Question

What stoichiometric concentration of the indicated substance is required to obtain an aqueous solution with the pH value shown: (a) \(\mathrm{Ba}(\mathrm{OH})_{2}\) for \(\mathrm{pH}=11.88 ;(\mathrm{b})\) \(\mathrm{CH}_{3} \mathrm{COOH}\) in \(0.294 \mathrm{M} \mathrm{NaCH}_{3} \mathrm{COO}\) for \(\mathrm{pH}=4.52 ?\)

Short answer

The stoichiometric concentrations of the substances required to obtain the specified pH levels are as follows: (a) the concentration of Ba(OH)2 is approximately 0.0038 M, and (b) the concentration of CH3COOH is approximately 0.396 M.

Step-by-step solution

Convert pH to [OH-] for Ba(OH)2

Firstly, the given pH needs to be converted to a pOH using the formula pOH = 14 - pH. The value of pOH is then converted to the concentration of OH- ions using the equation [OH-] = \(10^{-\text{pOH}}.\) The concentration of Ba(OH)2 is half of [OH-] because one formula unit of Ba(OH)2 produces two OH- ions.

Determine the Base Concentration

Firstly, the pH value given is 11.88. This is converted to pOH using the formula pOH = 14 - pH, which gives pOH = 14 - 11.88 = 2.12. Then, the pOH is converted to [OH-] using the equation [OH-] = \(10^{-\text{pOH}}\), so [OH-] = \(10^{-2.12} \approx 0.0076 \, \text{M}.\) Finally, the concentration of Ba(OH)2 is half this amount because it gives two OH- ions, so [Ba(OH)2] = 0.0076/2 ≈ 0.0038 M.

Use Henderson-Hasselbalch Equation for CH3COOH

For the second part of the problem, we are dealing with a buffer solution because it is a mixture of a weak acid and its conjugate base. In such situations, we can use the Henderson-Hasselbalch equation, pH = pKa + log([base]/[acid]). Given information in this problem allows us to use this equation to solve for the unknown concentration of the acid.

Determine the Acid Concentration

Firstly, CH3COOH is acetic acid, which has a pKa value of 4.76. Our pH value in this case is 4.52, and the concentration of the base (NaCH3COO its conjugate base) is given as 0.294 M. Inserting these values into the Henderson-Hasselbalch equation, 4.52 = 4.76 + log([0.294]/[CH3COOH]). This equation can be rearranged to solve for [CH3COOH], giving [CH3COOH] = 0.294 x \(10^{(4.76 - 4.52)} \approx 0.396 \, \text{M}\).

Key concepts

pH and pOH Relationship

The pH scale is a measure of how acidic or basic a solution is, with values typically ranging from 0 (very acidic) to 14 (very basic), and a pH of 7 being neutral. The relationship between pH and pOH is a key concept in understanding the acidity or basicity of a solution.

According to the equation:
\[ \text{pH} + \text{pOH} = 14 \]
this relationship allows us to find either the pH or pOH if the other is known. For instance, in the above exercise, given the pH, we can calculate the pOH by subtracting the pH from 14. These values are logarithmic; meaning a small change in pH or pOH represents a large change in ion concentration. Once the pOH is found, it can be used to calculate the concentration of hydroxide ions (OH-) in the solution by using the formula:
\[ [\text{OH}^-] = 10^{-\text{pOH}} \]

Understanding this inverse log relationship is vital for students as it forms the basis for precise calculations involving acidic and basic solutions in chemistry.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is indispensable when dealing with the pH of buffer solutions. It connects pH with the concentrations of an acid and its conjugate base through the formula:
\[ \text{pH} = \text{pKa} + \log\left(\frac{[\text{base}]}{[\text{acid}]}\right) \]
Here, pKa is the acid dissociation constant, pH is the measure of acidity, [base] is the concentration of the conjugate base, and [acid] is the concentration of the acid. This equation, derived from the acid dissociation constant (Ka) equation, assumes that the acid is only partially dissociated, which is true for weak acids.

In our example, the concentration of acetic acid (\(\text{CH}_3\text{COOH}\)) is determined using its known pKa and the pH of the solution. Additionally, we’re given the concentration of its conjugate base, thus simplifying the use of the equation to solve for the unknown acid concentration.

Acid-Base Equilibrium

Acid-base equilibrium is a fundamental aspect of chemistry that refers to the balance between acids and bases in a solution. It is governed by the equilibrium constants for the dissociation of acids and bases into their respective ions. For an acid, its dissociation constant (Ka) measures the strength of the acid in solution. The equation for an acid in water is:
\[ \text{Ka} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]
where \(\text{H}^+\) is the concentration of hydrogen ions, \(\text{A}^-\) is the concentration of the conjugate base, and \(\text{HA}\) is the concentration of the undissociated acid. Stronger acids have larger Ka values, leading to a lower pH.

The interplay between acid and base concentrations dictates the solution's overall pH. This concept is crucial when preparing buffer solutions that must maintain a constant pH despite small additions of acid or base.

Buffer Solutions

Buffer solutions play a pivotal role in maintaining a stable pH in a system. A buffer typically consists of a weak acid and its conjugate base, giving it the ability to neutralize small amounts of added strong acid or base. The remarkable property of buffers arises from their components' reactions with the added substances that would otherwise shift the solution's pH significantly.

To create a buffer, a weak acid (like acetic acid) is often mixed with a salt containing its conjugate base (like sodium acetate). When an external substance that would usually change the pH is added to the buffer solution, the acid-base equilibrium adjusts to counter this effect. The balanced reaction ensures the pH remains relatively stable, making buffers essential in processes like biochemical reactions that are sensitive to pH changes.