The dissociation constant of acetic acid is 0.000018 and...

The dissociation constant of acetic acid is $0.000018$ and that for cyanoacetic acid is $0.0036$ at $298 \mathrm{~K}$. What would be the ratio of volumes of the two acid solutions, each containing equal moles of the acids, so that the solutions becomes isohydric? (a) $1: 1$ (b) $1: \sqrt{200}$ (c) $1: 200$ (d) $200: 1$

The dissociation constant of acetic acid is $0.000018$ and that for cyanoacetic acid is $0.0036$ at $298 \mathrm{~K}$. What would be the ratio of volumes of the two acid solutions, each containing equal moles of the acids, so that the solutions becomes isohydric?
(a) $1: 1$
(b) $1: \sqrt{200}$
(c) $1: 200$
(d) $200: 1$

Key Concepts

Relationship Between Concentration, Volume, and Number of Moles

The concentration of an acid in a solution is defined as the number of moles per unit volume (C = n/V). When two acid solutions contain equal moles of acid yet need to achieve isohydric conditions, the volumes must be adjusted so that the resulting concentrations reflect the differences in acid strength. This relationship allows one to relate the volume ratio directly to the ratio of the concentrations needed to equalize the pH, as derived from the weak acid equilibrium approximation.

Isohydric Solutions

Isohydric solutions are those that have the same pH despite potentially having different acid compositions or concentrations. Achieving isohydric conditions requires adjusting the concentrations of the acids (and thus the volumes when moles are constant) so that the effects of different pKa values are counterbalanced. In this problem, matching pH values leads to a relationship between the acid concentrations of acetic acid and cyanoacetic acid, taking into account their respective pKa values.

Acid Dissociation Constant (Ka) and pKa

The acid dissociation constant (Ka) quantifies the extent to which an acid dissociates in solution, while pKa, defined as the negative logarithm of Ka, provides a more convenient scale for acid strength. A lower pKa indicates a stronger acid, as seen in the comparison between acetic acid and cyanoacetic acid in the problem. Understanding these parameters is fundamental in predicting the behavior of acids in solution and comparing their relative strengths.

Weak Acid Equilibrium and pH Approximation

For weak acids that only partially dissociate, an approximation can be used where the hydrogen ion concentration [H+] is given by the square root of the product of the acid dissociation constant and the acid concentration (i.e., [H+] = ?(Ka · C)). This leads to an approximate expression for the pH of a weak acid solution, pH ? 0.5(pKa ? log C). This approximation highlights the interplay between an acid's strength and its concentration in determining the solution’s pH.

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