Problem 231 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constants of \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(\mathrm{NH}_{4} \mathrm{OH}\) in aqueous solution are almost the same. The \(\mathrm{pH}\) of a solution of \(0.01 \mathrm{~N} \mathrm{CH}_{3} \mathrm{COOH}\) is \(4.0\) at \(25^{\circ} \mathrm{C}\). The \(\mathrm{pOH}\) of \(0.01 \mathrm{~N} \mathrm{NH}_{4} \mathrm{OH}\) solutions at the same temperature will be

Step-by-Step Solution

Answer

pOH = 10

Step 1: Write the Relationship Between pH and pKa

The pH of a solution is related to the acid dissociation constant, or Ka, through the equation: \[ ext{pH} = -rac{1}{2} imes ext{log}_{10}(C imes K_a)\]where \(C\) is the concentration and \(K_a\) is the acid dissociation constant. This equation is given for a weak acid in solution, specifically when the concentration is known, and we use the assumption that concentration of the acid compared to its dissociation remains practically the same due to its weak dissociation.

Step 2: Determine the Relationship for pOH

Similarly, for a weak base like \(NH_4OH\), we have the base dissociation constant, \(K_b\), related through\[ ext{pOH} = -rac{1}{2} imes ext{log}_{10}(C imes K_b)\]Here, \(K_b\) is the base dissociation constant and \(C\) is the concentration of the base, similar to what we did for the acid in step 1.

Step 3: Use the Given Information

Since it is given that the pH of 0.01 N \(CH_3COOH\) is 4.0, we determine that \[ ext{pH} = 4.0\]It follows that the dissociation constant, \(K_a\), if calculated, should satisfy this relationship with the concentration given.

Step 4: Relate pKa and pKb

Knowing the relationship between acids and bases, for conjugate acid-base pairs,\[pK_w = pK_a + pK_b = 14 \]Given that pH is given for equivalent concentrations and that their \(K_a\) and \(K_b\) are approximately equal, if \(pH = 4.0\) for \(CH_3COOH\), the \(pK_a\) would also align towards 4, given convergence. Then,\[pK_b = 14 - 4 = 10\]This means the \(pOH\) should reflect the equivalent strength of the conjugate base.

Step 5: Calculate the pOH Based on Assumptions

From the equal dissociation constants perspective and the equality noted,when \(pK_a = pK_b\), hence, the operational form is that \(pOH\) for the \(NH_4OH\) would then also be slightly different based on inverse, specifically translating operational difficulty, but is simplified by assuming basic equivalency for practical measure, keeping it simple and internally balanced around a mid-fold path. For concluding resort:\[pOH = 10\]

Key Concepts

Dissociation Constants

One of the fundamental aspects of understanding acids and bases in chemistry is knowing how they dissociate in solutions. Dissociation constants, like the acid dissociation constant (K_a) or the base dissociation constant (K_b), indicate the extent to which an acid or base can donate protons or accept them in aqueous solution. This constant is crucial because it tells us about the strength of the acid or base.

The larger the dissociation constant, the stronger the acid or base. Conversely, smaller constants signify weaker acids or bases. This relationship arises because a larger constant reflects a greater tendency to dissociate, releasing more hydrogen ions (H^+) in the case of acids or hydroxide ions (OH^-) for bases.

When dealing with weak acids or bases, the dissociation is not complete. Because only a small fraction of molecules dissociate, we can use a mathematical formula to represent this partial dissociation. It's expressed as:\[ K = \frac{{[H^+]^n [A^-]}}{{[HA]}} \]for acids or a similar formula for bases. Here, [A^-] represents the concentration of the dissociated species, and [HA] indicates the concentration of the un-dissociated acid. Keeping track of these concentrations helps us understand how the system moves toward equilibrium.

pH and pOH Relationships

The concepts of \( pH \) and \( pOH \) are closely intertwined in any discussion about acids and bases. They serve as scales to determine the acidity or basicity of a solution respectively. \( pH \) is the negative logarithm (base 10) of the hydrogen ion concentration, \( [H^+] \), whilst \( pOH \) is the negative logarithm of the hydroxide ion concentration, \( [OH^-] \).

A critical relationship to remember is \( pH + pOH = 14 \). This linkage enables us to transition between the acidity and basicity assessments of any aqueous solution easily. For example, if the \( pH \) of a solution is determined to be 4, like in our exercise, you can directly calculate \( pOH \) by subtracting \( pH \) from 14, giving a \( pOH \) of 10.

This equilibrium is vital when you have to switch between examining the properties of acids and bases, especially when their roles and functions overlap, as in water or similar solutions. Hence, understanding and using the relationship between \( pH \) and \( pOH \) helps not only measure but also predict the behavior of solutions under different conditions.

Weak Acids and Bases

Weak acids and bases play an essential role in many chemical processes, due to their incomplete dissociation in solution. Unlike strong acids or bases, which completely ionize in water, weak acids and bases dissociate only partially, leaving a significant amount of the original compound intact.

This limited ionization is why we often use \( K_a \) and \( K_b \) values to describe them. For instance, acetic acid (CH_3COOH) and ammonia solution (NH_4OH) are examples of weak acids and bases respectively. Their dissociation can be described by equilibrium expressions involving \[ CH_3COOH \rightleftharpoons H^+ + CH_3COO^- \] for acetic acid and \[ NH_4OH \rightleftharpoons NH_4^+ + OH^- \] for ammonia.

Because these substances do not release many ions, they serve well in buffering solutions where a stable \( pH \) is needed. Such buffer systems are important in many biological and chemical applications, including maintaining conditions in physiological contexts. They resist drastic changes in \( pH \) when small quantities of acids or bases are added, ensuring stability for many reactions to proceed optimally.