Question

How many milliliters of concentrated hydrochloric acid solution \((36.0 \%) \mathrm{HCl}\) 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 \(2.05 ?\)

Short answer

To produce 10.0 L of a solution with a pH of 2.05, we first find the concentration of H+ ions: \([H^{+}] = 10^{-2.05}\). Next, we calculate the moles of H+ ions: \(moles\: of\: H^{+} = [H^{+}] × 10.0\, L\). Then, determine the mass of HCl: \(mass\: of\: HCl = moles\: of\: H^{+} × 36.5\,g/mol\). After that, find the mass of the concentrated solution: \(mass\: of\: concentrated\: solution = mass\: of\: HCl / 0.36\). Lastly, calculate the volume of the concentrated HCl solution: \(volume\: of\: concentrated\: solution = mass\: of\: concentrated\: solution / 1.18\,g/mL\).

Step-by-step solution

Calculate the concentration of H+ ions from the pH value

The pH is defined as follows: \[pH = -\log_{10} [H^{+}]\] Where [H+] is the concentration of H+ ions in moles per liter (mol/L). To get the concentration of H+ ions in the new solution, we need to reverse this relationship: \[[H^{+}] = 10^{-pH}\] Using the given pH value of 2.05, we can find the concentration of H+ ions in the new solution: \[[H^{+}] = 10^{-2.05}\] Calculate the result to get the concentration of H+ ions in the new solution.

Calculate the moles of H+ ions in the new solution

Now that we have the concentration of H+ ions in the new solution, we can multiply this value by the volume of the new solution to find the moles of H+ ions required: \[moles\: of\: H^{+} = [H^{+}] × Volume\: of\: new\: solution\] Using the given volume of the new solution (10.0 L), calculate the moles of H+ ions in the new solution.

Calculate the mass of HCl in the new solution

With the moles of H+ ions required, we can now calculate the mass of HCl in the new solution using the molar mass of HCl (36.5 g/mol): \[mass\: of\: HCl = moles\: of\: H^{+} × molar\: mass\: of\: HCl\] Calculate the mass of HCl required for the new solution.

Calculate the mass of the concentrated HCl solution

We know that the concentrated HCl solution is 36.0% by mass. Therefore, we can find the mass of the concentrated solution that contains the required mass of HCl: \[mass\: of\: concentrated\: solution = \frac{mass\: of\: HCl}{percentage\: of\: HCl}\] Calculate the mass of the concentrated HCl solution required to prepare 10.0 L of the new solution with a pH of 2.05.

Calculate the volume of the concentrated HCl solution

Finally, we can use the density of the concentrated HCl solution (1.18 g/mL) to find the volume required: \[volume\: of\: concentrated\: solution = \frac{mass\: of\: concentrated\: solution}{density\: of\: concentrated\: solution}\] Calculate the volume of the concentrated HCl solution required to produce 10.0 L of a solution with a pH of 2.05.

Key concepts

pH Calculation

Understanding the pH calculation is crucial in many areas of chemistry, especially when dealing with acid-base reactions. The pH scale is a measure of the acidity or basicity of an aqueous solution. It is mathematically defined as the negative logarithm of the hydrogen ion concentration:
\[pH = -\log_{10} [H^{+}]\]
In our case, the pH is given as 2.05, reflecting a relatively acidic solution. By reversing the formula, \[ [H^{+}] = 10^{-pH} \]
we can calculate the hydrogen ion concentration required for the solution. Knowing the pH and understanding how to calculate the corresponding hydrogen ion concentration is the introductory step before we can prepare a solution with specific acidic properties.

Molarity and Molar Mass

The concepts of molarity and molar mass are fundamental in solution chemistry. Molarity (M) is a measure of the concentration of a solute in a solution, and is defined as the number of moles of solute per liter of solution.
\[Molarity = \frac{moles\: of\: solute}{volume\: of\: solution\: in\: liters}\]
Molar mass, on the other hand, is the mass of one mole of a substance, often measured in grams per mole (g/mol). For hydrochloric acid (HCl), the molar mass is 36.5 g/mol. In our exercise, we use the molar mass to convert between the mass of HCl needed and the moles required to achieve the desired pH level.

Solution Dilution

Solution dilution involves reducing the concentration of a solute in solution, usually by adding more solvent. It’s a critical process in preparing solutions of desired molarity. The key principle to remember is that the number of moles of solute remains constant before and after the dilution. This can be expressed as:
\[C_1V_1 = C_2V_2\]
where \(C_1\) and \(C_2\) are the initial and final concentrations, and \(V_1\) and \(V_2\) are the initial and final volumes, respectively. In the case of preparing a dilute hydrochloric acid solution, you would start with a known concentration of concentrated stock solution and calculate the volume required to achieve the new, lower concentration for a given final volume.

Acid-Base Chemistry

Acid-base chemistry deals with the properties and reactions of acids and bases. An acid is a substance that can donate a hydrogen ion (H+), while a base can accept a hydrogen ion. In aqueous solutions, acidic properties are determined by the concentration of hydrogen ions. Hydrochloric acid (HCl) is a strong acid, meaning it completely dissociates into its ions in water, which results in an increase of hydrogen ion concentration in the solution. The relationship between the acid's concentration and the pH of the solution is inverse and logarithmic, as discussed earlier. When preparing an acid solution, we are essentially manipulating this relationship to achieve a desired pH, underscoring the importance of a good grasp of acid-base chemistry principles.