Question

What volumes of \(0.50 \mathrm{M} \mathrm{HNO}_{2}\) and \(0.50 \mathrm{M} \mathrm{NaNO}_{2}\) must be mixed to prepare \(1.00 \mathrm{~L}\) of a solution buffered at \(\mathrm{pH}=3.55\) ?

Short answer

To prepare 1.00 L of a buffered solution at pH 3.55, mix 0.414 L of 0.50 M HNO2 and 0.586 L of 0.50 M NaNO2.

Step-by-step solution

Find the pKa of HNO2

First, we need to find the pKa of HNO2. The dissociation of HNO2 in water is given by: \[HNO_2 \rightleftharpoons H^+ + NO_2^-\] The acid dissociation constant, Ka, is: \[K_a = \frac{[H^+][NO_2^-]}{[HNO_2]}\] Taking the negative logarithm of both sides gives us: \[-\log K_a = -\log ([H^+][NO_2^-]/[HNO_2])\] The pKa is defined as: \[pK_a = -\log K_a\] For HNO2, the pKa is 3.4.

Set up the Henderson-Hasselbalch equation

Now that we have the pKa, we can set up the Henderson-Hasselbalch equation, which relates the pH, pKa, and the ratio of the conjugate acid/base concentrations: \[pH = pK_a + \log \frac{[NO_2^-]}{[HNO_2]}\] Given the desired pH is 3.55, substitute the values in the equation: \[3.55 = 3.4 + \log \frac{[NO_2^-]}{[HNO_2]}\]

Solve for the volume ratio

To solve for the volume ratio, we first rearrange the equation to isolate the concentration ratio: \[\log \frac{[NO_2^-]}{[HNO_2]} = 3.55 - 3.4\] \[\frac{[NO_2^-]}{[HNO_2]} = 10^{0.15}\] Let x be the volume of 0.50 M HNO2 required, and y be the volume of 0.50 M NaNO2. We need a total volume of 1.00 L, so x + y = 1. The concentrations of HNO2 and NaNO2 are equal to moles/volume: \[\frac{0.50x}{x+y} = 0.50\] Using the ratio equation derived earlier: \[\frac{[NO_2^-]}{[HNO_2]} = \frac{0.50y}{0.50x}\] \[10^{0.15} = \frac{y}{x}\] Substitute x + y = 1 and solve for x and y: \[x = \frac{1}{1 + 10^{0.15}} = 0.414 \mathrm{~L}\] \[y = 1 - x = 1 - 0.414 = 0.586 \mathrm{~L}\]

Provide the answer

The volumes of 0.50 M HNO2 and 0.50 M NaNO2 required to prepare 1.00 L of a buffered solution at pH 3.55 are 0.414 L and 0.586 L, respectively.

Key concepts

Acid Dissociation Constant (Ka)

Understanding the acid dissociation constant is essential when dealing with buffer solutions. The constant, denoted as Ka, provides a measure of an acid's strength in solution. It is calculated from the equilibrium concentrations of the acid (\( HA \)), its dissociated ions (\( H^+ \) and A^-), and the equilibrium equation:

\[K_a = \frac{[H^+][A^-]}{[HA]}\]
The stronger the acid, the higher the concentration of hydrogen ions in solution, resulting in a larger Ka value. For weak acids, like nitrous acid (\( HNO_2 \)), Ka values are typically small, indicating only a small portion of the acid dissociates.

In the case of preparing a buffer solution, knowing the Ka of the acid allows you to predict how the acid will behave in solution and how much acid and its conjugate base are needed to achieve a desired pH.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a critical formula used in buffer solution preparation, helping to relate the pH of the solution to the pKa of the acid and the ratio of the concentrations of its conjugate base and the acid. It is written as:

\[pH = pK_a + \text{log} \frac{[A^-]}{[HA]}\]
Where \( pK_a \) is the negative logarithm of the Ka value, and \( [A^-] \) and \( [HA] \) are the molar concentrations of the conjugate base and the acid, respectively. The equation simplifies the process of calculating the proportion of acid and conjugate base required to maintain a buffer solution at a certain pH. For a buffer at pH 3.55 with nitrous acid, you would adjust the ratio of nitrite ion (\( NO_2^- \) - the conjugate base) to the undissociated acid (\( HNO_2 \)) using this equation.

pH Calculation for Buffers

The process of pH calculation for buffer solutions hinges on the principles encapsulated by the Henderson-Hasselbalch equation. The pH is a scale used to quantify the acidity or basicity of an aqueous solution. Calculating the pH involves the logarithmic measurement of the hydrogen ion concentration:

\( pH = -\text{log} [H^+] \)
The pH scale ranges from 0 to 14, with 7 being neutral. pH values less than 7 indicate acidity, while values greater than 7 indicate alkalinity.

For buffer solutions, especially, maintaining a stable pH is crucial. By balancing the amounts of a weak acid and its conjugate base, you can achieve a buffer at a certain pH level. The prepared buffer can resist changes in pH when small amounts of acids or bases are added. During the preparation of a buffer solution, like one at a pH of 3.55 with nitrous acid, precise volumes of acid and its salt are calculated to create a stable environment and thus maintain the desired pH.