Calculate the \(\mathrm{pH}\) and \(\mathrm{pOH}\) of (a) a solution that is \(0.17 \mathrm{M} \mathrm{Na}_{2} \mathrm{HPO}_{4}\) (aq) and \(0.25 \mathrm{M} \mathrm{Na}_{3} \mathrm{PO}_{4}\) (aq); (b) a solution that is \(0.66 \mathrm{M} \mathrm{Na}_{2} \mathrm{HPO}_{4}(\mathrm{aq})\) and \(0.42 \mathrm{M}\) \(\mathrm{Na}_{3} \mathrm{PO}_{4}(\mathrm{aq}) ;\) (c) a solution that is \(0.12 \mathrm{M} \mathrm{Na}_{2} \mathrm{HPO}_{4}(\mathrm{aq})\) and \(0.12 \mathrm{M} \mathrm{Na}_{3} \mathrm{PO}_{4}(\mathrm{aq})\).

Understanding how to calculate the pH of a solution is essential when dealing with acid-base chemistry. The pH scale is a measure of how acidic or basic a solution is, ranging from 0 to 14. Solutions with a pH less than 7 are acidic, while those with a pH greater than 7 are basic or alkaline.

When you have a mixture of a weak acid and its salt with a strong base, such as in the given exercise with Na2HPO4 and Na3PO4, you're working with a buffer solution. In such cases, the pH can be found using the Henderson-Hasselbalch equation: \( \text{pH} = \text{pKa} + \text{log} \frac{[\text{A}^-]}{[\text{HA}]} \). Here, pKa is the acid's dissociation constant, [A^-] is the concentration of the base (the conjugate base of the acid), and [HA] is the concentration of the acid.

For example, with the given concentrations in part (a), you would substitute these values into the equation to find the pH. It's a straightforward process once you have the necessary pKa value and concentrations of the acidic and basic components of the buffer.

A buffer solution is like a chemical sponge, it resists changes in pH when small amounts of acid or base are added. They are made from a weak acid or a weak base and its corresponding salt. In this exercise, our buffer consists of a weak acid, HPO4^2-, and the salt of its conjugate base, PO4^3-.

Buffer solutions work because the weak acid and its salt establish an equilibrium that can shift to neutralize added acids or bases. The Henderson-Hasselbalch equation gives us a way to quantify the pH of these solutions based on the ratio of the concentrations of the acid and base forms.

Understanding Buffers

The ability of a buffer solution to maintain pH is based on the presence of both components: a weak acid that can react with added OH^- ions, and a salt that generates ions which can react with added H+ ions. This characteristic is vital in many biological systems and industrial applications where maintaining a constant pH is crucial.

Buffer solutions often involve mixtures of a weak acid and a salt formed from that acid and a strong base. This combination allows the buffer to stabilize the pH of the solution. Weak acids partially dissociate in water to form H+ ions and its conjugate base, while the salt completely dissociates, releasing anions that are the conjugate base of the acid.

In the scenario of Na2HPO4 and Na3PO4 mixtures, Na2HPO4 acts as the weak acid (HPO4^2-), while Na3PO4 provides the salt (PO4^3-) of the conjugate base. Since salts of weak acids and strong bases tend to be basic, their presence helps to establish the desired pH when used in the proper proportions with the corresponding weak acid.

Importance of the Conjugate Base

The role of the conjugate base is crucial because it is responsible for 'soaking up' H+ ions if acid is added, helping the buffer resist changes in pH. Similarly, if a base is added, the weak acid component in the mixture can donate H+ ions to counteract the increase in OH^- ions.

pOH is another important concept in acid-base chemistry. It is related to pH and is a measure of the hydroxide ion (OH^-) concentration in a solution. For any aqueous solution at 25°C, the sum of pH and pOH is always 14. This is an expression of the ion product constant for water.

Calculating the pOH of a solution can be done by subtracting the pH from 14 when the temperature is at standard conditions. If we have the pH, as calculated using the Henderson-Hasselbalch equation for a buffer system, we can easily determine the pOH. For example, if you calculate the pH of a buffer solution to be 8.2, the pOH would be \( 14 - 8.2 = 5.8 \).

Connection to pH

The relationship between pH and pOH is critical because it allows for the understanding of the concentration of both hydrogen ions and hydroxide ions in a solution. This is particularly useful when dealing with neutralization reactions or when needing to adjust the pH to a specific target in a chemical process.