The \(p K_{a}\) of acetic acid and \(p K_{b}\) of ammonium hydroxide are \(4.70\) and \(4.75\) respectively. Calculate the hydrolysis constant of ammonium acetate at \(298 \mathrm{~K}\) and also the degree of hydrolysis and \(\mathrm{pH}\) of its (a) \(0.01 M\) and (b) \(0.04 \mathrm{M}\) solutions.

To fully grasp the framework of hydrolysis constant calculations, one must first understand ionization constants, key players in this chemical plot. The ionization constants, often designated as 'Ka' and 'Kb', are values that measure the strength of an acid or a base in water, respectively. The lower the value of Ka or Kb, the weaker the acid or base.

These constants are used to predict the extent of ionization in solution; they are equilibrium constants for the acid and base dissociation reactions respectively. Essentially, they tell us how much of an acid or base will dissociate into ions in water. In a mathematical cloak, the relationships are thus: for an acid, HA dissociates into H+ and A-, represented by the equilibrium constant Ka = [H+][A-]/[HA]. Meanwhile, for a base BOH dissociating into B+ and OH-, Kb = [B+][OH-]/[BOH].

Moreover, the 'pKa' and 'pKb' are logarithmic measures of these constants, making it easier to work with the typically very small values of Ka and Kb. The equations connecting these are given by 'pKa = -log(Ka)' and 'pKb = -log(Kb)'. These constants are invaluable keys, unlocking the door to further calculations such as hydrolysis constants and pH levels in solutions of weak acids and bases.

Diving into the ocean of pH calculation, this is a crucial process in the study of solution chemistry. The pH is a measure of the acidity or basicity of an aqueous solution. It is defined as the negative logarithm of the hydrogen ion concentration: pH = -log[H+].

For weak acid or base solutions, calculating pH isn't a straightforward affair, as these weak electrolytes only partially dissociate in water. This is where the ionization constants and the degree of hydrolysis slink back into the picture. For the case of ammonium acetate solution discussed in our textbook exercise, the equation includes both the hydrolysis constant and the concentration of the solution.

For such buffered solutions, where hydrolysis plays a role, the pH can be derived from the expression pH = 7 + 1/2 (pKb - pKw + log C), invoking the ionic constants indirectly through the pKw and reflecting the concentration (C) and the ionization behavior of the salt's constituent ions.

Lastly, we unravel the degree of hydrolysis, which is a measure of the extent to which a salt's ions react with water to form the corresponding acid or base. Hydrolysis can be viewed as a dance of ions and water molecules, resulting in either an acidic or basic solution, depending on the nature of the salt.

The degree of hydrolysis (h) is expressed as a fraction of the salt that is hydrolyzed and is calculated using the formula h = √(Kh/C), where 'Kh' is the hydrolysis constant and 'C' is the concentration of the solution. Its value ranges from 0 (no hydrolysis) to 1 (complete hydrolysis), and it plays a pivotal role in the pH of the solution.

In the textbook solution, we gather that ammonium acetate undergoes hydrolysis in water to form acetic acid and ammonium hydroxide. By plugging in the calculated hydrolysis constant into the degree of hydrolysis formula, for each given concentration of solution, one can fathom the pH of this salt solution, completing the circle of understanding in this chemical conundrum.