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 \(\mathrm{pH}\) of the solution is \(4.89\) (option b).

Step 1: Identify the System

We have a solution containing equal volumes of sodium propionate and propanoic acid. This forms a buffer solution. The sodium propionate acts as the salt of the weak acid, propanoic acid.

Step 2: Use the Henderson-Hasselbalch Equation

For a buffer solution, we can use the Henderson-Hasselbalch equation: \[ \mathrm{pH} = \mathrm{pK}_a + \log \frac{[\text{salt}]}{[\text{acid}]} \]In this case, both the concentrations of the salt (sodium propionate) and the acid (propanoic acid) are the same: \(0.1 \mathrm{M}\). Therefore, the ratio \(\log \frac{[\text{salt}]}{[\text{acid}]}\) is \(\log 1 = 0\).

Step 3: Calculate pKa from Ka

The dissociation constant \(K_a\) of propanoic acid is given as \(1.3 \times 10^{-5}\). To find \(\mathrm{pK}_a\), use the formula: \[ \mathrm{pK}_a = -\log K_a \]Substituting the given \(K_a\) value:\[ \mathrm{pK}_a = -\log (1.3 \times 10^{-5}) = 4.89 \]

Step 4: Apply to Henderson-Hasselbalch Equation

Since the concentrations of acid and salt are equal, the \(\mathrm{pH}\) is just the \(\mathrm{pK}_a\) calculated:\[ \mathrm{pH} = 4.89 + 0 = 4.89 \]

Step 5: Conclusion

Comparing the calculated \(\mathrm{pH}\) with the provided options, the approximate \(\mathrm{pH}\) of the solution is option (b) \(4.89\).

Key Concepts

Henderson-Hasselbalch Equation

Buffer solutions help maintain a stable pH when acids or bases are added to them. When dealing with buffer solutions, the Henderson-Hasselbalch equation is a crucial tool for understanding and predicting the pH of the solution. The Henderson-Hasselbalch equation is expressed as:
\[ \mathrm{pH} = \mathrm{pK}_a + \log \frac{[\text{salt}]}{[\text{acid}]} \]
This equation relates the pH of a buffer solution to the concentration of the acid and its conjugate base (often a salt). It is especially useful when the concentrations of these components are known, allowing for straightforward pH calculation. In the exercise, since the concentrations of sodium propionate (the salt) and propanoic acid (the acid) were equal, it simplified the use of this equation tremendously.

• Key Element: The ability to predict pH from known concentrations.
• Application: Especially useful in buffering solutions such as biological buffers.

pH Calculation

Calculating the pH of a solution is central to acid-base chemistry. pH is a measure of how acidic or basic a solution is, on a scale from 0 to 14.
A lower pH (0-6) indicates acidity, while a higher pH (8-14) indicates basicity, with 7 being neutral.

• For strong acids and bases, pH can directly reflect the concentration of hydrogen or hydroxide ions in solution.
• For weak acids or buffers, as in our example, the calculation depends on both dissociation constant and concentrations.

In the provided exercise, by knowing the \(\mathrm{pK}_a\) of propanoic acid and using the Henderson-Hasselbalch equation, the pH was calculated to be 4.89, demonstrating the typical weakly acidic nature of the buffer solution consisting of propanoic acid and its salt.

Dissociation Constant

The dissociation constant, often denoted as \(K_a\) for acids, is an important parameter in acid-base chemistry. It quantifies the tendency of an acid to dissociate into its ions in solution.
For propanoic acid in the problem, the dissociation constant \(K_a\) was given as \(1.3 \times 10^{-5}\).

• High \(K_a\) value: The acid dissociates more in water, indicating a stronger acid.
• Low \(K_a\) value: Less dissociation, indicating a weaker acid.

To find \(\mathrm{pK}_a\), the negative logarithm of the \(K_a\) is taken:
\[ \mathrm{pK}_a = - \log K_a \]
For instance, the \(\mathrm{pK}_a\) for propanoic acid was calculated to be 4.89, showing it's a weak acid. Understanding the dissociation constant helps predict the behavior of acids in different chemical contexts, including reactivity and equilibrium.

Acid-Base Chemistry

Acid-base chemistry involves the study of how acids and bases react with one another, a fundamental area in chemistry. Acids are substances that can donate a proton (H+), while bases can accept a proton or donate an electron pair in some reactions.
The mixing of sodium propionate and propanoic acid in the exercise creates a classic acid-base environment at equilibrium.

• Concept: Acids, such as propanoic acid, interact with bases, leading to ion formation and potential pH change.
• Buffer Systems: A buffer resists changes to pH upon addition of small amounts of acid or base as seen in the solution.

Understanding this interaction is key to manipulating chemical reactions in laboratory and industrial settings, where control over reaction conditions is essential. In the given exercise, the acidity maintained by propanoic acid and the stabilizing effect of its salt characterizes an effective buffer system.