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The Henderson-Hasselbalch equation is a critical tool in chemistry for determining the pH of buffer solutions. It relates the pH of a buffer to the concentration of its acid and conjugate base components. For a basic solution, the equation is:

• \( \text{pH} = \text{p}K_b + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) \)

In this equation, \( \text{p}K_b \) is the negative logarithm of the base dissociation constant, \( K_b \), which measures the strength of a base in solution. This equation helps us understand how the pH changes depending on the ratio of the base to its conjugate acid. By applying this formula, we can predict how the buffer will respond to changes, like additions of acid or base, providing a stable pH environment.

Calculating the pH of a solution involves determining the concentration of hydrogen ions \( [\text{H}^+] \) in a solution. Normally, we use the formula:

• \( \text{pH} = -\log [\text{H}^+] \)

In buffer solutions, we often rely on ratios of acid to conjugate base, and conjugate base to acid, to decide the pH. In this exercise, the pH is calculated using the Henderson-Hasselbalch equation. The pK value (in this case, \( \text{p}K_b \)) is crucial because it shows the extent of dissociation of the acid or base, which directly relates to the pH. Knowing \( \text{pH} \), it becomes straightforward to calculate \( [\text{H}^+] \) using the inverse logarithmic function.

In chemical reactions, the limiting reactant is the substance that is completely used up first, thus determining the maximum amount of product that can be formed. In this problem, the limiting reactant is \( \text{HCl} \) because it has fewer moles (0.08 mol) compared to \( \text{CH}_3\text{NH}_2 \) (0.1 mol). The reaction involves a 1:1 ratio of \( \text{CH}_3\text{NH}_2 \) to \( \text{HCl} \):

• \(\text{CH}_3\text{NH}_2 + \text{HCl} \rightarrow \text{CH}_3\text{NH}_3^+ + \text{Cl}^-\)

Because \( \text{HCl} \) is the limiting reactant, it is completely consumed first, leaving 0.02 mol of \( \text{CH}_3\text{NH}_2 \) and generating 0.08 mol of \( \text{CH}_3\text{NH}_3^+ \). Identifying the limiting reactant is essential for accurate calculations in any stoichiometric reaction.

An acid-base reaction involves the transfer of protons from an acid to a base. In this exercise, \( \text{HCl} \), a strong acid, donates a proton to \( \text{CH}_3\text{NH}_2 \), a weak base, turning it into its conjugate acid \( \text{CH}_3\text{NH}_3^+ \). This process can be symbolized as:

• \(\text{CH}_3\text{NH}_2 + \text{HCl} \rightarrow \text{CH}_3\text{NH}_3^+ + \text{Cl}^-\)

Such reactions are fundamental in the formation of buffer solutions which are stable solutions resisting drastic pH changes. In a buffer, the presence of both the weak base and its conjugate acid allows the solution to neutralize added acids or bases by shifting the equilibrium, thus maintaining a consistent pH. This is why the resulting solution in the exercise acted as a buffer.