How many milliliters of 18.0 \(\mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}\) are required to react with 250 \(\mathrm{mL}\) of 2.50 \(\mathrm{M}\) \(\mathrm{Al}(\mathrm{OH})_{3}\) if the products are aluminum sulfate and water?

Understanding the balanced chemical equation is crucial in solving many chemistry problems, as it represents the proportional relationship among the reactants and products during a chemical reaction. A balanced equation has an equal number of each type of atom on both sides of the equation, adhering to the law of conservation of mass.

For instance, in the reaction between aluminum hydroxide, \(Al(OH)_3\), and sulfuric acid, \(H_{2}SO_{4}\), to produce aluminum sulfate, \(Al_{2}(SO_{4})_{3}\), and water, \(H_2O\), the equation is balanced as follows:
\[2Al(OH)_3 + 3H_{2}SO_{4} \rightarrow Al_{2}(SO_{4})_{3} + 6H_{2}O\]
The coefficients 2, 3, and 6 before the chemical formulas indicate the number of moles of each substance that are involved in the reaction. Learning to balance equations enables students to accurately determine the stoichiometry of the reaction, which is the foundation for further calculations, like the molarity and volume of solutions needed for a reaction.

Molarity is a measure of the concentration of a solution, defining the number of moles of solute per liter of solution. The formula for molarity is:
\[Molarity (M) = \frac{moles}{Liters of solution}\]
This concept is particularly important when dealing with chemical reactions in solution, as it allows chemists to calculate the amount of reactants needed to achieve a certain reaction.

For the purpose of solving our problem, we used the given molarity of aluminum hydroxide to find the number of moles in a 250 mL solution by rearranging the molarity formula:
\[\frac{2.50 moles}{1 L} = \frac{moles}{0.250 L}\]
By cross-multiplying and solving, we find that there are approximately 0.625 moles of aluminum hydroxide present. Similarly, we can use the molarity of sulfuric acid to find the volume needed once we have the number of moles required for the reaction. Such calculations are fundamental in preparing solutions and carrying out reactions in the lab.

The mole-to-mole ratio is derived from the coefficients of a balanced chemical equation and is used to convert moles of one substance in the reaction to moles of another. The ratio indicates the relative amounts of reactants needed to produce the desired products.

In the given exercise, the balanced equation shows that 2 moles of aluminum hydroxide react with 3 moles of sulfuric acid. This provides us with a mole-to-mole ratio of \(Al(OH)_3: H_{2}SO_{4}\) as 2:3. Therefore, to calculate how much sulfuric acid is required to react with a given amount of aluminum hydroxide, we use this ratio:
\[0.625 moles \times \frac{3 moles H_{2}SO_4}{2 moles Al(OH)_{3}} = 0.9375 moles H_{2}SO_{4}\]
This step is essential because it directly relates the moles of one substance to another, which allows us to then determine the volume of the sulfuric acid needed, given its molarity. Understanding mole-to-mole ratios is indispensable for stoichiometry as it ensures the right proportions of reactants are used to achieve a complete reaction without any excess of one reactant.