Problem 139 - A Complete Resource Book in Chemistry for JEE Main

Question

The approximate \(\mathrm{pH}\) of a solution formed by mixing equal volumes of solutions of \(0.1 \mathrm{M}\) sodium propionate and \(0.1 \mathrm{M}\) propanoic acid (the dissociation constant of propanoic acid is \(1.3 \times 10^{-5} \mathrm{~mol} \mathrm{dm}^{-3}\) ) will be (a) \(2.45\) (b) \(4.89\) (c) \(5.98\) (d) \(6.89\)

Step-by-Step Solution

Answer

The pH of the solution is approximately 4.89, so the correct answer is (b) 4.89.

Step 1: Identify the Components

We have a mixture of sodium propionate (a salt) and propanoic acid, forming a buffer solution. Sodium propionate is the conjugate base of propanoic acid. Propanoic acid, \( \text{CH}_3\text{CH}_2\text{COOH} \), partially dissociates in solution, and its dissociation constant \( K_a \) is given as \( 1.3 \times 10^{-5} \).

Step 2: Use the Henderson-Hasselbalch Equation

The pH of a buffer can be calculated using the Henderson-Hasselbalch equation:\[pH = pK_a + \log \frac{[A^-]}{[HA]}\]where \( A^- \) is the concentration of the conjugate base (sodium propionate), and \( HA \) is the concentration of the acid (propanoic acid). Since the volumes are equal, the concentrations remain unchanged at \( 0.1 \ M \) each.

Step 3: Calculate \( pK_a \)

Calculate \( pK_a \) for propanoic acid using the relationship \( pK_a = -\log(K_a) \):\[pK_a = -\log(1.3 \times 10^{-5}) = 4.89\]

Step 4: Calculate \( \frac{[A^-]}{[HA]} \)

Since the concentrations of the acid and its conjugate base are equal (both \(0.1 \ M \)), the ratio \( \frac{[A^-]}{[HA]} = \frac{0.1}{0.1} = 1 \).

Step 5: Substitute Values into the Equation

Substitute the values into the Henderson-Hasselbalch equation:\[pH = 4.89 + \log(1)\]Since \( \log(1) = 0 \), the equation simplifies to:\[pH = 4.89\]

Step 6: Determine the Answer

The calculated pH is \( 4.89 \). Therefore, the correct answer is option (b) \( 4.89 \).

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a critical tool in acid-base chemistry, especially when dealing with buffer solutions. Buffers are solutions that resist drastic pH changes when acids or bases are added. They are composed of a weak acid and its conjugate base or a weak base and its conjugate acid. The equation is given by:\[pH = pK_a + \log \frac{[A^-]}{[HA]}\]- **\(pK_a\)** is the negative logarithm of the acid dissociation constant \(K_a\), which quantifies a weak acid's strength.- **\([A^-]\)** represents the concentration of the conjugate base.- **\([HA]\)** denotes the concentration of the acid.

• By using this equation, we can determine the pH of a buffer solution when both the pK_a and the concentrations of the acid and conjugate base are known.
• It highlights the relationship between the pH, the acidity constant of the acid, and the relative concentrations of the acid and its conjugate base.

This mathematical relationship is invaluable for predicting how a buffer will react when subjected to changes in concentration.

Buffer pH Calculation

Calculating the pH of a buffer solution is often done using the Henderson-Hasselbalch equation which simplifies extensive calculations. This equation is most useful when dealing with solutions of weak acids and their salts (conjugate bases). In this case, we deal with a buffer composed of propanoic acid and sodium propionate.To determine the pH of a buffer, follow these steps:

• Identify the concentrations of the acid \([HA]\) and the conjugate base \([A^-]\).
• Calculate or use the given \(pK_a\) based on the dissociation constant \(K_a\) of the acid.
• Substitute these values into the Henderson-Hasselbalch equation.

In our example, since the concentrations of the acid and its conjugate base are equal, the ratio \(\frac{[A^-]}{[HA]}\) becomes 1. Thus simplifying the log term:\[pH = pK_a + \log(1) = pK_a\]This shows that the pH of the solution is directly equal to the \(pK_a\), demonstrating the stabilizing nature of buffers.

pKa and Dissociation Constant

The concepts of \(pK_a\) and the dissociation constant \(K_a\) are fundamental in understanding the behavior of acids in a solution. The \(K_a\) value of an acid reflects its strength; a higher \(K_a\) indicates a stronger acid, and a lower \(K_a\) depicts a weaker acid.The \(pK_a\) is simply a more manageable way to express \(K_a\), defined as:\[pK_a = -\log(K_a)\]

• Lower \(pK_a\) values indicate stronger acids that dissociate more completely in water.
• Higher \(pK_a\) values denote weaker acids, as seen in our example with propanoic acid’s dissociation constant of \(1.3 \times 10^{-5}\), yielding a \(pK_a\) of 4.89.

When calculating pH or planning buffer solutions, \(pK_a\) values allow for easier manipulation and integration into equations, providing clearer insight into acid-base reactions.

Acid-Base Chemistry

Understanding acid-base chemistry is key to grasping how buffer solutions work. At its core, acid-base chemistry involves the transfer of protons (H+) between substances. Acids are proton donors, while bases are proton acceptors. - **Types of acids and bases:** - *Strong acids and bases* dissociate completely in water, making them highly reactive. - *Weak acids and bases* partially dissociate and form equilibrium mixtures in solutions. - **Conjugate pairs**: When an acid donates a proton, it transforms into its conjugate base, and vice versa. A buffer solution, like the one formed by sodium propionate and propanoic acid, uses these principles to stabilize pH levels. The weak acid and its conjugate base pair work together to prevent significant shifts in pH when small amounts of acid or base are added. Buffers are crucial in many applications, such as biological systems, where maintaining constant pH is essential for enzyme function and metabolic processes. Understanding these interactions forms the backbone of many scientific disciplines, revealing the intricate balance of reactions occurring in various environments.