Question

What quantity (moles) of \(\mathrm{HCl}(g)\) must be added to \(1.0 \mathrm{~L}\) of 2.0 \(M \mathrm{NaOH}\) to achieve a pH of \(0.00 ?\) (Neglect any volume changes.)

Short answer

To achieve a pH of \(0.00\) in \(1.0 \mathrm{~L}\) of 2.0 \(M \mathrm{NaOH}\), you need to add \(3.0 \mathrm{~moles}\) of \(\mathrm{HCl}(g)\).

Step-by-step solution

Calculate initial moles of \(\mathrm{NaOH}\)

The initial moles of \(\mathrm{NaOH}\) can be calculated using the initial volume and molarity. Initial moles of \(\mathrm{NaOH}\) = Molarity × Volume Initial moles of \(\mathrm{NaOH}\) = \(2.0M \times 1.0L = 2.0\) moles

Calculate the final concentration of hydrogen ions \(([H^+])\)

Since we want a pH of \(0.00\), we will find out the final concentration of hydrogen ions using the pH formula: \(pH = -\log[H^+]\) We need to rearrange the formula to solve for \([H^+]\): \([H^+] = 10^{-pH}\) Hence, \([H^+] = 10^{-0} = 1 \mathrm{M}\)

Find moles of \(H^+\) ions required to achieve the desired pH

Now we will find the moles of \(H^+\) ions required to achieve the desired pH using the final concentration of \(H^+\) ions and the final volume of the solution: Moles of \(H^+\) ions = Concentration × Volume Moles of \(H^+\) ions = \(1 \mathrm{M} \times 1.0 \mathrm{L} = 1.0\) moles

Determine moles of \(HCl\) required to neutralize the given \(\mathrm{NaOH}\)

In order to neutralize the \(2.0\) moles of \(\mathrm{NaOH}\), we need an equal number of moles of \(\mathrm{HCl}\) (both being strong acid and strong base; they neutralize each other): Neutralization reaction: \(\mathrm{NaOH + HCl \rightarrow NaCl + H_2O}\) Moles of \(\mathrm{HCl}\) required for neutralization = Moles of \(\mathrm{NaOH}\) Moles of \(\mathrm{HCl}\) required for neutralization = \(2.0\) moles

Calculate the total moles of \(\mathrm{HCl}\) to be added

Now, we need to add the moles of \(\mathrm{HCl}\) required for neutralization and the moles of \(H^+\) ions required to achieve the desired pH: Total moles of \(\mathrm{HCl}\) = Moles of \(\mathrm{HCl}\) required for neutralization + Moles of \(H^+\) ions required for desired pH Total moles of \(\mathrm{HCl}\) = \(2.0 \mathrm{~moles} + 1.0 \mathrm{~moles} = 3.0 \mathrm{~moles}\) Therefore, \(3.0 \mathrm{~moles}\) of \(\mathrm{HCl}(g)\) must be added to \(1.0 \mathrm{~L}\) of 2.0 \(M \mathrm{NaOH}\) to achieve a pH of \(0.00\).

Key concepts

pH calculation

pH is a measure of how acidic or basic a solution is. It is calculated using the formula:\[pH = -\log[H^+]\]Where

• \([H^+]\) is the concentration of hydrogen ions in the solution.
• pH values less than 7 indicate an acidic solution, a value of 7 is neutral, and values greater than 7 are basic.

To find the concentration of hydrogen ions based on the desired pH, you can rearrange the formula:\[[H^+] = 10^{-pH}\]For example, if you need a pH of 0.00, the hydrogen ion concentration becomes:\[[H^+] = 10^{-0} = 1 \text{ M}\]This means you need a 1 mole per liter concentration of hydrogen ions to achieve this acidity.

molarity and volume

Molarity is a way to express the concentration of a solution in terms of the amount of solute per volume of solution. It's defined as:\[\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}}\]In many chemical calculations, combining molarity with the volume of the solution helps you find out how many moles of a solute you have. For instance, if you have a solution with a molarity of \(2.0 \text{ M NaOH}\) and a volume of \(1.0 \text{ L}\), you can calculate the initial moles of sodium hydroxide (NaOH) involved:\[\text{Moles of } NaOH = 2.0 \text{ M} \times 1.0 \text{ L} = 2.0 \text{ moles}\]This principle is essential for planning reactions, such as ensuring enough neutralization can occur when an acid is introduced.

neutralization reaction

A neutralization reaction occurs when an acid and a base react to form water and salt, effectively cancelling each other's reactive properties. The reaction can be represented as:\[\text{Base (e.g., NaOH) + Acid (e.g., HCl) } \rightarrow \text{ Salt (e.g., NaCl) + Water } (H_2O)\]In our example, \(\text{HCl}(g)\) is added to \(\text{NaOH}\), a strong acid reacting with a strong base, producing sodium chloride and water.

• The goal is to reach a state where there is no excess of hydrogen ions \([H^+]\) or hydroxide ions \([OH^-]\) left in the solution.
• The number of moles of hydrochloric acid required is equal to the moles of sodium hydroxide, due to their 1:1 molar ratio in the balanced equation.
• In our specific problem, \(2.0 \text{ moles of } HCl\) are needed to neutralize \(2.0 \text{ moles of } NaOH\).

After neutralizing NaOH, additional HCl is added to achieve the desired pH, illustrating the thorough application of stoichiometric principles to achieve specific chemical properties in a solution.