\(30 \mathrm{~mL}\) of \(0.1 \mathrm{M} \mathrm{BaCl}_{2}\) is mixed with \(40 \mathrm{~mL}\) of \(0.2 \mathrm{M} \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\). What is the weight of \(\mathrm{BaSO}_{4}\) formed? $$ \mathrm{BaCl}_{2}+\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \longrightarrow \mathrm{BaSO}_{4}+\mathrm{AlCl}_{3} $$

Chemical stoichiometry is the area of chemistry that pertains to the calculation of reactants and products in chemical reactions. It relies on the principle of conservation of mass where the mass of the reactants must equal the mass of the products in a chemical equation. Essential to stoichiometry is the balanced chemical equation, which shows the stoichiometric coefficients that indicate the ratios in which chemicals react and are produced.

For instance, in the reaction \(\mathrm{BaCl}_2 + \mathrm{Al}_2(\mathrm{SO}_4)_3 \rightarrow \mathrm{BaSO}_4 + \mathrm{AlCl}_3\), the coefficients before the chemical formulas imply a 1:1:1:2 ratio respectively. Thus, for every mole of \(\mathrm{BaCl}_2\), a mole of \(\mathrm{Al}_2(\mathrm{SO}_4)_3\), \(\mathrm{BaSO}_4\), and two moles of \(\mathrm{AlCl}_3\) are involved. When one of the reagents is used up completely, the reaction stops, and this reagent is known as the limiting reactant. The stoichiometry helps us identify this and calculate the amount of each substance needed or produced.

The mole concept is a fundamental principle in chemistry that aids in the quantification of substances. One mole represents Avogadro's number, approximately \(6.022 \times 10^{23}\), of particles (which could be atoms, molecules, ions, etc.).

This concept allows chemists to measure substances in the macroscopic world when considered in terms of moles, facilitating the use of balanced chemical equations to determine how much reactants will yield a certain amount of product. To illustrate, in the given exercise, moles of \(\mathrm{BaCl}_2\) were calculated by multiplying volume by molarity resulting in 3 mmol (or \(3 \times 10^{-3}\) moles), and similarly, for \(\mathrm{Al}_2(\mathrm{SO}_4)_3\), we found 8 mmol. This quantification is vital for predicting the course of a reaction and is intrinsically linked to stoichiometry; both are key when it comes to identifying the limiting reactant in a chemical reaction.

Precipitation reactions occur when two soluble ionic compounds form an insoluble product, known as a precipitate, when mixed in solution. It is important to understand the solubility rules to predict which combinations of ions will result in a precipitate. In our exercise, when \(\mathrm{BaCl}_2\) and \(\mathrm{Al}_2(\mathrm{SO}_4)_3\) are mixed, they undergo a double replacement reaction, forming \(\mathrm{BaSO}_4\) as a precipitate.

The process of this reaction involves the exchange of ions between the two reactants to form new products, one of which does not dissolve in the solution and falls out as a solid. This is a classic example of a precipitation reaction where \(\mathrm{BaSO}_4\) is the solid formed. This type of reaction is often used in qualitative chemical analysis, water treatment, and extracting certain metals from their ores. The ability to calculate the weight of the precipitate formed, as shown in the exercise, is a practical application of understanding precipitation reactions in conjunction with stoichiometry and the mole concept.