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 \(0.57 M \mathrm{NH}_{3}\). 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.

The ideal gas law is a fundamental equation in chemistry and physics that describes the behavior of gases. It is represented as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \((T\) is the temperature in Kelvins.

When solving the given exercise, we can use the ideal gas law to calculate the amount of \(\mathrm{HCl}\) in moles, since it starts as a gas before being dissolved in a solution. To do this, we plug in the given values for pressure, volume, and temperature, and use the specific value of \(R\) suited for the pressure unit mmHg. Always make sure to convert temperatures to Kelvin by adding 273.15 to the Celsius temperature and ensure all units are consistent with the value of \(R\) being used.

Stoichiometry involves the calculation of reactants and products in chemical reactions. It's fundamental for predicting how much reactants are needed to produce a certain amount of product and vice versa.

In the exercise, stoichiometry helps us understand the relationship between \(\mathrm{NH}_3\) and \(\mathrm{HCl}\) by providing the balanced chemical equation. This gives us a 1:1 mole ratio, meaning one mole of \(\mathrm{NH}_3\) reacts with one mole of \(\mathrm{HCl}\) to produce \(\mathrm{NH}_4^+\) and \(\mathrm{Cl}^-\). For the calculated moles of \(\mathrm{NH}_3\) and \(\mathrm{HCl}\), this stoichiometric ratio is crucial to identify the limiting reactant and to effectively move to the next steps in finding the consequent pH of the solution.

The limiting reactant in a chemical reaction is the substance that is completely consumed first, thereby determining the maximum amount of product that can be formed. When working through an exercise like ours, identifying the limiting reactant is an essential step.

To find the limiting reactant between \(\mathrm{NH}_3\) and \(\mathrm{HCl}\), we compare the moles of each reactant. The one with the fewer moles dictates the extent of the reaction because it will run out first, preventing any further reaction from occurring. In this case, after applying stoichiometry, it's a matter of simple arithmetic to figure out which reactant is the limiting one. Understanding which one is the limiting reactant helps us to calculate the concentrations of ions after the reaction is complete.

The Henderson-Hasselbalch equation provides a quantitative relationship between the pH, pKa, and the ratio of the concentrations of the protonated and unprotonated forms of a conjugate acid-base pair. It is expressed as \[\text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\], where \(\text{pKa}\) is the acid dissociation constant, \(\text{A}^-\) is the concentration of the base form, and \(\text{HA}\) is the concentration of the acid form.

For our exercise, after identifying the limiting reactant and calculating the resulting concentrations of \(\mathrm{NH}_3\) (the base) and \(\mathrm{NH}_4^+\) (the acid), this equation allows us to find the pH of the ammonia solution after the reaction with hydrochloric acid. This is done by inserting the known pKa value of ammonium (9.25) and the ratio of \(\mathrm{NH}_4^+\) to \(\mathrm{NH}_3\) into the equation. Providing a connection between a solution's chemistry and its pH is vital as it practically applies one of the most important properties in acid-base chemistry.