Problem 3 - Chemistry The Central Science In Si Units

Question

A buffer contains a weak acid, HA, and its conjugate base. The weak acid has a \(\mathrm{pK}_{a}\) of 4.5 , and the buffer has a pH of 4.7 . Without doing a calculation, state which of these possibilities are correct at pH 4.7 . (a) \([\mathrm{HA}]>\left[\mathrm{A}^{-}\right]\) (b) \([\mathrm{HA}]=\left[\mathrm{A}^{-}\right],\) or \((\mathbf{c})[\mathrm{HA}]<\left[\mathrm{A}^{-}\right]\).

Step-by-Step Solution

Answer

At pH 4.7, \([\mathrm{HA}] < [\mathrm{A}^-]\) is correct.

Step 1: Understanding the Henderson-Hasselbalch Equation

The pH of a buffer can be related to the concentrations of acid and its conjugate base using the Henderson-Hasselbalch equation: \[ \text{pH} = \text{pK}_a + \log\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right) \] The equation helps to determine the relative concentrations depending on the given pH and pK_a.

Step 2: Analyze Given Values

It is given that the pH of the buffer is 4.7 and the pK_a of the weak acid is 4.5. Using the equation, we need to analyze the relationship between \([\mathrm{A}^-]\) and \([\mathrm{HA}]\).

Step 3: Determine Concentration Relationship

Since the pH (4.7) is greater than the pK_a (4.5), according to the Henderson-Hasselbalch equation:\[\log\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}>\right) >0\]Thus, \([\mathrm{A}^-] > [\mathrm{HA}]\).

Step 4: Select the Correct Option

Based on the analysis, the correct choice is option (c) \([\mathrm{HA}]<\left[\mathrm{A}^-\right]\). This indicates that at pH 4.7, there is a higher concentration of the conjugate base than the acid.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental formula in chemistry that connects the pH of a buffer solution to the ratio of the concentration of a weak acid and its conjugate base. This equation is expressed as follows: \[ \text{pH} = \text{pK}_a + \log\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right) \]Here's how it works:

• \( [\mathrm{HA}] \): the concentration of the weak acid.
• \( [\mathrm{A}^-] \): the concentration of the conjugate base, which is the acid after donating its hydrogen ion.
• \( \text{pK}_a \): the acid dissociation constant, which indicates the acid's strength.

When the pH of a solution equals \( \text{pK}_a \), the concentrations of the acid and its conjugate base are equal. Knowing this balance helps determine how the solution will react to the addition of acids or bases. If the pH is greater than the \( \text{pK}_a \), it indicates a greater concentration of the conjugate base, \( [\mathrm{A}^-] \), compared to the weak acid, \( [\mathrm{HA}] \). This makes the Henderson-Hasselbalch equation a simple yet powerful tool for predicting the behavior of buffer solutions.

Acid-Base Equilibria

Acid-base equilibria involve the balance between acids and bases in a solution. Understanding this balance is critical for predicting how changes in concentration affect the overall pH of the solution. In a typical acid-base reaction:

• Acids donate hydrogen ions (H⁺).
• Bases accept these hydrogen ions.

A buffer solution consists of a weak acid and its conjugate base. These components maintain the pH when small amounts of acids or bases are added to the system. The equilibrium position will shift depending on whether more acids or bases are introduced:

• If more acid is added, the equilibrium shifts to the left, increasing the concentration of the weak acid.
• If more base is added, the equilibrium shifts to the right, increasing the concentration of the conjugate base.

This mechanism allows buffers to neutralize small additions of acids or bases, keeping the pH relatively stable which is essential in various biological and chemical systems.

pH and pKa

The concepts of pH and \(\text{pK}_a\) are interrelated but serve different purposes. is a measure of the acidity or basicity of an aqueous solution. It is calculated as:\[ \text{pH} = -\log[\text{H}^+] \]where \([\text{H}^+]\) is the hydrogen ion concentration. It ranges from 0 to 14, with 7 being neutral. Values below 7 indicate acidity, and values above 7 indicate basicity. on the other hand, represents the acid dissociation constant and reflects the strength of an acid in solution. The lower the \( \text{pK}_a \), the stronger the acid, meaning it is more likely to donate a proton. It is mathematically defined as:\[ \text{pK}_a = -\log K_a \]where \(K_a\) is the actual dissociation constant of the acid. When the pH of a solution is equal to the \(\text{pK}_a\), it indicates that the concentrations of the protonated (acid) and deprotonated (conjugate base) forms are equal. These relationships help us understand the buffering capacity of a solution to resist changes in pH, which is crucial for maintaining stable environments in both experimental settings and living organisms.