Problem 92 - A Complete Resource Book in Chemistry for JEE Main

Question

The pKa of a weak acid is \(4.8\). What should be the ratio of \([\) acid \(] /[\mathrm{salt}]\), if a buffer of \(\mathrm{pH}=5.8\) is required? (a) \(0.1\) (b) 10 (c) 1 (d) 2

Step-by-Step Solution

Answer

The ratio \([\text{acid}]/[\text{salt}]\) should be 0.1; option (a).

Step 1: Understand the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation used to calculate the pH of a buffer solution is given by: \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right) \]. This equation relates the pH of a buffer containing a weak acid and its conjugate base (salt) to the pKa of the acid.

Step 2: Set Up the Equation with Given Values

The pH of the buffer is given as 5.8 and the pKa of the weak acid is 4.8. Substitute these values into the Henderson-Hasselbalch equation: \[ 5.8 = 4.8 + \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right) \].

Step 3: Solve for the Ratio of [Salt]/[Acid]

Subtract 4.8 from both sides of the equation to isolate the log term: \[ 5.8 - 4.8 = \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right) \]. Simplify this to get: \[ 1 = \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right) \].

Step 4: Exponentiate to Solve for the Ratio

Convert the logarithmic equation to its exponential form to solve for the ratio: \[ 10^1 = \frac{[\text{salt}]}{[\text{acid}]} \]. Therefore, the ratio is \( \frac{[\text{salt}]}{[\text{acid}]} = 10 \), which implies \( \frac{[\text{acid}]}{[\text{salt}]} = \frac{1}{10} \).

Step 5: Interpret the Result

The inverse of the ratio \( \frac{[\text{acid}]}{[\text{salt}]} \) is what we need per the question, which equals \( 0.1 \). Hence, the ratio \( \frac{[\text{acid}]}{[\text{salt}]} \) is 0.1.

Key Concepts

Buffer Solution

A buffer solution is a special type of solution that resists changes in its pH when small amounts of an acid or a base are added. This stability is what makes buffers incredibly useful in a variety of chemical and biological settings. The fascinating aspect of buffer solutions lies in their ability to maintain a relatively constant pH, despite external influences.

In a typical buffer system, you'll have a mixture of a weak acid (and its conjugate base, which could be a salt) or a weak base and its conjugate acid. The weak acid or base neutralizes any added acid or base, thus maintaining the pH. Such solutions are often used in biological systems, as many biochemical processes require a constant pH to function correctly.

• Think of a buffer as a chemical cushion – absorbing the impact of added acids or bases and keeping pH changes in check.
• The effectiveness of a buffer depends on its concentration and the relative amounts of the acid and conjugate base it contains.

Understanding buffer solutions is crucial for chemistry students because it allows them to grasp how the principal compound and its conjugate form interact to stabilize a solution's pH.

Weak Acid

A weak acid is one that does not fully dissociate in an aqueous solution. Unlike strong acids, which completely ionize, weak acids reach a state of equilibrium between the undissociated acid and the ions they form. This partial dissociation is an important characteristic when considering buffer solutions.

In a weak acid equilibrium, for every molecule of acid that ionizes, a molecule of its conjugate base is formed. The formula of a weak acid in water can be written generally as:

• \[ \text{HA} ightleftharpoons \text{H}^+ + \text{A}^- \]
• \( \text{HA} \) represents the weak acid.
• \( \text{H}^+ \) and \( \text{A}^- \) represent the hydrogen ions and conjugate base, respectively.

The extent to which the weak acid dissociates is quantitatively described by its dissociation constant, \( K_a \). The behavior of weak acids in solutions is essential to understand optimal buffer system setup.

pKa

The term \( pK_a \) refers to the acid dissociation constant on a logarithmic scale, which is used to express the strength of an acid. It indicates how readily an acid releases protons (\( H^+ \)) in a solution. A smaller \( pK_a \) value corresponds to a stronger acid, which means it ionizes more completely.

For buffer solutions, \( pK_a \) is vital because it provides insight into the effectiveness of a weak acid in stabilizing the pH within a certain range. The Henderson-Hasselbalch equation uses \( pK_a \) to relate the pH of a solution to the ratio of concentrations of conjugate base to acid:

• The equation is: \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right)\]
• Choosing a buffer with a \( pK_a \) value close to the desired pH allows for more effective pH control.

The \( pK_a \) is insightful for predicting at what pH the acid is half dissociated, which maps the strongest buffering action around this pH value.

pH Calculation

Calculating pH in a buffer solution requires an understanding of both the concentration of the components involved and the equilibrium established between them. The Henderson-Hasselbalch equation proves extremely useful in this context:

The equation \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{salt}]}{[\text{acid}]} \right) \] defines the relationship between pH, \( pK_a \), and the log ratio of the concentrations of the conjugate base (often referred to as salt) and the weak acid.

• The equation simplifies the pH calculation, as long as the \( pK_a \) is known along with the concentrations of the acid and its conjugate base.
• It's crucial to remember that this equation is especially valid for buffer solutions, where the pH is relatively close to the \( pK_a \).

These calculations help chemists design solutions with precisely controlled pH levels suitable for specific reactions or biological environments. Understanding and applying this calculation is crucial for areas of research, pharmaceutical development, and various industrial processes.