Problem 117 - Chemistry: The Central Science in SI Units, Global Edition

Question

How many milliliters of concentrated hydrochloric acid solution \((36.0 \%\) HCl by mass, density \(=1.18 \mathrm{~g} / \mathrm{mL}\) ) are required to produce \(10.0 \mathrm{~L}\) of a solution that has a pH of \(2.05 ?\)

Step-by-Step Solution

Answer

Approximately, 210 mL of the concentrated hydrochloric acid solution (36.0% HCl by mass, density = 1.18 g/mL) is required to produce 10.0 L of the solution with a pH of 2.05.

Step 1: Convert the pH to hydrogen ion concentration

We need to find the hydrogen ion concentration (\( [H^+] \)) of the solution with a pH of 2.05. We can use the formula: pH = - log\(_{10}\)(\[H^+\]) \[H^+\] = 10\(^{-\text{pH}}\) So, substitute the pH value into the formula to find \([H^+]\): \[H^+\] = 10\(^{-2.05}\) \[H^+\] = 8.91 × 10\(^{-3}\) mol/L

Step 2: Calculate the moles of HCl needed

Now, we have to find the moles of HCl needed to produce 10.0 L of the desired solution. We can use the formula: moles = volume × concentration With the given values: moles = 8.91 × 10\(^{-3}\) mol/L × 10.0 L moles = 0.0891 mol

Step 3: Determine the mass of HCl in the concentrated solution

We have been given that the concentrated HCl solution is 36.0% by mass and has a density of 1.18 g/mL. Therefore, we can calculate the mass of HCl per mL of concentrated solution: mass of HCl per mL = density × percentage by mass mass of HCl per mL = (1.18 g/mL) × 0.36 mass of HCl per mL = 0.4248 g/mL

Step 4: Calculate the volume of concentrated HCl solution

Finally, we can now calculate the volume of the concentrated HCl solution required to produce 10.0 L of the desired solution. We will use the formula: volume of concentrated HCl = (moles of HCl needed) / (mass of HCl per mL) Substitute the values: volume of concentrated HCl = (0.0891 mol) / (0.4248 g/mL) volume of concentrated HCl ≈ 0.210 L or 210 mL Approximately, 210 mL of the concentrated hydrochloric acid solution is required to produce 10.0 L of the solution with a pH of 2.05.

Key Concepts

Hydrogen Ion Concentration

In chemistry, understanding the concentration of hydrogen ions \( [H^+] \) in a solution is essential for determining its acidity. The pH scale is a convenient way to express this concentration, with lower pH values indicating more acidic solutions. The relation between pH and \( [H^+] \) is given by the formula: \( \text{pH} = -\log_{10}([H^+]) \).
To find the hydrogen ion concentration for a solution with a known pH, you rearrange the formula as follows: \[ H^+ = 10^{-\text{pH}}. \]
For the exercise, a solution with a pH of 2.05 has a hydrogen ion concentration of \( 8.91 \times 10^{-3} \text{ mol/L} \). This calculation tells us how many moles of hydrogen ions are present per liter of solution. Understanding this concept is fundamental when working with acids and bases, as it quantifies the solution's level of acidity. This kind of calculation is crucial in various applications in laboratories and industries where precise pH levels are required for specific processes.

Moles Calculation

Moles are a way to express the amount of a chemical substance. To solve this problem, we need to determine how many moles of hydrochloric acid \( \text{HCl} \) are necessary to achieve the desired hydrogen ion concentration in a large volume of solution. The formula used is:

• Moles = Volume \( \times \) Concentration

Given the hydrogen ion concentration \( [H^+] \) of \( 8.91 \times 10^{-3} \text{ mol/L} \) and a solution volume of 10 liters, you multiply these two values to find the moles of acid needed. Therefore, \( \text{Moles} = 10 \times 8.91 \times 10^{-3} \), resulting in 0.0891 moles of \( \text{HCl} \).
This calculation is vital because it determines how much of our concentrated hydrochloric acid will be used to prepare the requested solution while maintaining the correct pH level. Moles calculations are fundamental in chemistry since they provide a bridge between the macroscopic quantities we measure and the molecular-scale events we want to understand.

Solution Preparation

Solution preparation involves several steps, each requiring accurate measurements to achieve the desired solution characteristics. In this exercise, although we knew the \( \text{moles} \) of hydrochloric acid needed, we also had to consider the solution's concentration and density. The concentrated \( \text{HCl} \) solution is 36% hydrochloric acid by mass, and its density is 1.18 g/mL.
To find the amount of acid in grams per milliliter of concentrated solution, use:

• Density \( \times \) Percentage by Mass

The outcome is 0.4248 g/mL. Next, we need the volume of this concentrated solution that holds 0.0891 moles of \( \text{HCl} \). Calculating the volume required is straightforward: divide the moles by the mass per mL:

• Volume = Moles / Mass per mL

So, \( \text{Volume} = 0.0891 \text{ mol} / 0.4248 \text{ g/mL} \), which approximately equals 0.210 liters or 210 milliliters. This calculation ensures that we have the right volume of concentrated acid to prepare a 10-liter solution of the desired pH. Accurately preparing solutions is a common task in laboratory settings, enabling scientists to create conditions necessary for experiments and various applications.