Problem 82 - Chemistry The Science in Context

Question

Given the following equation, how many grams of \(\mathrm{PbCO}_{3}\) will dissolve when \(1.00 \mathrm{L}\) of \(1.00 \mathrm{M} \mathrm{H}^{+}\) is added to \(5.00 \mathrm{g}\) of $\mathrm{PbCO}_{3} ?$$$ \mathrm{PbCO}_{3}(s)+2 \mathrm{H}^{+}(a q) \rightarrow \mathrm{Pb}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(\ell)+\mathrm{CO}_{2}(g)$$

Step-by-Step Solution

Answer

5.00 grams of PbCO3 will dissolve.

Step 1: Calculate initial moles of H+ and PbCO3

To find the initial moles of H+ ions and PbCO3, we can use the equation: moles = (volume x concentration) for H+ ions and (mass / molar mass) for PbCO3. For H+ ions: moles= volume x concentration moles= 1.00 L x 1.00 M moles= 1.00 mol For PbCO3: molar mass = 207.2(Pb) + 12.0(C) + 3 x 16.0(O) = 267.2 g/mol moles= mass / molar mass moles= 5.00 g / 267.2 g/mol moles= 0.0187 mol

Step 2: Determine the limiting reagent

We will now use stoichiometry to determine the limiting reagent. For the given reaction, $$ \mathrm{PbCO}_{3} + 2\mathrm{H}^{+} \rightarrow \mathrm{Pb}^{2+} +\mathrm{H}_{2}\mathrm{O} + \mathrm{CO}_{2} $$ 1 mole of PbCO3 reacts with 2 moles of H+ ions. From the previous step, we have 0.0187 mol of PbCO3 and 1.00 mol of H+ ions. To determine the limiting reagent, we check if the quantity of H+ ions is enough to react with all the PbCO3 moles. H+ ions required = 0.0187 mol x (2 mol H+ / 1 mol PbCO3) = 0.0374 mol Since there is enough H+ ions (1.00 mol > 0.0374 mol), PbCO3 is the limiting reagent.

Step 3: Calculate the remaining moles of PbCO3 and convert to grams

All the PbCO3 will react with some of the H+ ions, and the reaction will stop when PbCO3 is spent. Since all the PbCO3 reacts, the remaining moles of PbCO3 is 0. Now, we will find grams of dissolved PbCO3 which is basically the whole 5.00 g. Total dissolved PbCO3 = Initial mass - Remaining mass Dissolved PbCO3 = 5.00 g - 0 g Dissolved PbCO3 = 5.00 g So, 5.00 grams of PbCO3 will dissolve when 1.00 L of 1.00 M H+ ions is added to 5.00 g of PbCO3.

Key Concepts

Limiting Reagent

Understanding the concept of the limiting reagent is crucial in the field of chemistry, especially when it comes to performing reactions. It refers to the reactant that will be completely used up first in a chemical reaction, thus determining the amount of product that will be formed. When you have unequal quantities of reactants, it's not always obvious which one will limit the reaction. To find out, you compare the mole ratio of the reactants to the ratio given by the balanced chemical equation.

Let's say you're baking cookies, and the recipe requires one egg for every cup of flour. If you have ten eggs but only five cups of flour, your flour is the limiting reagent because you can't make more than five batches of cookies, regardless of how many eggs you have. Similarly, in the exercise, we determined that the limiting reagent was PbCO₃ because despite having an excess of H⁺ ions, the reaction couldn't proceed once all the PbCO₃ was consumed.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. One mole is defined as the amount of substance that contains as many entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of pure carbon-12. This number is known as Avogadro's number, approximately equal to 6.022 x 10²³ entities per mole.

In stoichiometry problems, we use the mole concept to relate the mass of substances to the number of particles and the volume of gases at standard temperature and pressure. For instance, in the given exercise, we calculate the moles of PbCO₃ by dividing its mass by its molar mass. This molar mass is based on the atomic weights of the constituent atoms, allowing us to use mass (a macroscopic property) to find the number of moles (and thus the number of molecules) in an easily measurable way.

Chemical Reactions

A chemical reaction involves the transformation of one or more substances into different substances. To predict the outcome of a chemical reaction, we use a balanced chemical equation. This equation follows the law of conservation of mass, indicating that atoms are neither created nor destroyed in a chemical reaction.

The equation provided in the exercise, PbCO₃ + 2H⁺ → Pb²⁺ + H₂O + CO₂, shows us the stoichiometry or the exact proportions of reactants that will react with each other. In real-world applications, knowledge of chemical reactions allows us to predict product formation, reactant usage, as well as energy changes and other aspects of reactions. This is essential in industries such as pharmaceuticals, agriculture, and manufacturing.

Solubility

Solubility is the property of a substance to dissolve in a solvent, forming a homogeneous mixture at a specified temperature and pressure. It's often expressed in terms of the maximum amount of solute that can dissolve in a given amount of solvent. However, solubility is not just binary – it's not a matter of soluble or insoluble. Instead, it's a spectrum, with some compounds having very high solubility in a particular solvent, while others have very low solubility.

In the context of the exercise, when we add H⁺ ions to PbCO₃, a reaction occurs that effectively dissolves the PbCO₃, transitioning it from the solid phase to an aqueous phase, where Pb²⁺ ions and CO₂ gas are produced. Understanding solubility is important for a myriad of processes, including the purification of compounds, the synthesis of medicines, and even the biological availability of minerals in the body.