Question

Calculate the \(\left[\mathrm{H}^{+}\right]\)and pH of a buffer solution that is \(0.20 M\) in \(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) and contains sufficient sodium acetate to make the \(\left[\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}^{-}\right.\)] equal to \(0.20 M\). ( \(K_{\mathrm{a}}\) for \(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}=1.8 \times 10^{-5}\) )

Short answer

The pH of the buffer solution is equal to the pKa of acetic acid. After calculating pKa, pH is found to be approximately 4.74. The concentration of hydrogen ions \( [\mathrm{H}^{+}] \) is \( 1.8 \times 10^{-5} M \).

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

For buffer solutions, the Henderson-Hasselbalch equation relates the pH of the solution to the pKa of the acid and the concentrations of the acid and its conjugate base. It is given by: \( pH = pKa + \log \left( \frac{[\text{Conjugate base}]}{[\text{Acid}]} \right) \). In this case, the conjugate base is acetate ion \( \mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^{-} \) and the acid is acetic acid \( \mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2} \).

Calculate the pKa

The pKa is the negative logarithm of the Ka value. Calculate it using the formula: \( pKa = -\log(Ka) \). For acetic acid, \( Ka = 1.8 \times 10^{-5} \), so \( pKa = -\log(1.8 \times 10^{-5}) \).

Use the Henderson-Hasselbalch Equation to find pH

Since the concentrations of the acid and its conjugate base are equal (both are \(0.20 M\)), the \( \log(\frac{[\text{Conjugate base}]}{[\text{Acid}]}) \) term becomes \( \log(1) \), which is 0. The pH will then be simply the pKa of the acid. Substitute the calculated pKa into the Henderson-Hasselbalch equation to find the pH.

Calculate the hydrogen ion concentration

The hydrogen ion concentration \( [\mathrm{H}^{+}] \) can be found from the pH using the formula: \( [\mathrm{H}^{+}] = 10^{-\text{pH}} \). Use the previously calculated pH value to find \( [\mathrm{H}^{+}] \).

Key concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is crucial for students venturing into the world of acid-base chemistry. At its most basic, this equation offers a way to estimate the pH of a buffer solution—a solution that resists changes in pH upon the addition of small amounts of acid or base.

The equation is given by: \[ pH = pKa + \log \left( \frac{[\text{Conjugate base}]}{[\text{Acid}]} \right) \].
When you see the term 'buffer solution', you can think of it as a mixture containing appreciable amounts of both an acid and its conjugate base, or vice versa. It's the balance of these two components that maintains the stability of a buffer's pH. For example, in a vinegar solution, the acid is acetic acid and its partner, the conjugate base, is the acetate ion.

pKa Calculation

Diving deeper, the Henderson-Hasselbalch equation incorporates the pKa value of the acid involved. So, what is pKa? It's a parameter that reflects the strength of an acid. Simply speaking, the lower the pKa value, the stronger the acid. To compute the pKa, you take the negative logarithm of the acid dissociation constant (Ka): \[ pKa = -\log(Ka) \].

This mathematical transformation provides us with a more digestible number to work with, rather than grappling with the often very small Ka values. In our exercise, since acetic acid has a Ka of \( 1.8 \times 10^{-5} \), it's just a matter of plugging this value into the pKa calculation to find the number we need for further calculations.

Hydrogen Ion Concentration

Following the establishment of pH, a natural progression is to explore the hydrogen ion concentration, represented as \( [\mathrm{H}^{+}] \). This figure is a direct measure of a solution's acidity—more hydrogen ions mean a lower pH and a more acidic solution.

To convert from pH back to hydrogen ion concentration, reverse logic is applied with the formula: \[ [\mathrm{H}^{+}] = 10^{-\text{pH}} \].
Calculation of the hydrogen ion concentration is a straightforward application of this formula. Knowing \( [\mathrm{H}^{+}] \) is especially helpful for predicting how the solution will behave when interacting with other substances.

Acid-Base Chemistry

Finally, the underlying principles of acid-base chemistry contextualize the above concepts. An acid is a substance that donates hydrogen ions, and a base is a substance that accepts them. The dance between hydrogen ion donors and acceptors is the essence of acid-base reactions.

In buffer solutions, this dance is carefully choreographed to maintain equilibrium, thus keeping the pH relatively constant even in the face of added acids or bases. Understanding this equilibrium, and how the components of a buffer interact, is not only pivotal for mastering chemistry but also for practical applications, such as formulating blood buffers in medical treatments.