A \(500.0-\mathrm{mL}\) sample of \(0.200 M\) sodium phosphate is mixed with \(400.0 \mathrm{~mL}\) of \(0.289 M\) barium chloride. What is the mass of the solid produced?

Understanding the concept of a limiting reactant is crucial in stoichiometry, which is the area of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. The limiting reactant is the substance that is entirely consumed first in a chemical reaction and thus determines the maximum amount of product that can be formed. Calculating the limiting reactant involves a few essential steps.

First, we need to begin with a balanced chemical equation to know the stoichiometric ratios of reactants that react to form the products. Then, using the provided concentrations and volumes, we calculate the moles of each reactant to determine which one will run out first. We compare the mole ratio of the reactants to the stoichiometric ratio from the balanced equation to identify the reactant in shortage. This approach is used because reactions proceed based on defined mole ratios as dictated by the balanced equation.

In our example, by performing a mole ratio comparison, we've established that sodium phosphate (Na₃PO₄) is the limiting reactant. This information is pivotal because it allows us to predict how much product will be made. Without identifying the limiting reactant, we cannot accurately proceed to calculate the mass of the product formed in the reaction.

Once the limiting reactant is known, mole-to-mass conversion becomes the next step to find the mass of the product. This step is crucial since laboratory measurements are often made in grams rather than moles. To convert moles to mass, we use the molar mass of the substance, which is the mass of one mole of a compound or element and is expressed in grams per mole (g/mol).

The molar mass can be calculated by summing the atomic masses of all the atoms in a single molecule of the substance. Atomic masses can be found on the periodic table and are usually averaged to account for isotopic abundances. In the case of barium phosphate (Ba₃(PO₄)₂), the molar mass is determined by adding together the molar masses of barium (Ba), phosphorus (P), and oxygen (O) atoms in the compound, as shown in the exercise.

With the molar mass at hand, we perform the conversion from moles to grams by multiplying the number of moles of the compound by its molar mass. This calculation yields the mass of the product that can be produced from the limiting reactant; in our example, it is approximately 30.1 grams of barium phosphate.

The foundation of stoichiometry lies in the balanced chemical equation. This is a representation of a chemical reaction in which the number of atoms for each element is the same on both the reactant and product sides of the equation. This balance is necessary because matter cannot be created or destroyed per the law of conservation of mass.

Writing a balanced equation involves identifying all the reactants and products and then adjusting their coefficients to balance the atoms. The coefficients indicate the proportion of molecules (or moles) of each substance involved. In our problem, the balanced equation is:
2 Na₃PO₄(aq) + 3 BaCl₂(aq) → Ba₃(PO₄)₂(s) + 6 NaCl(aq)

This equation tells us that two moles of sodium phosphate react with three moles of barium chloride to yield one mole of barium phosphate and six moles of sodium chloride.

It is vital to start with a correctly balanced equation because all subsequent stoichiometric calculations hinge on the mole ratios derived from it. An unbalanced equation would lead to inaccurate results and a misunderstanding of the reaction's chemistry. Therefore, achieving a balanced chemical equation is the first and foremost step in solving any stoichiometry problem.