Question

An aqueous solution at room temperature is \(0.1 M\) with respect to ammonium chloride and \(0.01 M\) with respect to \(\mathrm{NH}_{4} \mathrm{OH}\left(\mathrm{pK}_{b}=5\right)\), the \(\mathrm{pH}\) of the solution is : (a) 75 (b) \(6.8\) (c) \(6.5\) (d) \(8.0\)

Short answer

The pH of the solution is 8.0.

Step-by-step solution

Identify the Relevant Equations

To find the \(pH\) of the solution, use the equation for \(pOH\) which is given by \(pOH = pK_b + \log\left(\frac{[Salt]}{[Base]}\right)\). Then, convert \(pOH\) to \(pH\) using the relationship \(pH + pOH = 14\).

Calculate the pOH

Insert the given concentrations into the pOH equation: \(pOH = 5 + \log\left(\frac{0.1}{0.01}\right)\). Calculate the logarithm term, which is \log(10) = 1. Thus, \(pOH = 5 + 1 = 6\).

Calculate the pH

Convert the \(pOH\) to \(pH\) using the relationship between \(pH\) and \(pOH\): \(pH = 14 - pOH\). Substitute the calculated value of \(pOH\) to get \(pH = 14 - 6 = 8\).

Key concepts

pH and pOH Relationship

Understanding the relationship between pH (potential of hydrogen) and pOH (potential of hydroxide) is crucial in the realm of physical chemistry, particularly when it comes to aqueous solutions. pH measures the acidity of a solution, while pOH measures its basicity. The pH scale ranges from 0 to 14, with 0 being very acidic, 7 being neutral, and 14 very basic or alkaline.

The key to this relationship is that pH and pOH are inversely related and together always add up to 14 at 25°C, which is the common temperature reference for this calculation. This connection is elegantly described by the equation:
\( pH + pOH = 14 \).

When we know the concentration of either hydrogen ions ([H+]) or hydroxide ions ([OH-]) in a solution, we can calculate one value and then easily find the other. For instance, in the exercise provided, calculating the pOH first allowed us to easily find the pH by subtracting the pOH from 14. This simple yet powerful relationship is a foundation for solving numerous problems in acid-base chemistry.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a cornerstone in understanding acid-base equilibrium in buffered solutions, which plays a pivotal role in maintaining a stable pH. This equation elegantly connects the pH of a solution with the pKa (acid dissociation constant) value and the concentrations of the acid and its conjugate base. It can be written as:
\( pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right) \),

where [A-] represents the concentration of the conjugate base, and [HA] represents the concentration of the acid. While the given exercise involves a weak base and its conjugate acid, we can still apply a similar logic. If we had the pKa of the conjugate acid, we could use this equation directly to find the pH. In typical scenarios, the Henderson-Hasselbalch equation is pivotal in buffer calculations and in predicting the pH when small amounts of acid or base are added to a solution. It essentially provides a shortcut to estimate the pH without resorting to complex equilibrium calculations.

Acid-Base Equilibrium

Acid-base equilibrium refers to the state where the rates of the forward and reverse reactions of an acid-base reaction are equal, resulting in a stable pH. In any aqueous solution of an acid or a base, an equilibrium is established between the undissociated molecules and the ions they form. The equilibrium constant for this reaction is known as the acid dissociation constant (Ka) for an acid or the base dissociation constant (Kb) for a base.

For the exercise provided, we are working with ammonium chloride (NH4Cl), a salt derived from a weak base (NH4OH) and a strong acid (HCl). Dissolution of NH4Cl in water results in the formation of NH4+ and Cl-. The NH4+ reacts with water to re-form NH4OH and liberate H+ ions, which affect the pH. This exercise demonstrates the dynamic nature of acid-base equilibrium and highlights the importance of understanding how salts of weak bases or acids behave in solution.

Predicting the resulting pH from these equilibria often involves using the ion concentrations in equations like that for pOH and pH, as shown in the exercise steps. Grasping acid-base equilibrium concepts is essential not only for solving chemistry problems but also for practical applications in biological systems, environmental science, and industrial processes.