Plot the dissociation fraction \(\alpha\) of \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(\mathrm{NH}_{4} \mathrm{OH}\) as a function of PH. Use pK \(=4.6\) for \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(\mathrm{pK}=9.3\) for \(\mathrm{NH}_{4} \mathrm{OH}\).

The relationship between pKa and pH is foundational in understanding the behavior of weak acids and bases in solution.
The pKa value of a substance is the negative logarithm of its acid dissociation constant (Ka).
It represents a specific pH at which the species is half dissociated.
When the pH of a solution equals the pKa of the acid, the concentrations of the acid (\text{HA}) and its conjugate base (\text{A-}) are equal.
This is an important point:

• If the pH is lower than the pKa, the environment is more acidic, and the acid remains mostly in its undissociated form (\text{HA}).
• If the pH is higher than the pKa, the environment is more basic, and the acid is mostly in its dissociated form (\text{A-}).

For the weak acid \text{CH\(_3\)COOH}, with a pKa of 4.6, a pH below 4.6 means less dissociation, while a pH above 4.6 means more dissociation.
Similarly, for the weak base \text{NH\(_4\)OH}, with a pKa of 9.3, it predominantly remains undissociated at pH levels below 9.3 and mostly dissociated at pH levels above 9.3.

The dissociation fraction, denoted as \text{alpha} (\text{alpha}), represents the fraction of the substance that has dissociated in solution.
For weak acids, the dissociation fraction \text{alpha} can be calculated using:
\text{\text{alpha} = \frac{1}{1 + 10^{(\text{pKa} - \text{pH})}}}
This formula tells us how much of the weak acid has dissociated based on the pH of the solution and its pKa.
For \text{CH\(_3\)COOH}, with a pKa of 4.6, the fraction \text{alpha} decreases as the pH falls below 4.6 and increases as the pH rises above 4.6.

In the case of weak bases, the dissociation fraction follows a similar formula but accounts for the reverse relationship between pH and pKa:
\text{\text{alpha} = \frac{1}{1 + 10^{(\text{pH} - \text{pKa})}}}
This version of the formula is used for \text{NH\(_4\)OH}, which has a pKa of 9.3.
As the pH increases above 9.3, the dissociation fraction grows, indicating higher dissociation, while at pH levels below 9.3, the dissociation fraction drops.

Weak acids and bases do not fully dissociate in water; instead, they reach an equilibrium state where the undissociated form coexists with the dissociated ions.
This partial dissociation is characterized by the acid dissociation constant (Ka) for acids and the base dissociation constant (Kb) for bases.
The strength of the acid or base and the extent of its dissociation can be assessed through these constants.
\text{CH\(_3\)COOH}, a weak acid, dissociates in water to form acetate ions (\text{CH\(_3\)COO-}) and hydrogen ions (\text{H+}).
\text{NH\(_4\)OH}, a weak base, dissociates to ammonium ions (\text{NH\(_4\)+}) and hydroxide ions (\text{OH-}).
For a given pH:

• The extent of dissociation depends on the comparison between the pH and the pKa of the acid or base.
• The dissociation fraction formula helps quantify this extent.
• Understanding these relationships allows for predicting and controlling the behavior of acids and bases in various chemical environments.

In practical applications like buffer solutions, this knowledge is crucial for maintaining a desired pH by choosing acids and bases with appropriate pKa values.