Question

The dissociation constant of a weak acid \(\mathrm{HX}\) is, \(10^{-5}\). The buffer \(\mathrm{HX}+\mathrm{NaX}\) can be best used to maintain the \(\mathrm{pH}\) in the range (a) \(9-11\) (b) \(2-4\) (c) \(11-13\) (d) \(4-6\)

Short answer

(d) 4-6

Step-by-step solution

Understanding the Henderson-Hasselbalch Equation

To determine the pH range a buffer can maintain, utilize the Henderson-Hasselbalch equation for a weak acid and its salt: \[\text{pH} = \text{p}K_a + \log\left(\frac{\text{[A^-]}}{\text{[HA]}}\right)\]where \(\text{p}K_a = -\log(K_a)\), \([A^-]\) is the concentration of the conjugate base, and \([HA]\) is the concentration of the acid.

Calculating pKa

The dissociation constant (Ka) for the weak acid HX is given as \(10^{-5}\). Calculate the pKa using the formula: \[\text{p}K_a = -\log(K_a)\]Plug in the given value: \[\text{p}K_a = -\log(10^{-5})\]

Determine the pKa Value

Perform the calculation to determine the pKa value: \[\text{p}K_a = 5\]The pKa value is 5.

Assessing the Effective Buffer Range

A buffer is most effective at a pH close to its pKa value, typically within one pH unit above and below the pKa. Since the pKa is 5, the effective buffer range for HX would be from pH 4 to 6.

Selecting the Answer

With the effective range of the buffer being from pH 4 to 6, the correct answer is option (d) \(4-6\).

Key concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is critical when studying buffer solutions and their pH. This equation provides a direct relationship between the pH of a solution and the concentration of the acid and its conjugate base. It's expressed as:
\[\begin{equation}\text{pH} = \text{p}K_a + \log\left(\frac{\text{[A^-]}}{\text{[HA]}}\right)\end{equation}\]
where \(\text{p}K_a\) is the negative logarithm of the acid dissociation constant (Ka), \([A^-]\) is the concentration of the conjugate base, and \([HA]\) is the concentration of the acid. This equation is particularly useful for predicting the pH of buffer solutions, which are mixtures of a weak acid and its conjugate base that resist changes in pH when small amounts of acid or base are introduced.
To improve the understanding of the Henderson-Hasselbalch equation, consider that the equation assumes that the concentration of the acid and its conjugate base are those present at equilibrium. Also, the equation is most reliable when the concentrations of the acid and base are similar and when the solution pH is within ±1 pH unit of the pKa value of the acid. Buffer systems work best when the pH is close to the pKa of the weak acid, and their range is usually within one pH unit above or below the pKa.

Weak Acid Dissociation Constant

The dissociation constant (Ka) of a weak acid is a measure of its tendency to donate hydrogen ions in solution. It's a crucial component in understanding the strength and behaviour of a weak acid in aqueous solutions. A large Ka value indicates a strong acid, as it readily dissociates to release hydrogen ions, whereas a smaller Ka value signifies a weaker acid. For the weak acid \(\mathrm{HX}\), with a Ka value of \(10^{-5}\), this indicates a low tendency to dissociate and thus classifies \(\mathrm{HX}\) as a weak acid.

In the context of a buffer solution, knowing the Ka value allows for the calculation of the \(\text{p}K_a\), which helps determine the effective pH range the buffer can maintain. The ideal pH range for a buffer is typically within one pH unit of the acid's \(\text{p}K_a\) value. This range is where the buffer has its greatest capacity to neutralize added acids or bases without a significant change in pH. The concentration ratio of the conjugate base to acid, which appears in the Henderson-Hasselbalch equation, is instrumental in calculating the exact pH.

pKa Calculation

The pKa is the negative base-10 logarithm of the Ka value and is used to determine the strength of an acid in solution. It's simpler to work with because it converts the exponentially scaled Ka to a more manageable linear scale. For instance, with the given Ka value of \(10^{-5}\) for the weak acid \(\mathrm{HX}\), we calculate the pKa as follows:
\[\begin{equation}\text{p}K_a = -\log(K_a) \Rightarrow \text{p}K_a = -\log(10^{-5}) \Rightarrow \text{p}K_a = 5\end{equation}\]
A pKa of 5 indicates that the acid has a moderate level of acidity. When using this value in relation to a buffer system, we can understand that the buffer comprised of \(\mathrm{HX}\) and its conjugate base \(\mathrm{NaX}\) will be most effective in the pH range of 4 to 6. This is because a buffer's capacity to maintain pH is strongest when the pH is close to the pKa value of the weak acid component, facilitating the equilibrium between the acid and its conjugate base in the buffer solution.
To aid in grasping this concept, remember that a pKa value close to the desired pH range of a solution makes the corresponding acid an excellent candidate for a buffer component. This close relationship between pKa and pH range is fundamental in buffer solution design and effectiveness.