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Unformatted text preview: Pictu re 1 When may an expression be simplified and when must the quadratic equation be used? In solving a problem such as determining the pH of a weak acid solution, a quadratic form is obtained: HA + H 2 O H 3 O + + A- K a = [H 3 O + ][A- ] = x 2 where M is the initial molarity of the weak acid and x is the amount of weak acid that ionizes in mol/L [HA] M - x If x is small enough compared to M , then the approximation M-x M can be made and the expression can be simplified to: K a = x 2 M We will assume that the approximation can be made if x is less than about 10% of M , but we need to be able to predict when this will be the case. The magnitude of x depends on the values of both K a and M . For solutions of a given concentration, as the value of K a gets larger, the amount of weak acid that ionizes, or x , gets larger. For example if we compare 0.10 M solutions of hydrofluoric acid and acetic acid we see that the one with the larger value of K a has a larger percent ionization. However, in both of these cases the percent ionized is less than 10% and the approximation can be made. Acid K a % ionized, 0.1 M % ionized, 0.01 M HF 7.4 x 10-4 8.2% 23.7% CH 3 COOH 1.8 x 10-5 1.3% 4.2% As the initial concentration of a weak acid decreases, the percent ionization increases. If we compare 0.01 M solutions of hydrofluoric acid and acetic acid we see that the percent ionization has increased to 23.7% and 4.2%, respectively. In these more dilute solutions, the approximation still works for CH 3 COOH, but it is no longer good for HF. For HF, the initial concentration is too small for such a relatively large K a . The effects of both the initial acid concentration and K a can be combined in the following criterion: If M > 100K a , the approximation can be made. The larger the value of K a , the higher the minimum initial concentration of the acid must be in order for the approximation to be used. This criterion ensures that the percent ionization will be less than about 9.5%. Pictu re 1 0 If the initial concentration is smaller than 100K a , x will be too large to ignore compared to M, and the quadratic equation must be used....
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