- Determine the moles of sulfuric acid: \(n_{H_2SO_4} = 2.00 \text{ M} \times 0.25 \text{ L} = 0.50 \text{ mol}\).
- Apply reaction stoichiometry to find the moles of sodium hydroxide: \(n_{NaOH} = 2 \times 0.50 \text{ mol} = 1.0 \text{ mol}\).
- Calculate the concentration of the sodium hydroxide solution: \(M_{NaOH} = \frac{1.0 \text{ mol}}{2.00 \text{ L}} = 0.50 \text{ M}\).
- Conclude that the correct concentration is \(\boxed{0.50 \text{ M}}\).
Explanation
- Problem Analysis and Data Overview
To determine the concentration of the NaOH solution, we first analyze the provided chemical reaction:
\[2 NaOH + H_2SO_4 \rightarrow Na_2SO_4 + 2 H_2O\]
From this balanced equation, we observe the stoichiometry of the reaction: 1 mole of sulfuric acid (\(H_2SO_4\)) reacts completely with 2 moles of sodium hydroxide (\(NaOH\)). We are given that \(V_{H_2SO_4} = 0.25 \text{ L}\), \(M_{H_2SO_4} = 2.00 \text{ M}\), and \(V_{NaOH} = 2.00 \text{ L}\). We need to find the molarity (\(M\)) of the \(NaOH\) solution, which is defined as the number of moles of solute divided by the volume of the solution in liters, \(M = \frac{n}{V}\).
- Calculation of NaOH Concentration
First, calculate the number of moles of \(H_2SO_4\) that were used in the neutralization:
\[n_{H_2SO_4} = M_{H_2SO_4} \times V_{H_2SO_4} = 2.00 \text{ mol/L} \times 0.25 \text{ L} = 0.50 \text{ mol}\]
According to the balanced chemical equation, 2 moles of \(NaOH\) react for every 1 mole of \(H_2SO_4\). Therefore, the number of moles of \(NaOH\) required for this reaction is:
\[n_{NaOH} = 2 \times n_{H_2SO_4} = 2 \times 0.50 \text{ mol} = 1.0 \text{ mol}\]
Now, we can determine the concentration of the \(NaOH\) solution by dividing the moles of \(NaOH\) by its total volume:
\[M_{NaOH} = \frac{n_{NaOH}}{V_{NaOH}} = \frac{1.0 \text{ mol}}{2.00 \text{ L}} = 0.50 \text{ M}\]
Comparing this result to the given options, we find that the concentration of the \(NaOH\) solution is \(0.50 \text{ M}\), which corresponds to option b.
Examples
In real life, titration is a common laboratory technique used to determine the concentration of an unknown solution. For example, if a food scientist needs to determine the acidity (concentration of acetic acid) in a batch of vinegar, they would perform a neutralization reaction using a known base, such as sodium hydroxide. By measuring the exact volume of base needed to neutralize a specific volume of the acid, the scientist can calculate the concentration of the acid, ensuring the product meets quality and safety standards. This proportional relationship between reactants is fundamental to chemistry and quality control in manufacturing.