Question

The active ingredients in a particular antacid tablet are aluminum hydroxide, $\mathrm{Al}(\mathrm{OH}{)}_3,$ and magnesium hydroxide, $\mathrm{Mg}(\mathrm{OH}{)}_2.\mathrm{A}5.00\times {10}^2\mathrm{mg}$ sample of the active ingredients was dissolved in $50.0\mathrm{mL}$ of $0.500\mathrm{M}\mathrm{HCl}.$ The resulting solution, which was still acidic, required $16.5\mathrm{mL}$ of $0.377\mathrm{M}\mathrm{NaOH}$ for neutralization. What are the mass percentages of $\mathrm{Al}(\mathrm{OH}{)}_3$ and $\mathrm{Mg}(\mathrm{OH}{)}_2$ in the sample?

Short answer

To solve this problem, the number of moles of H+ ions that were neutralized by the tablet needs to be calculated, followed by calculation of the number of moles of the active ingredients, and finally the mass percentages of the active ingredients. As it is not possible to solve $3x+2y=0.025-0.006205(x\ge 0,y\ge 0)$ without more information, the exact numerical answer cannot be given, but the procedure to find the masses and mass percentages is described.

Step-by-step solution

Calculate Hydrogen Ion Concentration

First, the total number of moles of H+ ions neutralized by both the active ingredients and NaOH needs to be calculated. This is done using the molarity and volume of HCl and NaOH. Molarity ($M$) is moles/liter, so moles = Molarity * Volume (in liters). For HCl, moles = $0.500M\ast 0.050L=0.025mol$. For NaOH, moles = $0.377M\ast 0.0165L=0.006205mol$.

Calculate Number of Moles of $Al(OH{)}_3$ and $Mg(OH{)}_2$

Next, using stoichiometry, calculate the moles of the active ingredients that reacted with the H+ ions. This requires understanding that each mole of $Al(OH{)}_3$ reacts with 3 moles of H+ ions and each mole of $Mg(OH{)}_2$ reacts with 2 moles of H+ ions. Let the moles of $Al(OH{)}_3$ be $x$ and $Mg(OH{)}_2$ be $y$, then from the reaction it can be written as $3x+2y=0.025-0.006205(x\ge 0,y\ge 0)$. The left side of the equation represents the total amount of H+ ions, which must be equal to the total number of H+ ions neutralized (that is, the moles from HCl minus the moles from NaOH). While $x$ and $y$ are real numbers because number of moles cannot be negative.

Calculate the Mass Percentages

Convert the moles of $Al(OH{)}_3$ and $Mg(OH{)}_2$ to mass using their molar masses (Al = 26.98 g/mol, O = 16.00 g/mol, H = 1.01 g/mol, Mg = 24.31 g/mol). The molar masses of $Al(OH{)}_3$ and $Mg(OH{)}_2$ are 78.00 g/mol and 58.32 g/mol respectively. The masses are $x\ast$ 78.00 g/mol for $Al(OH{)}_3$ and $y\ast$ 58.32 g/mol for $Mg(OH{)}_2$. Then, calculate the mass percentage by dividing the mass of each compound by the total mass of the tablet and multiplying by 100. The mass percentage of $Al(OH{)}_3$ and $Mg(OH{)}_2$ are $\frac{x\ast 78.00}{500}\ast 100\mathrm{\%}$ and $\frac{y\ast 58.32}{500}\ast 100\mathrm{\%}$ respectively.

Key concepts

Solution Concentration

Solution concentration refers to how much solute is present in a given amount of solution. It is often expressed in molarity (M), which is moles of solute per liter of solution. This is an essential concept in stoichiometry when dealing with chemical reactions in solutions, such as neutralization reactions.
To find the number of moles of hydrogen ions ( H^ + ) in this exercise, we used the formula:

• Moles of H^ + = Molarity (M) × Volume (L)

In our example, the molarity of HCl was 0.500 M, and the volume was 0.050 L, resulting in 0.025 moles of H^ + ions. Understanding concentration helps us determine how chemically active an ingredient is in a solution, impacting the effectiveness of reactions and neutralizations.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products through the breaking and forming of bonds. In this exercise, we explored neutralization reactions. Here's a breakdown of how they work:

• Al(OH)_3 and Mg(OH)_2 are bases that react with the acid HCl . Each mole of Al(OH)_3 can neutralize three moles of H+ ions, while each mole of Mg(OH)_2 can neutralize two moles of H+ ions.

By using the equation:
3x + 2y = 0.025 - 0.006205
We can solve for (x) and (y) , which represent the number of moles of Al(OH)_3 and Mg(OH)_2 , respectively, based on their neutralization capacity. This equation is a representation of stoichiometry, which allows us to predict how much of each reactant is needed to complete the reaction.

Mass Percentage

Mass percentage is a way to express concentration by comparing the mass of a component to the total mass of the mixture. In stoichiometry, it's helpful to determine the composition of mixtures, such as antacid tablets.
For our exercise:

• The molar masses: Al(OH)_3 is 78.00 g/mol, and Mg(OH)_2 is 58.32 g/mol.
• To find the mass percentage, covert moles of each compound to grams, then divide by the total sample mass.

The formula used was:
Mass Percentage = (mass of component/total mass) × 100%
This calculation helps see how much of each active ingredient is in the antacid, ensuring the correct formulation for effectiveness in neutralizing stomach acid.