Question

What volume of \(5 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) must be added to \(25 \mathrm{~mL}\) of \(1 \mathrm{M} \mathrm{BaCl}_{2}\) to produce \(10 \mathrm{~g}\) of \(\mathrm{BaSO}_{4} ?\) (a) \(8.58 \mathrm{~mL}\) (b) \(7.2 \mathrm{~mL}\) (c) \(10 \mathrm{~mL}\) (d) \(12 \mathrm{~mL}\)

Short answer

7.2 mL

Step-by-step solution

Calculate moles of BaSO4

To determine the number of moles of BaSO4 that can be produced from 10 g, use the molar mass. The molar mass of BaSO4 is 233.39 g/mol. Number of moles = mass (g) / molar mass (g/mol)Number of moles of BaSO4 = 10 g / 233.39 g/mol

Calculate moles of Na2SO4 required

BaCl2 reacts with Na2SO4 in a 1:1 molar ratio to form BaSO4, according to the equation BaCl2 + Na2SO4 → BaSO4 + 2 NaCl. The moles of Na2SO4 required are equal to the moles of BaSO4 needed.

Calculate volume of 5M Na2SO4

To find the volume of 5M Na2SO4 required to get the calculated moles, use the equation: Molarity = moles of solute / volume of solution (L)Volume of solution (L) = moles of solute / MolarityMake sure to convert this volume from liters to milliliters by multiplying by 1000.

Key concepts

Molarity

When we talk about molarity, we're referring to the concentration of a solution, which tells us how much of a substance is dissolved in a certain volume of solvent. Molarity is defined as the number of moles of solute (the substance being dissolved) per liter of solution. The equation for calculating molarity (M) is:
\[\begin{equation}M = \frac{\text{moles of solute}}{\text{liters of solution}}\end{equation}\]
In stoichiometry problems, molarity can be used to determine how much reactant is needed or how much product will be formed. For instance, in the textbook exercise, knowing the molarity of a sodium sulfate solution helps calculate the exact volume required to produce a certain mass of barium sulfate precipitate.

Molar Mass

Molar mass is a critical concept in stoichiometry, defined as the mass of one mole of a substance, usually in grams per mole (g/mol). Each element has a unique molar mass, which can be found on the periodic table, and the molar mass of a compound is the sum of the molar masses of its constituent elements.
For the practical application of molar mass, consider the original problem: to find out how many moles of barium sulfate are in 10 grams, we divide the mass of barium sulfate by its molar mass (233.39 g/mol). This allows us to transition from grams, which is a lab-measurable quantity, to moles, which connects directly to chemical reaction equations.

Chemical Reaction Equations

Chemical reaction equations are not just symbolic representations of chemical reactions; they are the blueprints that allow chemists to quantify the reactants and products involved in reactions. These equations are balanced to obey the Law of Conservation of Mass and often include states of matter (solid, liquid, gas, aqueous). Importantly, the coefficients in a balanced equation denote the molar ratios of the substances involved.
For example, the equation
\[\begin{equation}\text{BaCl}_2 (aq) + \text{Na}_2\text{SO}_4 (aq) \rightarrow \text{BaSO}_4(s) + 2 \text{NaCl} (aq)\end{equation}\]
illustrates a reaction with a 1:1 molar ratio between BaCl2 and Na2SO4. This molar ratio is crucial to solving stoichiometry problems because it allows the calculation of how much of one reactant is needed to react completely with a given amount of the other.

Limiting Reactant

The concept of the limiting reactant is vital in stoichiometry, as it determines the maximum amount of product that can be formed in a chemical reaction. The limiting reactant is the substance that is completely consumed first, thus stopping the reaction and leaving the other reactants in excess. To identify which reactant is limiting, one must compare the molar amount of each reactant with the ratios provided in the balanced chemical equation.
In the problem we are examining, the ratio of BaCl2 to Na2SO4 is 1:1, meaning they react equally in molar terms to produce BaSO4. Here, knowing the molar amount of BaSO4 allows us to work backward to determine how much Na2SO4 is required. If the quantities or molarity of BaCl2 were different, we would have to re-evaluate to identify the limiting reactant to ensure our calculations for the desired product, BaSO4, are accurate.