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 approximate pH is 4.89, option (b).

Step 1: Identify the type of solution

The question deals with a solution formed by mixing equal volumes of sodium propionate and propanoic acid, which constitutes a buffer solution. This is because sodium propionate is a salt that provides the conjugate base (propionate ion), and propanoic acid is the weak acid.

Step 2: Use the Henderson-Hasselbalch equation

For buffer solutions, we use the Henderson-Hasselbalch equation, which is given by: \[ \text{pH} = \text{pK}_a + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \] here, \([\text{A}^-]\) is the concentration of the conjugate base (propionate ion) and \([\text{HA}]\) is the concentration of the weak acid (propanoic acid).

Step 3: Calculate the pKa value

First, calculate \(\text{pKa}\) from the given dissociation constant \(K_a\) of propanoic acid:\[ \text{pK}_a = -\log(K_a) = -\log(1.3 \times 10^{-5}) \] Calculating this gives: \[ \text{pK}_a = 4.89 \]

Step 4: Apply the Henderson-Hasselbalch equation

Since the volumes are equal and the concentrations of both the weak acid and its conjugate base are equal at \(0.1 \text{ M}\), the ratio \(\frac{[\text{A}^-]}{[\text{HA}]} = 1\). Now, substitute \(\text{pK}_a\) and the ratio into the Henderson-Hasselbalch equation:\[ \text{pH} = 4.89 + \log(1) \] Since \(\log(1) = 0\), we have \[ \text{pH} = 4.89 \]

Step 5: Compare with answer choices

The calculated \(\text{pH}\) is 4.89, which corresponds to option (b) \(4.89\) in the given choices.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch Equation is a crucial formula in chemistry that helps us figure out the pH of buffer solutions. A buffer is a special type of solution that resists drastic changes in pH when small amounts of acid or base are added. Why is this formula important? Because it makes understanding the pH of common biological and chemical systems easier.

The equation is written as:\[\text{pH} = \text{pK}_a + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]

• \([\text{A}^-]\) represents the concentration of the conjugate base in the buffer (the part that neutralizes acids).
• \([\text{HA}]\) stands for the concentration of the weak acid in the buffer (the part that neutralizes bases).
• \(\text{pK}_a\) is the negative logarithm of the acid dissociation constant, \(K_a\), which tells us how strong or weak an acid is.

When you have equal concentrations of acid and its conjugate base, the equation becomes even more straightforward:
\[\text{pH} = \text{pK}_a\] This simplification happens because the ratio \(\frac{[\text{A}^-]}{[\text{HA}]} = 1\), and \(\log(1)\) is zero.

So, in buffer solutions, like the one with sodium propionate and propanoic acid, this equation helps us predict the pH accurately.

Propanoic Acid

Propanoic acid, also known as propionic acid, is an organic compound with the chemical formula \(\text{C}_3\text{H}_6\text{O}_2\). It is a simple carboxylic acid, which means it has a carboxyl group (\(-\text{COOH}\)) attached. In terms of smell, it has a pungent, unpleasant odor, often described as somewhat like body odor.

Propanoic acid plays a significant role in food preservation and as an antimicrobial agent. It prevents the growth of molds and bacteria in baked goods and animal feed. But in chemistry, it's more interesting for its behavior as a weak acid.

Weak Acid Properties

Being a weak acid, propanoic acid only partially dissociates in water. This partial dissociation is why it can act as part of a buffer system with its conjugate base, the propionate ion \((\text{CH}_3\text{CH}_2\text{COO}^-)\). This buffered action is particularly valuable in maintaining a stable pH in both biological and chemical systems.

Understanding the behavior of propanoic acid in buffer solutions is crucial because it helps in the formulation of dietary plans, pharmaceutical applications, and even in various industrial processes.

In summary, propanoic acid is more than just another molecule; it’s a versatile component in a variety of applications, supporting systems that require stable pH levels.

Dissociation Constant

The dissociation constant, often symbolized as \(K_a\), is a measure that tells us how easily an acid donates its proton in an aqueous solution. For weak acids, like propanoic acid, \(K_a\) helps us understand their partial dissociation behavior.

The given \(K_a\) for propanoic acid is \(1.3 \times 10^{-5}\, \text{mol}\, \text{dm}^{-3}\), which might seem small, but it is crucial for calculating the pH of buffer solutions. This constant directly relates to the strength of the acid—lower \(K_a\) values indicate weaker acids, which do not dissociate much in solution.

Calculating \(\text{pK}_a\)

One important property derived from \(K_a\) is \(\text{pK}_a\), calculated using:\[\text{pK}_a = -\log(K_a)\]For propanoic acid, this calculation gives us a \(\text{pK}_a\) of 4.89, which allows us to understand and predict the behavior of the buffer solution when using the Henderson-Hasselbalch Equation.

The significance of the \(\text{pK}_a\) value is its indication of how a specific acid-base pair will buffer against changes in pH. With this knowledge, chemists can design buffers specific to different needs, from pharmaceuticals to industrial processes.