Problem 85 - Chemistry: The Central Science

Question

A sample of solid \(\mathrm{Ca}(\mathrm{OH})_{2}\) is stirred in water at \(30^{\circ} \mathrm{C}\) until the solution contains as much dissolved \(\mathrm{Ca}(\mathrm{OH})_{2}\) as it can hold. A \(100-\mathrm{mL}\) sample of this solution is withdrawn and titrated with \(5.00 \times 10^{-2} \mathrm{M} \mathrm{HBr}\). It requires \(48.8 \mathrm{~mL}\) of the acid solution for neutralization. What is the molarity of the \(\mathrm{Ca}(\mathrm{OH})_{2}\) solution? What is the solubility of \(\mathrm{Ca}(\mathrm{OH})_{2}\) in water, at \(30^{\circ} \mathrm{C}\), in grams of \(\mathrm{Ca}(\mathrm{OH})_{2}\) per \(100 \mathrm{~mL}\) of solution?

Step-by-Step Solution

Answer

The molarity of the saturated Ca(OH)₂ solution is \(1.22 \times 10^{-2}\) M, and its solubility in water at 30°C is 0.0906 g per 100 mL of solution.

Step 1: Write the balanced chemical equation for the titration reaction

The balanced chemical equation for the reaction between Calcium Hydroxide and Hydrobromic Acid is: \[Ca(OH)_2 + 2 HBr \rightarrow CaBr_2 + 2 H_2O\]

Step 2: Calculate the moles of HBr used in the titration

Given that 48.8 mL of 5.00 x 10⁻² M HBr is used for the titration, we first need to calculate the moles of HBr used: Moles of HBr = molarity × volume_of_HBr Moles of HBr = (5.00 x 10⁻² mol/L) × (48.8 mL × 0.001 L/mL) Moles of HBr = 2.44 x 10⁻³ mol

Step 3: Use stoichiometry to determine the moles of Ca(OH)₂ in the sample

From the balanced chemical equation in Step 1, we can see that the stoichiometric ratio between Ca(OH)₂ and HBr is 1:2. Therefore, we can determine the moles of Ca(OH)₂ in the 100 mL sample: Moles of Ca(OH)₂ = moles of HBr / 2 Moles of Ca(OH)₂ = 2.44 x 10⁻³ mol / 2 Moles of Ca(OH)₂ = 1.22 x 10⁻³ mol

Step 4: Find the molarity of Ca(OH)₂ solution

We can now use the moles of Ca(OH)₂ in the sample to find its molarity: Molarity of Ca(OH)₂ = moles of Ca(OH)₂ / volume of sample Molarity of Ca(OH)₂ = 1.22 x 10⁻³ mol / 0.100 L Molarity of Ca(OH)₂ = 1.22 x 10⁻² M

Step 5: Find the solubility of Ca(OH)₂ in water at 30°C

To find the solubility, we first need to calculate the mass of Ca(OH)₂ dissolved in the 100 mL sample. The molar mass of Ca(OH)₂ is 40.08 + 2(16.00 + 1.01) = 74.1 g/mol. Mass of Ca(OH)₂ = moles of Ca(OH)₂ × molar mass Mass of Ca(OH)₂ = (1.22 x 10⁻³ mol)×(74.1 g/mol) Mass of Ca(OH)₂ = 0.0906 g Therefore, the solubility of Ca(OH)₂ in water at 30°C is 0.0906 g per 100 mL of solution.

Key Concepts

Titration

Understanding titration is crucial for analyzing the concentration of an unknown solution. It is a process where a solution of known concentration, the titrant, is added to a specific volume of a solution with an unknown concentration, the analyte, until the chemical reaction between them is complete—typically indicated by a color change, referred to as the end point.

In our case, hydrobromic acid (HBr) was the titrant, and the calcium hydroxide (Ca(OH)₂) solution was the analyte. The titration helps to determine the molarity of the Ca(OH)₂ solution. By using the volume of the titrant and its molarity, you can find the number of moles of HBr used. This information is then applied to find the stoichiometrically equivalent amount of Ca(OH)₂ present in the solution.

Stoichiometry

Stoichiometry is a section of chemistry that involves calculations based on the relationships of reactants and products in balanced chemical reactions. For every chemical process, stoichiometry can be used to predict the amounts of products and reactants that are consumed or produced.

In our exercise, stoichiometry comes into play after establishing the moles of HBr. The balanced equation shows us the ratio of Ca(OH)₂ to HBr is 1:2, meaning one mole of calcium hydroxide reacts with two moles of hydrobromic acid. Using this ratio, we can calculate the moles of Ca(OH)₂, once we know the moles of HBr. These calculations are fundamental to understanding chemical reactions and are key to executing titrations correctly.

Molarity

Molarity is defined as the number of moles of a solute divided by the volume of the solution in liters. It's expressed in units of moles per liter (mol/L) and is used to quantify the concentration of a solution.

Molarity is integral to the problem we're examining because it allows us to describe the concentration of Ca(OH)₂ in quantitative terms. After finding the moles of Ca(OH)₂ through stoichiometry, we use the molarity formula to convert these moles into a molarity value, considering the volume of the sample in liters. This step provides us with a clear picture of how concentrated the analyte is in the solution.

Balanced Chemical Equation

A balanced chemical equation is crucial in chemistry; it ensures that the law of conservation of mass is respected. This means that the number of atoms of each element is conserved and the same on both the reactant and product sides of the equation.

The balanced equation provided: \[Ca(OH)_2 + 2 HBr \rightarrow CaBr_2 + 2 H_2O\] represents the reaction that occurs during the titration process. Each component of this reaction is vital for stoichiometry calculations. For instance, balancing the equation allows us to know that two moles of HBr are needed for every one mole of Ca(OH)₂, which was used to determine the molarity of the calcium hydroxide solution from the known molarity and volume of the hydrobromic acid.