Question

A 0.25 M solution of a salt NaA has $\mathrm{pH}=9.29.$ What is the value of $K_a$ for the parent acid HA?

Short answer

The value of $K_a$ for the parent acid HA is approximately $1.51\times {10}^{-9}$.

Step-by-step solution

Calculate the concentration of OH⁻ ions

First, we'll determine the concentration of ${\mathrm{OH}}^{-1}$ ions in the solution using the formula: $pOH=14-pH$ Given that the $\mathrm{pH}$ is 9.29, we can find the pOH: $pOH=14-9.29=4.71$ Now, we can find the concentration of ${\mathrm{OH}}^{-1}$ ions using the formula: $[{\mathrm{OH}}^{-}]={10}^{-pOH}$ $[{\mathrm{OH}}^{-}]={10}^{-4.71}\approx 1.95\times {10}^{-5}\mathrm{M}$

Determine the concentration of A⁻ ions

Now, we need to find the concentration of ${\mathrm{A}}^{-}$ ions in the solution. This can be done by the fact that both ${\mathrm{A}}^{-}$ and ${\mathrm{OH}}^{-}$ ions come from the dissociation of the salt $\mathrm{NaA}$. Since the initial concentration of the salt NaA is 0.25 M, the concentration of ${\mathrm{A}}^{-}$ ions in the solution is also 0.25 M.

Calculate the concentration of H₃O⁺ ions

We can determine the concentration of ${\mathrm{H}}_3{\mathrm{O}}^{+}$ ions using the relationship between ${\mathrm{OH}}^{-}$ ions and ${\mathrm{A}}^{-}$ ions. In a basic solution, the following equilibrium is established: ${\mathrm{A}}^{-}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons \mathrm{HA}+{\mathrm{OH}}^{-}$ Since the equilibrium constant for this reaction is small, we can assume that the change in the concentrations of ${\mathrm{A}}^{-}$ and ${\mathrm{OH}}^{-}$ ions is negligible. Therefore, the concentration of H₃O⁺ ions can be found using the following equation: $[{\mathrm{H}}_3{\mathrm{O}}^{+}]={\displaystyle \frac{[{\mathrm{OH}}^{-}{]}^2}{[{\mathrm{A}}^{-}]}}$ Calculating the concentration of ${\mathrm{H}}_3{\mathrm{O}}^{+}$ ions: $[{\mathrm{H}}_3{\mathrm{O}}^{+}]={\displaystyle \frac{(1.95\times {10}^{-5}{)}^2}{0.25}}\approx 1.51\times {10}^{-9}\mathrm{M}$

Calculate the Ka value for the parent acid HA

Finally, we can determine the value of $K_a$ for the parent acid HA using the equilibrium equation for the dissociation of the parent acid: $\mathrm{HA}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{A}}^{-}+{\mathrm{H}}_3{\mathrm{O}}^{+}$ The $K_a$ can be expressed as: $K_a={\displaystyle \frac{[{\mathrm{A}}^{-}][{\mathrm{H}}_3{\mathrm{O}}^{+}]}{[\mathrm{HA}]}}$ Since the change in the concentration of ${\mathrm{A}}^{-}$ ions is negligible as mentioned earlier, the initial concentration of HA can be considered nearly equal to the concentration of ${\mathrm{A}}^{-}$ ions. Therefore, the $K_a$ formula can be rewritten as: $K_a={\displaystyle \frac{[{\mathrm{A}}^{-}][{\mathrm{H}}_3{\mathrm{O}}^{+}]}{[{\mathrm{A}}^{-}]}}$ Calculating the value of $K_a$: $K_a={\displaystyle \frac{0.25\times 1.51\times {10}^{-9}}{0.25}}\approx 1.51\times {10}^{-9}$ The value of $K_a$ for the parent acid HA is approximately $1.51\times {10}^{-9}$.

Key concepts

Chemical Equilibrium

When a weak acid dissolves in water, it doesn't completely separate into its constituent ions. Instead, a dynamic state known as chemical equilibrium is reached, where the forward reaction (acid dissociating) and the reverse reaction (ion recombination) occur at the same rate. This is represented by a double arrow in the chemical equation. The key to understanding this concept lies in the equilibrium constant ($K$), which expresses the ratio of products to reactants at equilibrium. For acid dissociation, we specifically refer to this constant as the acid dissociation constant ($K_a$).

When calculations involve weak acids or bases, the assumption often made is that the change in the concentration of the ions due to dissociation is so small that it can be neglected. This simplifies the calculation, still yielding an accurate approximation of $K_a$ without complex iterative methods. In practice, this assumption is valid for weak acids or bases where the equilibrium constant is much less than 1, signifying limited dissociation.

pH Calculation

The pH scale is a logarithmic representation of the hydrogen ion concentration in a solution and is highly useful in indicating whether a solution is acidic (pH < 7), neutral (pH = 7), or basic (pH > 7). To calculate the pH of a solution, one first needs to know the concentration of the hydrogen ions ($[{\text{H}}_3{\text{O}}^{+}]$ or simply $[{\text{H}}^{+}]$).

The relationship between $[{\text{H}}_3{\text{O}}^{+}]$ and pH is given by the formula:
$pH=-\text{log}([{\text{H}}_3{\text{O}}^{+}])$
Similarly, for basic solutions with hydroxide ions, the pOH can be found using the hydroxide ion concentration and is given by the formula:
$pOH=-\text{log}([{\text{OH}}^{-}])$
The sum of pH and pOH always equals 14, which is useful in moving between the two scales and ultimately finding the pH from a known pOH.

Weak Acid

A weak acid is a compound that does not fully dissociate into ions in an aqueous solution. This incomplete dissociation results in the dynamic equilibrium between the undissociated acid molecules and the ions formed in the dissociation process. The strength of a weak acid is quantified by its acid dissociation constant ($K_a$), which is low compared to that of a strong acid.

For instance, the exercise provided involves a weak acid, HA, whose dissociation in water produces hydronium ions (\text{H}_3\text{O}^+\text{)} and its conjugate base (\text{A}^-). The value of $K_a$ is determined through an understanding of the equilibrium established in the solution and knowing the concentrations of the ions present. A larger $K_a$ value corresponds to a stronger acid, indicating a greater tendency to lose a proton. Conversely, a smaller $K_a$ value suggests a weaker acid, which is less likely to part with a proton.