Question

Calculate the relative concentrations of the amine aniline \(\left(\mathrm{p} K_{\mathrm{b}}=9.41\right)\) and anilinium chloride that are required to prepare a buffer of \(\mathrm{pH} 5.00\)

Short answer

The ratio \( \frac{[BH^+]}{[B]} \) is approximately 0.3909.

Step-by-step solution

Understand the relationship

To find the relative concentrations of aniline (the base, \(B\)) and anilinium chloride (the conjugate acid, \(BH^+\)), we can use the Henderson-Hasselbalch equation for basic buffers. For a basic buffer, the equation is given by: \[pOH = pK_b + \log\left( \frac{[BH^+]}{[B]} \right) \]

Convert pH to pOH

We know the pH of the buffer is 5.00. Since \(pH + pOH = 14.00\), we can calculate the pOH as follows:\[ pOH = 14.00 - pH = 14.00 - 5.00 = 9.00 \]

Rearrange the equation

Now, substitute \(pK_b = 9.41\) and \(pOH = 9.00\) into the Henderson-Hasselbalch equation:\[9.00 = 9.41 + \log\left( \frac{[BH^+]}{[B]} \right) \]Rearrange the equation to solve for the concentration ratio:\[\log\left( \frac{[BH^+]}{[B]} \right) = 9.00 - 9.41 = -0.41 \]

Solve for concentration ratio

To find the ratio \( \frac{[BH^+]}{[B]} \), take the antilogarithm:\[\frac{[BH^+]}{[B]} = 10^{-0.41} \approx 0.3909 \]

Interpret the result

The result means that the concentration of the conjugate acid \([BH^+]\) is approximately 0.3909 times the concentration of the base \([B]\). Therefore, for every 1 unit of aniline, there should be about 0.3909 units of anilinium chloride to achieve the desired pH.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is an important tool in pH calculation when dealing with buffer solutions. It relates the pH or pOH of a solution to the pK value and the ratio of concentrations of the species in the buffer. Specifically, for basic buffers, the equation can be written as:

• \[ pOH = pK_b + \log\left( \frac{[BH^+]}{[B]} \right) \]

This equation helps to connect the chemical equilibrium of a weak base and its conjugate acid with the pH we observe. Breaking it down:

• The term \(pK_b\) represents the base's strength. Lower values of \(pK_b\) mean a stronger base.
• The ratio \(\frac{[BH^+]}{[B]}\) tells us how much of the base is in its conjugate acid form compared to its base form.
• This logarithmic relationship facilitates easy calculation of pH when concentrations are known.

Understanding this equation allows us to effectively prepare and adjust buffer solutions to maintain a specific pH.

pK_b

In acid-base chemistry, the term "\(pK_b\)" refers to the negative logarithm of the base dissociation constant \(K_b\). It offers insight into the strength of a base in a solution.

• A lower \(pK_b\) value indicates a stronger base, while a higher \(pK_b\) value points to a weaker base.
• This measure is crucial when determining how a base and its conjugate acid will behave in solution, particularly in buffer solutions.

In the original exercise, knowing that aniline has a \(pK_b = 9.41\) suggests it is a relatively weak base.
The concept of \(pK_b\) is closely tied to the concept of \(pK_a\), the acid dissociation constant, via the relation \(pK_w = pK_a + pK_b\), where \(pK_w\) of water at 25°C is 14.00.
This relation assists in the conversion between acids and bases, which aids in buffer calculations involving both acids and bases.

Buffer Solution

A buffer solution is a special type of solution that resists changes in pH when small amounts of an acid or a base are added.

• Buffers typically consist of a weak acid and its conjugate base or a weak base and its conjugate acid.
• These solutions are vital in biological systems, chemical applications, and industrial processes where maintaining a stable pH is critical.

The buffer described in the original exercise involves aniline, a weak base, and anilinium chloride, its conjugate acid.
To prepare this buffer, we adjust the concentrations of the weak base and its conjugate acid based on the desired pH. Using the Henderson-Hasselbalch equation, we can determine the required concentration ratios to achieve the necessary pH.
Such systems depend on the equilibrium established between the weak base and its conjugate acid, providing the solution's ability to buffer against pH fluctuations.