As a part of your research program on formic acid, you need to titrate a solution of formic acid with sodium hydroxide solution and want to know what to expect. Calculate the pH of

(a) 0.100 m HCOOH(aq) and

(b) The solution resulting when 5.00 mL of 0.150 m NaOH(aq) is added to 25.00 mL of the acid. Use K_a = 1.8 * 10^–4 for HCOOH.

ANTICIPATE When the strong base is added, some of the weak acid is neutralized, so you should expect the pH to rise from part (a) to part (b).

PLAN For part (a), use the procedure in Toolbox 6D.1. For part (b), find the pH using the procedure in Toolbox 6H.2.

What should you assume? Assume that the autoprotolysis of water does not contribute significantly to the pH and that the weak acid formic acid undergoes only a small degree of deprotonation.

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Toolbox 6D.1 HOW TO CALCULATE THE pH OF A SOLUTION OF A WEAK ACID CONCEPTUAL BASIS Because the proton transfer equilibrium is established as soon as a weak acid is dissolved in water, the concentrations of acid, hydro- nium ion, and conjugate base of the acid must jointly correspond to the acidity constant of the acid. Any of these quantities can be cal- culated by setting up an equilibrium table like that in Toolbox 51.1. PROCEDURE Step 1 Write the chemical equation and the expression for K₂ for the proton transfer equilibrium. Set up a table with columns labeled by the acid (HA), H₂O*, and the conjugate base of the acid (A). In the first row below the headings, show the initial concentration of each species. For this step, assume that no acid molecules have been deprotonated. In the second row, write the changes in the concentration needed for the reaction to reach equilibrium. Assume that the concentra- tion of the acid decreases by x mol-L¹ as a result of deprotonation. The reaction stoichiometry gives us the other changes in terms of x. In the third row, write the equilibrium concentration for each substance by adding the change in its concentration (line 2) to its initial concentration (line 1). Acid, HA H₂O 0 +x x [HA]Initial -X [HA] initial - X Conjugate base, A initial concentration change in concentration equilibrium concentration Although a change in concentration may be positive (an increase) or negative (a decrease), the value of the concentration itself must always be positive. 0 +x Step 2 Substitute the equilibrium concentrations into the expression for Ka Step 3 Solve for the value of x, which (from line 3) gives [H₂O*]. The calculation of x can often be simplified, as shown in Toolbox 51.1, by ignoring changes of less than 5% of the initial concentra- tion of the acid. A simple way to anticipate that the approximation can be used is to compare the values of Ka and the initial con- centration of weak acid. If the numerical value of [HA]inmal is at least two orders of magnitude greater than the value of K₂ (is more than 10² times as large), then the approximation is probably valid. However, at the end of the calculation, check that x is consistent with the approximation, by calculating the percentage of acid de- protonated. If this percentage is greater than about 5%, then the exact expression for K, must be solved for x. An exact calculation requires solving a quadratic equation. If the pH is greater than 6 (but less than 7), the acid is so dilute or so weak that the autopro- tolysis of water contributes significantly to the pH. In such cases, use the procedures described in Topic 6F, which take autoprotoly- sis into account. The contribution of the autoprotolysis of water in an acidic solution can be ignored only when the calculated H₂O* concentration is substantially (about 10 times) higher than 107 mol-L¹, corresponding to a pH of 6 or less. Step 4 Calculate the pH from pH = -log [H₂O*]. Although pH should be calculated to the number of significant figures appropriate to the data, the answers are often considerably less reliable than that. One reason for this poor reliability is that in- teractions between the ions in solution are not taken into account. This procedure is illustrated in Examples 6D.1 and 6D.2.