Question 18.7: Finding Ka of a Weak Acid from the Solution pH Problem Pheny......

Plan We are given [HPAc]_{\text{init}} (0.12 M) and the pH (2.62) and must find K_a. As always, we first write the equation for HPAc dissociation and the expression for K_a to see which values we need. We set up a reaction table (Table 1) and use the given pH to find [H_3O^+], which equals [PAc^−] and [HPAc]_{\text{dissoc}} (we assume that [H_3O^+]_{\text{from }H_2O} is negligible). To find [HPAc], we assume that, because it is a weak acid, very little dissociates, so [HPAc]_{\text{init}}  −  [HPAc]_{\text{dissoc}}  =  [HPAc]  ≈  [HPAc]_{\text{init}}. We make these assumptions, substitute the equilibrium values, solve for K_a, and then check the assumptions using the 5% rule (Sample Problem 17.9).

Solution Writing the dissociation equation and K_a expression:

Setting up a reaction table (Table 1), with x  =  [HPAc]_{\text{dissoc}}  =  [H_3O^+]_{\text{from HPAc}}  =  [PAc^−]:

Calculating [H_3O^+]:
              [H_3O^+]  =  10^{−pH}  =  10^{−2.62}  =  2.4×10^{−3}  M
Making the assumptions:

1. The calculated [H_3O^+]  (2.4×10^{−3}  M)  >>  [H_3O^+]_{\text{from }H_2O}  (1.0×10^{−7}  M), so we assume that [H_3O^+]  ≈  [H_3O^+]_{\text{from HPAc}}  =  [PAc^−]  =  x (the change in [HPAc], or [HPAc]_{\text{dissoc}}).
2. HPAc is a weak acid, so we assume that [HPAc] = 0.12 M − x ≈ 0.12 M.
Solving for the equilibrium concentrations:

             x  ≈  [H_3O^+]  =  [PAc^−]  =  2.4×10^{−3}  M
             [HPAc]  =  0.12  M  −  x  =  0.12  M  −  (2.4×10^{−3}  M)  ≈  0.12  M  \text{(to 2 sf)}

Substituting these values into K_a:
             K_a  =  \frac{[H_3O^+][PAc^−]}{[HPAc]}  ≈  \frac{(2.4×10^{−3})(2.4×10^{−3})}{0.12}  =  4.8×10^{−5}
Checking the assumptions by finding the percent error in concentration:

1. For [H_3O^+]_{\text{from }H_2O}:  \frac{1×10^{−7}  M }{ 2.4×10^{−3}  M}  ×  100  =  4×10^{−3}% (<5%; assumption is justified.)

2. For [HPAc]_{\text{dissoc}}:  \frac{2.4×10^{−3}  M}{0.12  M}  ×  100  =  2.0% (<5%; assumption is justified.)

Check The [H_3O^+] makes sense: pH 2.62 should give [H_3O^+] between 10^{−2} and 10^{−3} M. The K_a calculation also seems in the correct range: (10^{−3})^2/10^{−1}  =  10^{−5}, and this value seems reasonable for a weak acid.