentage of Aluminum chlorohydrate = \(\frac{Mass ~of ~Aluminum ~chlorohydrate}{Total ~Mass ~of ~Antiperspirant ~Sample} \times 100\) Mass percentage of Aluminum chlorohydrate = \(\frac{1.302 \mathrm{~g}}{2.000 \mathrm{~g}}\times 100\) Mass percentage of Aluminum chlorohydrate = \(65.1 \%\) (rounded to one decimal place) Thus, the mass percentage of aluminum chlorohydrate in the antiperspirant is approximately 65.1%.