Question

Calculate the volume of \(1.50 \times 10^{-2} \mathrm{M} \mathrm{NaOH}\) that must be added to \(500.0 \mathrm{~mL}\) of \(0.200 \mathrm{M} \mathrm{HCl}\) to give a solution that has \(\mathrm{pH}=2.15\)

Short answer

To find the volume of \(1.50 \times 10^{-2} M\) NaOH solution that must be added to \(500.0 mL\) of \(0.200 M\) HCl to give a solution with a pH of 2.15, we carry out these main steps: (1) Calculate the concentration of H+ ions after addition of NaOH, (2) Calculate the moles of H+ ions, (3) Determine the moles of OH- ions needed to neutralize H+ ions, and (4) Calculate the volume of NaOH solution needed. After completing all steps, we find the volume of NaOH solution required to achieve the desired pH.

Step-by-step solution

Determine the concentration of H+ after adding NaOH

Since the pH is given as 2.15, we can calculate the concentration of H+ ions ([H+]) remaining in the solution after adding NaOH using the formula: \[pH = -\log\text{[H+]}\] Rearrange the formula to find [H+]: \[[H+] = 10^{-pH}\] Substitute the given pH value, and calculate the remaining concentration of H+ ions.

Calculate the moles of H+ ions

To calculate the volume of NaOH required to reach the desired pH, we first need to find the moles of H+ ions in the final solution. We can use the formula: \[moles\;of\;H+ = [H+]\times V_{final}\] Where the final volume of the solution, \(V_{final}\), is the sum of the initial volume of HCl solution and the volume of NaOH added. Note that since we don't know the volume of NaOH added yet, let's denote it as V_NaOH. Now we can write the equation: \[moles\;of\;H+ = [H+]\times(V_{initial}+V_{NaOH})\] Use the values of [H+] calculated in Step 1, and given initial volume of the HCl solution, 500.0 mL, to calculate moles of H+.

Calculate moles of OH- ions needed to neutralize H+ ions

To increase pH, we need to neutralize some of the H+ ions present, by adding the required amount of NaOH. The balanced chemical equation for the neutralization is: \[\mathrm{HCl + NaOH \rightarrow NaCl + H_2O}\] This shows that one mole of HCl reacts with one mole of NaOH to produce one mole of water and one mole of NaCl. Knowing the moles of H+ ions from Step 2, we can now calculate the moles of OH- ions needed. As the molar ratio between HCl and NaOH in the reaction is 1:1, the moles of OH- required to react with the moles of H+ ions is equal to the calculated moles of H+.

Calculate the volume of NaOH needed

Now, we can calculate the volume of NaOH needed to provide the required moles of OH- ions. We can use the formula: \[V_{NaOH} = \frac{moles\;of\;OH-}{[\mathrm{OH^-}]}\] where [\(\mathrm{OH^-}\)] is the concentration of the NaOH solution, and moles of OH- is the amount we calculated in Step 3. Use the given value of NaOH concentration, \(1.50 \times 10^{-2} M\), to calculate the volume of the NaOH solution. In conclusion, the calculated volume of NaOH solution in step 4 needs to be added to the 500.0 mL of 0.200 M HCl solution to achieve a pH of 2.15.

Key concepts

Neutralization Reaction

A neutralization reaction is a chemical process in which an acid and a base react to form water and a salt. In the equation given, hydrochloric acid (HCl) reacts with sodium hydroxide (NaOH), a strong base.
This neutralization reaction can be written as:- \[\text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O}\]This equation shows that one mole of HCl reacts with one mole of NaOH.
The balanced reaction implies that both the acid and base perfectly combine in equal moles to neutralize each other.

• The process involves the production of water, a common solvent, and NaCl, which remains in the solution as dissolved ions.
• Neutralization is fundamental in pH calculations because it helps determine how much base is needed to achieve a desired pH in an acidic solution.

Understanding the stoichiometry, or the ratio of reactants, is crucial in calculating the amount of base required for the neutralization reaction.

Molar Concentration

Molar concentration, also known as molarity, is the measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution (mol/L). It is a key concept in understanding reaction stoichiometry and is vital in the steps for calculating the volume of NaOH needed.
In our exercise, the concentration of the HCl solution is 0.200 M, meaning there are 0.200 moles of HCl in every liter of solution.

• Similarly, the concentration of NaOH is given as 1.50 × 10^{-2} M, which implies there are 0.0150 moles of NaOH in every liter of solution.
• Molar concentration helps determine how much of a base or acid is needed to achieve a particular pH level.

Molarity plays a crucial role in calculating moles from volume and vice versa, impacting the overall balance of a neutralization reaction.

Acid-Base Titration

Acid-base titration is a laboratory method used to determine the concentration of an unknown acid or base by gradually adding a reagent of known concentration until the reaction reaches its equivalence point.
In our scenario, NaOH is added to HCl until the solution reaches the desired pH of 2.15.

• The titration is complete when the amount of base added perfectly neutralizes a portion of the acid, achieving a specific pH level.
• It involves measuring the volume of the titrant (here, NaOH) required to reach the equivalence or endpoint.

This technique is widely used in chemistry to determine the concentration of a substance or to follow the course of a reaction, and it is directly applicable to this exercise since it involves calculating how much base is necessary to adjust the pH of an acidic solution.

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction reaches a state where the concentrations of reactants and products remain constant over time, meaning the forward and reverse reactions occur at the same rate. Although our problem focuses on reaching a specific pH, understanding equilibrium helps to appreciate the balanced interactions during the neutralization.
In the reaction between HCl and NaOH, reaching equilibrium means the reaction has progressed to a point where no additional net change occurs in the concentration of reactants and products.

• This balance involves both strong acids and bases, which typically react to completion, hence reaching a stable state quickly.
• However, understanding equilibrium concepts aids in recognizing how reactions progress and stabilize around specific concentrations.

Chemical equilibrium concepts underscore important principles governing reaction rates, and even though it may not seem directly applicable to simple titrations, it helps explain the consistent behavior of reactions.