You are given \(250.0 \mathrm{mL}\) of \(0.100 \mathrm{M} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COOH}\) (propionic acid, \(\left.K_{\mathrm{a}}=1.35 \times 10^{-5}\right) .\) You want to adjust its \(\mathrm{pH}\) by adding an appropriate solution. What volume would you add of (a) \(1.00 \mathrm{M} \mathrm{HCl}\) to lower the \(\mathrm{pH}\) to \(1.00 ;\) (b) \(1.00 \mathrm{M} \mathrm{NaCH}_{3} \mathrm{CH}_{2} \mathrm{COO}\) to raise the \(\mathrm{pH}\) to \(4.00 ;(\mathrm{c})\) water to raise the \(\mathrm{pH}\) by 0.15 unit?

Understanding the behavior of weak acids is fundamental in chemistry, especially when calculating pH levels. Unlike strong acids that fully dissociate in water, weak acids do not completely ionize. This partial ionization is characterized by an equilibrium between the undissociated acid (HA) and its ions (H+ and A-). This process can be represented by an equilibrium constant known as the acid dissociation constant, Ka.
For example, with propionic acid (CH3CH2COOH), its ionization in water can be expressed as:
\[ CH3CH2COOH (l) + H2O (l) \rightleftharpoons CH3CH2COO^- (aq) + H^+ (aq) \]
The equilibrium constant for this reaction, Ka, can be calculated using the concentrations of the products and reactants at equilibrium. This constant is critical for calculating the initial pH of a weak acid solution and understanding how the pH will change when other substances are added.

The Henderson-Hasselbalch equation establishes a relationship between pKa, pH, and the ratio of the concentration of an acid to its conjugate base. This equation greatly simplifies the process of calculating pH for buffer solutions, which consist of a weak acid and its conjugate base.
The equation is given as:
\[ pH = pKa + log \left( \frac{\text{[A^-]}}{\text{[HA]}} \right)\]
This relationship allows for the determination of the pH of a solution when the concentrations of an acid and its conjugate base are known or can be calculated. For instance, to raise the pH of a weak acid solution, one can add its conjugate base. The change in pH can be calculated by determining the new ratio of the base to the acid and then using the Henderson-Hasselbalch equation.

Acid-base stoichiometry involves the calculations based on the quantitative relationships between the amounts of reactants and products in an acid-base reaction. Knowing the molarity and volume of acidic and basic solutions allows for precise calculations of how much of one solution is needed to react completely with another.
When dealing with strong acids like HCl, the stoichiometry is straightforward since they fully dissociate into H+ and Cl-. To find the volume of HCl needed to lower the pH of a weak acid solution to a certain level, we apply the stoichiometry of the neutralization reaction:
\[ HA + HCl \to A^- + H_2O \]
Through this stoichiometric relationship, we calculate how much strong acid is required to achieve the desired pH, taking into account the concentration of hydrogen ions at that pH level.

The pH of a solution is a measure of its acidity or alkalinity, expressed as the negative logarithm of the hydrogen ion concentration. For any acid, pKa is the negative logarithm of the acid dissociation constant, Ka. The relationship between pH and pKa is important for understanding how a solution's pH changes with the addition of acid or base.
A solution where the pH is equal to the pKa of the weak acid is where the concentrations of the acid and its conjugate base are equal. This point represents the maximum buffering capacity of the solution. Knowing this relationship helps in designing buffer solutions and predicting how the buffer will respond to the addition of acids or bases. For instance, adjusting the pH to a certain level involves adding substances that will shift the ratio of the weak acid to its conjugate base, thereby changing the pH according to the Henderson-Hasselbalch equation.

Molarity (M) is the number of moles of solute per liter of solution, and it plays a crucial role in the dilution and concentration of solutions. The concept of molarity is essential when performing dilutions, which involve adding solvent to a solution to decrease the concentration of solutes.
The molarity and volume relationship is captured in the dilution equation:
\[ M_1V_1 = M_2V_2 \]
where \(M_1\) and \(M_2\) represent the initial and final molarities, respectively, while \(V_1\) and \(V_2\) represent the initial and final volumes. This equation is particularly useful when calculating how much water to add to a solution to change its pH, as in the case of raising the pH of the propionic acid solution by a small amount. By applying this dilution principle, one can determine the final volume of the solution needed to achieve the targeted molarity, and as a result, the desired pH.