Question

How many grams of \(\mathrm{HCl}\) gas are dissolved in \(7.50 \mathrm{~L}\) of an aqueous \(\mathrm{HCl}\) solution that has a \(\mathrm{pH}\) of \(2.40 ?\)

Short answer

The mass of HCl gas dissolved in the solution can be calculated by the following steps: 1. Convert the pH to the concentration of HCl ions: \([H^+] = 10^{-2.40}\) 2. Calculate the moles of HCl gas: \(n = 3.98 \times 10^{-3} \mathrm{M} \times 7.50 \mathrm{L}\) 3. Convert moles to grams: \(mass = (3.98 \times 10^{-3} \mathrm{M} \times 7.50 \mathrm{L}) \times 36.46 \, \frac{\mathrm{g}}{\mathrm{mol}}\) By performing these calculations, we find that there are approximately 1.09 grams of HCl gas dissolved in the 7.50 L aqueous HCl solution with a pH of 2.40.

Step-by-step solution

Convert the pH to the concentration of HCl ions

The pH of a solution is defined as the negative logarithm of the concentration of H+ ions in the solution. We can use this formula to find the concentration of HCl ions: \[pH = -\log_{10}[H^+]\] Given, pH = 2.40 Now, let's solve for the concentration of HCl ions: \[[H^+] = 10^{-pH} = 10^{-2.40}\]

Calculate the moles of HCl gas

Once we have the concentration of HCl ions in the solution, we can proceed to calculate the moles of HCl gas dissolved in the solution. From step 1, we have: \[[H^+] = 3.98 × 10^{-3} \mathrm{M}\] The solution volume is given as 7.50 L. Therefore, the moles (n) of HCl gas can be calculated as: \[n = [H^+] × Volume = 3.98 × 10^{-3} \mathrm{M} × 7.50 \mathrm{L}\]

Convert moles to grams

Finally, we can convert the moles of HCl gas into grams using its molar mass. The molar mass of HCl is approximately 36.46 g/mol. To find the grams of HCl gas, we can use the formula: \[mass = moles × molar\,mass\] Now calculate the mass of HCl gas dissolved in the solution: \[mass = (3.98 × 10^{-3} \mathrm{M} × 7.50 \mathrm{L}) × 36.46 \, \frac{\mathrm{g}}{\mathrm{mol}}\] After performing this calculation, we get the mass of dissolved HCl gas.

Key concepts

Molar Concentration

Understanding molar concentration, often referred to as molarity, is pivotal in various chemistry calculations, especially when working with solutions. Molarity is defined as the number of moles of solute present per liter of solution. It's indicated in moles per liter (\text{mol/L}) or molarity (M). To calculate molarity, you need to know the amount of substance in moles and the volume of the solution in liters. The formula for molarity is:
\[ M = \frac{n}{V} \]
where \text{M} is the molarity, \text{n} is the number of moles, and \text{V} is the volume of the solution in liters. To solve the textbook problem, we determined the molarity of HCl from the given pH value. When dealing with acid-base reactions especially, knowing how to work with molarity is crucial as it allows you to understand the concentration of ions in solution, which directly affects pH levels.

Molar Mass

The molar mass is a fundamental concept in chemistry that facilitates the conversion between moles and grams. It represents the mass of one mole of a substance and is expressed in grams per mole (g/mol). This property is unique to each element or compound and is calculated by summing up the atomic masses of all the atoms constituting a molecule. In our exercise, the molar mass of hydrochloric acid (HCl) is approximately 36.46 g/mol, as found on the periodic table: the atomic mass of hydrogen (H) is about 1.01 g/mol, and that of chlorine (Cl) is about 35.45 g/mol. With this information, we can convert moles of a substance to grams, making it a stepping stone in stoichiometry and practical chemistry applications.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in chemical reactions. It involves calculations based on the laws of conservation of mass and fixed proportions. In essence, stoichiometry deals with the calculation of quantities of substances consumed and produced in chemical reactions. An understanding of stoichiometry involves being familiar with concepts like molar ratios, the limiting reactant, and yield calculations. For our exercise, we used stoichiometry to calculate the grams of HCl gas by knowing the moles from the molar concentration and the molar mass. Stoichiometry comes into play after determining the required moles of HCl, allowing us to find the mass of HCl gas that was dissolved in the solution.

Acid-Base Chemistry

Acid-base chemistry is a central theme in chemical sciences that explains the properties and reactions of acids and bases. An important aspect of acid-base chemistry is the pH scale, which measures how acidic or basic a solution is. The pH scale ranges from 0 to 14, with 7 being neutral. Values lower than 7 indicate an acidic solution, while values above 7 indicate a basic one.
In the context of our exercise, to find the concentration of HCl in the solution, we first needed to convert the provided pH value into the concentration of hydronium ions (\text{H}^+). This is achieved using the following relationship:
\[ pH = -\log_{10}[H^+] \]
As pH decreases, the [H+] increases, indicating a more acidic solution. By solving the inverse calculation, we can determine the molar concentration of the acid in the solution, which is the foundation for subsequent stoichiometric calculations.