Question

When \(1.0 \mathrm{~mL}\) of \(1.0 \mathrm{M} \mathrm{HCl}\) is added to \(50 . \mathrm{mL}\) of a buffer solution that is \(1.0 \mathrm{M}\) in \(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) and \(1.0 \mathrm{M}\) in \(\mathrm{NaC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\), the \(\left[\mathrm{H}^{+}\right.\)] changes from \(1.8 \times 10^{-5} M\) to \(1.9 \times 10^{-5} \mathrm{M}\). Calculate the initial \(\mathrm{pH}\) and the \(\mathrm{pH}\) change in the solution.

Short answer

The initial pH of the buffer solution is 4.74. The exact pH change calculation requires using a calculator to find the difference between the initial and final pH values, which are dependent on the final hydrogen ion concentration after adding HCl.

Step-by-step solution

Calculate the initial pH of the buffer solution

The initial pH of the buffer solution can be calculated using the Henderson-Hasselbalch equation, which is given by: \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \] For the acetic acid buffer system, the pKa value is approximately 4.74 (since acetic acid, \( \text{HC}_2 \text{H}_3 \text{O}_2 \), is being used). As the buffer is equimolar in acid (HA) and its conjugate base (A^-), the ratio \( \frac{[\text{A}^-]}{[\text{HA}]} \) is 1. Using this information: \[ \text{pH} = 4.74 + \log(1) \] Simplifying the log of 1, which is 0: \[ \text{pH} = 4.74 \]

Calculate the pH change

The change in hydrogen ion concentration \( \Delta [\text{H}^+] \) is the difference between the final and initial hydrogen ion concentrations. Given that the initial \( [\text{H}^+] \) is \( 1.8 \times 10^{-5} \text{M} \) and the final \( [\text{H}^+] \) is \( 1.9 \times 10^{-5} \text{M} \), calculate the change: \[ \Delta [\text{H}^+] = 1.9 \times 10^{-5} \text{M} - 1.8 \times 10^{-5} \text{M} = 1.0 \times 10^{-6} \text{M} \] The change in \( [\text{H}^+] \) can then be used to find the pH change. The pH change is the difference between the initial and final pH values, which can be calculated using the formula \( \text{pH} = -\log [\text{H}^+] \) for both the initial and final hydrogen ion concentrations.

Calculate the final pH of the solution

Using the same formula for pH calculation: \[ \text{pH}_{\text{final}} = -\log(1.9 \times 10^{-5}) \] To find the pH change, subtract the initial pH from the final pH: \[ \text{pH change} = \text{pH}_{\text{final}} - \text{pH}_{\text{initial}} \] Use a calculator to find the pH values and the pH change.

Key concepts

Henderson-Hasselbalch Equation

Understanding how to manage the pH of a buffer solution is critical when performing chemical reactions which are sensitive to acidity or alkalinity. The Henderson-Hasselbalch equation provides a straightforward way to do just that. This equation connects the pH of a solution with the concentration of an acid and its conjugate base, through the formula:
\[\begin{equation} \text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\end{equation}\]
Here, \( \text{pKa} \) is the negative logarithm of the acid dissociation constant, \( [\text{A}^-] \) is the concentration of the conjugate base, and \( [\text{HA}] \) is the concentration of the acid. When the acid and its conjugate base are at equal concentrations, the logarithmic term becomes zero because the log of one is zero. Therefore, the pH equals the pKa when the acid and its conjugate base are in equal molar concentrations, which is often the ideal point for a buffer.

Acid-Base Equilibrium

The state of balance between an acid and its conjugate base in a solution is known as acid-base equilibrium. It's an essential concept in understanding buffer solutions, as it relies on the reversible reaction between the acid (HA) and its conjugate base (A-). The acid dissociates to a certain extent to give protons (H+) and its conjugate base, concurrently the conjugate base can accept protons to reform the acid, establishing a dynamic equilibrium.

The strength and stability of a buffer solution depend on this equilibrium because the buffer can 'absorb' additions of acids or bases without a significant change in pH. This aspect was illustrated in our exercise where the addition of HCl, a strong acid, resulted in a relatively minor change in the hydrogen ion concentration and hence the pH.

Logarithms in Chemistry

Logarithms are a powerful mathematical tool used frequently in chemistry to deal with the wide range of concentrations and reaction rates. In the context of acids and bases, they are used to express the acid dissociation constant (Ka) as pKa, and hydrogen ion concentration ([H+]) as pH, with the p indicating the negative logarithm. That means:

\[\begin{equation}\text{pH} = -\log[\text{H}^+]\end{equation}\]
\[\begin{equation}\text{pKa} = -\log(\text{Ka})\end{equation}\]
The logarithmic scale provides a convenient way to represent these values, which can span many orders of magnitude, in a more handleable range, typically between 0 and 14 for pH. Using logarithms simplifies the process of calculating relative concentrations and understanding the chemistry behind buffer solutions.

pH and pKa Relationship

The pH and pKa values are closely related to each other. The pKa value is inherent to the properties of an acid, signifying the pH at which half of the acid is dissociated. At this point, the concentrations of the acid and its conjugate base are equal, leading to the buffer's maximum buffering capacity. Understanding the relationship between pH and pKa is vital to buffer solution preparation.

Knowing the pKa allows chemists to predict how an acid will behave at a certain pH level, and vice versa. When the pH is lower than the pKa, the acid form predominates, and when the pH is higher, the conjugate base form is more prevalent. This concept is especially important when you consider the addition of strong acids or bases to a buffer system, like in the given exercise, and predict how the pH will change.