Problem 105 - General Chemistry: Principles and Modern Applications

Question

Calculate the pH of the buffer formed by mixing equal volumes \(\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\right]=1.49 \mathrm{M} \quad\) with \(\quad\left[\mathrm{HClO}_{4}\right]=\) 1.001 M. \(K_{\mathrm{b}}=4.3 \times 10^{-4}\)

Step-by-Step Solution

Answer

The pH of the buffer solution is found to be approximately 8.92 after performing all calculations.

Step 1: Calculate the final concentrations

In our case, equal volumes of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) and \(\mathrm{HClO}_{4}\) are mixed together. Therefore, the final concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) will be 0.745 M and the final concentration of \(\mathrm{HClO}_{4}\) will be 0.5005 M.

Step 2: Identify the buffer reaction

The reaction between \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) and \(\mathrm{HClO}_{4}\) can be written as: \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2} + \mathrm{HClO}_{4} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3}^{+} + \mathrm{ClO}_{4}^{-}\). During this reaction, \(\mathrm{HClO}_{4}\) reacts completely with \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\), forming the \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3}^{+}\) ion.

Step 3: Calculate new concentrations after reaction

After the reaction, the final concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3}^{+}\) will be 0.5005 M and the remaining concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) will be 0.2445 M.

Step 4: Apply the Henderson-Hasselbalch equation

We can calculate pH by using the Henderson-Hasselbalch equation: \(pH = pK_{b} + \text{log} \left( \frac{[base]}{[acid]} \right)\). Here, \(pK_{b} = -\text{log} \left(4.3 \times 10^{-4}\right)\), [base] is the remaining concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) 0.2445 M, and [acid] is the concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3}^{+}\) 0.5005 M.

Step 5: Calculate pH

Substituting the given values into the equation, we will get the pH value of the buffer solution.

Key Concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a straightforward tool used to determine the pH of buffer solutions. It's especially efficient when dealing with weak acid-base equilibria. The equation is written in terms of pKa or pKb, which are the negative logarithms of the acid or base dissociation constants, respectively. For buffers that involve weak bases and their conjugate acids, the equation is:
\[ pH = pK_{b} + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) \]The strength of this equation lies in its ability to give you the pH directly from known concentrations of a base and its conjugate acid—or vice versa. For accurate results, make sure the ratio of base to acid isn't too high or low. An optimal buffer capacity generally occurs when the ratio is close to 1, meaning that the concentrations of the acid and base are similar or equal.

• pH: A measure of the acidity or basicity of a solution.
• pKa/pKb: Key components reflecting the acid/base strength.
• [Base]: Concentration of the weak base in the buffer.
• [Acid]: Concentration of the conjugate acid.

Understanding and using this equation can help with predicting how buffers will respond to pH changes upon adding small amounts of acid or base.

buffer solution

A buffer solution is an amazing chemical concoction that helps maintain a stable pH when acids or bases are added to it. Buffers are crucial in many physiological and industrial processes where maintaining a particular pH range is essential. They are typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid.
Let's break it down:

• Conjugate Pairs: A weak acid or base paired with its conjugate partner is the backbone of a buffer.
• Resilience: Buffers resist pH changes when small amounts of acid or base are added because of the equilibrium established between the conjugate pairs.
• Application: Buffers are used in biological systems like blood and in various chemical formulations.

In the exercise, our buffer comprises \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2} \), a weak base, and its conjugate acid \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3}^{+} \) formed after reacting with \( \mathrm{HClO}_{4} \). This combination ensures that the pH remains relatively constant under different conditions.

pH calculation

Calculating pH is an essential skill in chemistry, especially when it comes to buffering solutions. pH is a scale that ranges from 0 to 14 and indicates the acidity or basicity of a solution, where lower values are more acidic and higher values are more basic.
In the buffer solution exercise:

• We first calculated the concentrations of the components after the reaction.
• The Henderson-Hasselbalch equation was then applied with the values of the conjugate base and acid.
• Finally, by substituting these values into the equation, we obtained the pH of the buffer.

A simple and consistent approach like this makes pH calculation manageable even when dealing with complex solutions. Remember:

• Equilibrium Consideration: Always ensure that the solution has reached equilibrium before calculation.
• Temperature Influence: Temperature can affect the dissociation constants and, therefore, slightly influence pH values.

By understanding these principles, determining pH can be much clearer and more predictable.