Question

A \(200-\mathrm{mL}\) volume of \(\mathrm{NaOH}\) solution was added to \(400 \mathrm{~mL}\) of a \(2.00 \mathrm{M} \mathrm{HNO}_{2}\) solution. The \(\mathrm{pH}\) of the mixed solution was 1.50 units greater than that of the original acid solution. Calculate the molarity of the \(\mathrm{NaOH}\) solution.

Short answer

The molarity of the NaOH solution is approximately 0.211 M.

Step-by-step solution

Calculate initial pH of HNO2 solution

The initial pH of the HNO2 solution can be found using the formula for weak acids:\[pH = 0.5(pK_a - \log [ ext{initial concentration}])\]Where pKa is the dissociation constant for HNO2. Using the given HNO2 molarity of \(2.00 \, M\), we identify that the specific pKa is approximately 3.25. Substituting these into the formula gives:\[pH_{ ext{initial}} = 0.5(3.25 - \log 2.00) \approx 1.35\]

Determine final pH of the mixed solution

We know the pH of the mixed solution increases by 1.50 units from the initial value of 1.35:\[pH_{ ext{final}} = pH_{ ext{initial}} + 1.50 = 1.35 + 1.50 = 2.85\]

Calculate moles of HNO2

Using the initial concentration of the HNO2 solution, calculate the number of moles:\[ ext{moles of HNO2} = 2.00 \, M \times 0.400 \, L = 0.800 \, ext{moles}\]

Find equilibrium H+ concentration at final pH

At pH 2.85, the hydrogen ion concentration can be calculated using:\[[H^+] = 10^{-2.85} \approx 1.41 \times 10^{-3} \, M\]

Use neutralization equation to find NaOH molarity

From the increase in pH, we assume that the NaOH neutralizes some HNO2 and calculate its moles using the reaction equation. Subtracting the equilibrium hydrogen ion concentration from initial gives NaOH moles:\[ ext{moles of } ext{NaOH} = ext{initial moles } HNO2 - ext{remaining moles } H^+\]Calculating remainder HNO2 is \[0.800 \, ext{moles} - rac{(1.41 \times 10^{-3}) \times (0.600)}{2} \approx 0.800 - 0.0422 \approx 0.758\]

Calculate molarity of the NaOH solution

Given the volume of NaOH solution used (\(0.200 \, L\)), calculate its molarity using the result from step 5:\[\text{Molarity of } ext{NaOH} = \frac{0.0422 \, ext{moles}}{0.200 \, L} = 0.211 \, M\]

Key concepts

pH Calculation

Understanding how to calculate the pH of a solution is essential in chemistry, especially when dealing with acids and bases. The pH scale, which ranges from 0 to 14, indicates the acidity or basicity of a solution. Lower values mean more acidic, while higher values mean more basic. Pure water has a neutral pH of 7. When working with weak acids like nitrous acid (HNO_2), the pH can be determined using a specific formula:\[pH = 0.5(pK_a - \log [\text{concentration}])\]In this exercise, the initial pH of the HNO_2 solution was calculated by using its known dissociation constant (pK_a) and the concentration of HNO_2. Knowing the starting pH helps us understand how the solution changes after more components, like NaOH, are introduced.

Neutralization Reaction

Neutralization reactions occur when an acid and a base react to form water and a salt, effectively neutralizing each other. In this exercise, the HNO_2 (an acid) reacted with NaOH (a base). This type of reaction can be represented by the formula:\[\text{Acid} + \text{Base} \rightarrow \text{Water} + \text{Salt}\]As NaOH is added to HNO_2, they react and reduce the concentration of hydrogen ions (H^+) in the solution, resulting in an increased pH. When the pH increased by 1.50 units, it indicated that the NaOH successfully neutralized a significant portion of the HNO_2. Understanding this helps us calculate the necessary NaOH concentration to achieve such a pH change.

Weak Acid

Weak acids, like nitrous acid ( HNO_2 ), do not completely disassociate in water, meaning they only partially release H^+ ions. This incomplete dissociation explains why their pH calculation is slightly more complex and involves the pK_a , a measure of the acid's strength. For weak acids, the equilibrium between the dissociated and undissociated forms is an important characteristic. This equilibrium impacts the ability of the weak acid to react in neutralization reactions effectively. Knowing how to handle weak acids is crucial in predicting how they will behave in various chemical reactions, such as neutralization with NaOH .

NaOH Solution

Sodium hydroxide ( NaOH ) is a common strong base used in neutralization reactions. It is known for completely dissociating in water to provide hydroxide ions ( OH^- ). These ions readily react with the H^+ ions from acids, such as HNO_2 , in a neutralization process. In this exercise, the goal was to find the molarity of the NaOH solution that would increase the pH of the initial HNO_2 solution by 1.50 units. The molarity was determined by calculating the amount of NaOH required to react with the given amount of HNO_2 and subtracting remaining H^+ ions at the new pH level. This understanding of NaOH in reactions allows chemists to design and predict the outcome of experiments and industrial processes.