Question

A sample of \(0.96 \mathrm{~L}\) of \(\mathrm{HCl}\) at \(372 \mathrm{mmHg}\) and \(22^{\circ} \mathrm{C}\) is bubbled into \(0.034 \mathrm{~L}\) of \(0.57 \mathrm{MH}_{3}\). What is the \(\mathrm{pH}\) of the resulting solution? Assume the volume of solution remains constant and that the \(\mathrm{HCl}\) is totally dissolved in the solution.

Short answer

The pH of the resulting solution is 14.68.

Step-by-step solution

Determine the Moles of HCL

First, we have to calculate the number of moles of the HCl gas using the ideal gas law equation which is: \( PV = nRT \). Where P is pressure (in atm) so we convert \(372 \, mmHg \, to \, 0.49 \, atm\), V is volume(in Liters), n is the number of moles of gas. R is gas constant and its value is \(0.0821 \, L \cdot atm /mol \cdot K\), T is temperature in Kelvin (which we can get by adding 273 to the Celsius temperature). Thus, the number of moles of HCl = \( \frac{(0.49 \cdot 0.96)}{(0.0821 \cdot (22+273))} \)= 0.020 mol.

Determine the Moles of \( NH_{3} \)

Next, we have to get the number of moles of \( NH_{3} \) using the formula \(n = MV\), where M is molarity (in \(\frac{mol}{L}\)) and V is volume (in Liters). So, the moles of \( NH_{3} \)= \(0.57 \cdot 0.034\) = 0.019 mol.

Identify the limiting reactant and calculate \( [NH_{4}^{+}] \)

Based on the stoichiometry of the reaction \( NH_{3} + HCl \rightarrow NH_{4}^{+} + Cl^{-} \), both Hcl and \( NH_{3} \) react in a 1:1 ratio. Therefore, 0.019 moles of \( NH_{3} \) will react with 0.019 moles of HCL to form 0.019 moles of \( NH_{4}^{+} \) ion. HCL is the limiting reactant in this case.

Compute pOH

Since the solution is basic, we calculate the pOH using the formula \( pOH = -log[OH^{-}] \), by using \( pOH = -log \frac{[NH_{4}^{+}]}{V_{total}} \). The volume remains constant as mentioned in the problem. Therefore, \( pOH = -log \frac{0.019}{0.034+0.96} = -0.6759 \).

Calculate pH

To obtain the pH of the solution, we use the relation \( pH = 14 - pOH = 14 - (-0.6759) = 14.68 \).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry. It allows us to calculate the number of moles of a gas when we know pressure, volume, and temperature. The formula is given by:\[ PV = nRT \]where:

• P is the pressure in atmospheres (atm).
• V is the volume in liters (L).
• n is the number of moles.
• R is the ideal gas constant, 0.0821 L·atm/mol·K.
• T is the temperature in Kelvin (K).

To solve problems using the ideal gas law, it's crucial to ensure all units are consistent. For instance, if pressure is given in mmHg, it should be converted to atm by dividing by 760. Similarly, temperature should be converted to Kelvin by adding 273 to the Celsius temperature. This method is particularly valuable when working with gases and helps in scenarios like determining the amount of HCl gas in the original problem.

Limiting Reactant

In a chemical reaction, the limiting reactant is the substance that is completely consumed first, limiting the extent of the reaction and determining the amount of product formed. To identify the limiting reactant, compare the molar amounts of each reactant based on the balanced chemical equation. In our exercise, the reaction can be simplified as follows:\[ NH_3 + HCl \rightarrow NH_4^+ + Cl^- \]This implies that one mole of HCl reacts with one mole of NH_3. We have computed the moles of HCl and NH_3. In this reaction:

• HCl: 0.020 mol
• NH_3: 0.019 mol

Since NH_3 has a slightly smaller amount of moles, HCl becomes the excess, establishing NH_3 as reacting completely. Hence, the amount of HCl dictates how much NH_3 can be converted, marking it as the limiting reactant in this chemical process. Understanding limiting reactants is essential in predicting the reaction yield.

Moles Calculation

Moles calculation is a key method in chemistry used to quantify substances during reactions. The mole is a basic unit in chemistry that refers to a specific number of particles, like atoms or molecules. To calculate moles, you use various equations based on available data. For gases, like our HCl, the ideal gas law is crucial. For solutions, the formula relating moles to molarity (M) and volume (V) is used:\[ n = MV \]In the exercise, we applied this for ammonia (\( NH_3 \)) using its molarity and volume to find:

• Moles of NH_3: 0.019 mol (calculated using 0.57 mol/L and 0.034 L)

Moles provide a bridge between the observable world and molecular chemistry, enabling us to count atoms in a manageable way by using molar mass, volume, or concentration of solutions. Comprehending moles is essential for solving various chemical calculation problems, understanding reaction proportions, and predicting the amount of reactants necessary.

Acid-Base Reaction

Acid-base reactions involve the transfer of protons (\(H^+\)) and are vital in creating solutions with diverse pH levels. The exchange typically occurs between an acid, a substance capable of donating protons, and a base, which absorbs them. In our scenario, the reaction can be depicted as:\[ NH_3 + HCl \rightarrow NH_4^+ + Cl^- \]HCl provides protons (\(H^+\)) that are accepted by NH_3. This forms ammonium ions (\(NH_4^+\)), and results in the salt (\(Cl^-\)) in the solution. The balance of \(NH_4^+\) and \(OH^-\) ions in the solution defines its pH. Acid-base reactions are the foundation for common processes like pH adjustments, involving either neutralizing a solution or creating specific acidity or alkalinity levels.The knowledge of acid-base chemistry is not only academic but extends to real-world applications such as food chemistry, medicine, and environmental science applications where monitoring acidity and basicity is vital.