Question

Some \(\mathrm{K}_{2} \mathrm{SO}_{3}\) and \(\mathrm{KHSO}_{3}\) are dissolved in 250.0 \(\mathrm{mL}\) of solution and the resulting \(\mathrm{pH}\) is \(7.25 .\) Which is greater in this buffer solution, the concentration of \(\mathrm{SO}_{3}^{2-}\) or the concentration of \(\mathrm{HSO}_{3}-7\) If \(\left[\mathrm{SO}_{3}^{2-}\right]=1.0 \mathrm{M}\) in this solution, calculate the concentration of \(\mathrm{HSO}_{3}\) .

Short answer

In conclusion, the concentration of SO3^2- is greater than the concentration of HSO3- in the given buffer solution, and the concentration of HSO3- is 0.316 M.

Step-by-step solution

Write down the Henderson-Hasselbalch equation for a buffer solution

The Henderson-Hasselbalch equation for a buffer solution is given by: pH = pKa + log\(\frac{[\text{A}^-]}{[\text{HA}]}\) Where pH is the solution's pH, pKa is the negative logarithm of the acid dissociation constant (Ka) of the weak acid, and [\text{A}^-] and [\text{HA}] are the molar concentrations of the conjugate acid/base pair. In our case, the weak acid is HSO3- and its conjugate base is SO3^2-.

Calculate pKa using the given pH value

We are given that the pH of the solution is 7.25. To find the pKa of HSO3-, we can use the relationship between pH, pOH, pKw, pKa, and pKb: pOH = 14 - pH = 14 - 7.25 = 6.75 We can say \(pK_a = p_{OH}\), That gives, \(pK_a = 6.75\)

Use the Henderson-Hasselbalch equation to find the concentration of HSO3-

Now we can plug the values of pH, pKa, and [\(\text{SO}_{3}^{2-}\)] into the Henderson-Hasselbalch equation: 7.25 = 6.75 + log\(\frac{[\text{SO}_{3}^{2-}]}{[\text{HSO}_{3}^{-}]}\) Rearrange the equation to solve for [\(\text{HSO}_{3}^{-}\)]: [\(\text{HSO}_{3}^{-}\)] = [\(\text{SO}_{3}^{2-}\)] / 10^(\(pH - pK_a\)) [\(\text{HSO}_{3}^{-}\)] = \(1.0 M\) / 10^(\(7.25 - 6.75\)) [\(\text{HSO}_{3}^{-}\)] = \(1.0 M\) / 10^(\(0.5\)) [\(\text{HSO}_{3}^{-}\)] = 0.316 M

Compare the concentrations and determine which ion has a higher concentration

Now that we have the concentration of HSO3- (0.316 M), we can compare it to the given concentration of SO3^2- (1.0 M): [\(\text{SO}_{3}^{2-}\)] = 1.0 M [\(\text{HSO}_{3}^{-}\)] = 0.316 M Since the concentration of SO3^2- (1.0 M) is greater than the concentration of HSO3- (0.316 M), the concentration of SO3^2- is greater in this buffer solution. In conclusion, the concentration of SO3^2- is greater than the concentration of HSO3- in the given buffer solution, and the concentration of HSO3- is 0.316 M.

Key concepts

Buffer Solution

A buffer solution is a special kind of solution that resists drastic changes in pH when small amounts of acids or bases are added. This amazing property helps maintain a stable environment, which is crucial in many chemical and biological processes. Buffer solutions are typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid.

For example, in our context, the buffer solution consists of the weak acid \( ext{HSO}_3^-\) and its conjugate base \( ext{SO}_3^{2-}\). This pair helps the solution maintain a relatively constant pH even when minor amounts of other substances are introduced. When an acid \(\text{H}^+\) is added, it reacts with the \(\text{SO}_3^{2-}\) ion forming \(\text{HSO}_3^-\), and when a base \(\text{OH}^-\) is added, it will react with \(\text{HSO}_3^-\) forming \(\text{SO}_3^{2-}\). This ability to neutralize small amounts of added acid or base is what defines a buffer solution.

The behavior of buffer solutions can be explained and calculated using the Henderson-Hasselbalch equation, a widely used mathematical relationship that connects pH, the concentration of acid-base pairs, and acid dissociation constant.

Acid Dissociation Constant

The acid dissociation constant, represented as \(K_a\), is a measure of the strength of an acid in a solution. It reflects the tendency of an acid to donate protons to the base. The larger the \(K_a\) value, the stronger the acid, implying it dissociates more in solution.

In this particular exercise, the value \(pK_a\) is more commonly used, which is simply the negative logarithm of the \(K_a\) value: \(pK_a = -\log(K_a)\). This transformation is useful because it makes it easier to work with the numbers, especially when they are very large or very small. In our buffer solution context, \(pK_a\) tells us about the strength of the weak acid \(\text{HSO}_3^-\).

Using the Henderson-Hasselbalch equation, we were able to deduce the \(pK_a\) from the given pH of the solution, which is 7.25. Essentially, it shows how the \(pH\) of a buffer is related to the relative concentrations of the acid and base pair, which is crucial in determining how the buffer functions.

Conjugate Acid-Base Pair

A conjugate acid-base pair consists of two species that transform into each other by the gain or loss of a proton \((\text{H}^+)\). When an acid donates its proton, it forms its conjugate base, and when a base accepts a proton, it forms its conjugate acid. This dynamic is central to understanding buffer solutions and acid-base chemistry.

In this exercise, the conjugate acid-base pair is \(\text{HSO}_3^-\) and \(\text{SO}_3^{2-}\). The weak acid \(\text{HSO}_3^-\) loses a proton to become its conjugate base \(\text{SO}_3^{2-}\). Conversely, \(\text{SO}_3^{2-}\) can gain a proton to become \(\text{HSO}_3^-\). This reversible transformation is what makes them a conjugate pair.

The ratio of the concentrations of the components of the conjugate acid-base pair determines the pH of the solution, as expressed through the Henderson-Hasselbalch equation. Their interaction ensures that the solution can resist changes in pH, maintaining stability, which is vital in many biochemical and industrial processes where pH stability is required.