Problem 112 - Chemistry and Chemical Reactivity

Question

A solution of hydrochloric acid has a volume of \(250 .\) mL and a pH of \(1.92 .\) Exactly \(250 .\) mL of \(0.0105 \mathrm{M} \mathrm{NaOH}\) is added. What is the \(\mathrm{pH}\) of the resulting solution?

Step-by-Step Solution

Answer

The pH of the resulting solution is approximately 3.12.

Step 1: Determine the initial concentration of hydrochloric acid

The initial pH of the hydrochloric acid solution is 1.92. We use the pH formula:\[ \text{pH} = -\log[\text{H}^+] \rightarrow [\text{H}^+] = 10^{-\text{pH}} \]Substituting the given pH:\[ [\text{H}^+] = 10^{-1.92} \approx 1.202 \times 10^{-2} \text{ M} \]

Step 2: Calculate the moles of HCl

The volume of the HCl solution is given as 250 mL, which is 0.250 L. Use the concentration from Step 1 to find the moles:\[ \text{Moles of HCl} = 1.202 \times 10^{-2} \text{ M} \times 0.250 \text{ L} \approx 3.005 \times 10^{-3} \text{ mol} \]

Step 3: Calculate the moles of NaOH

The NaOH solution has a concentration of 0.0105 M and a volume of 250 mL (0.250 L). Calculating the moles:\[ \text{Moles of NaOH} = 0.0105 \text{ M} \times 0.250 \text{ L} = 2.625 \times 10^{-3} \text{ mol} \]

Step 4: Determine the reaction between HCl and NaOH

HCl and NaOH will react in a 1:1 mole ratio:\[ \text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O} \]The moles of HCl present after the reaction will be:\[ \text{Moles of HCl leftover} = 3.005 \times 10^{-3} - 2.625 \times 10^{-3} = 0.380 \times 10^{-3} \text{ mol} \]

Step 5: Find the concentration of excess HCl

The total volume of the solution after mixing is 250 mL + 250 mL = 500 mL = 0.500 L.The concentration of excess HCl is:\[ [\text{HCl}] = \frac{0.380 \times 10^{-3} \text{ mol}}{0.500 \text{ L}} = 7.6 \times 10^{-4} \text{ M} \]

Step 6: Calculate the new pH of the solution

Using the formula for pH:\[ \text{pH} = -\log[\text{H}^+] = -\log(7.6 \times 10^{-4}) \approx 3.12 \]

Key Concepts

pH calculation

The concept of pH calculation is essential in understanding acid-base reactions. The pH scale operates as a measure of acidity or basicity of a solution. Ranging from 0 to 14, it utilizes the negative logarithm of the hydrogen ion concentration, \(\text{H}^+\).
The formula to find the pH is:

• \[ \text{pH} = -\log[\text{H}^+] \]

In our exercise, the initial pH of the hydrochloric acid solution is given as 1.92. By substituting into the pH formula, we find the \(\text{H}^+\) concentration. This concentration tells us how acidic the solution is before adding any base. pH calculations provide insights into the chemical properties and reactivity of the solution. Knowing how to calculate it is crucial for predicting the outcome of reactions involving acids and bases.

mole concept

Understanding the mole concept is vital for chemistry, particularly when dealing with reactions and concentrations. A mole is a unit that measures the amount of substance. It is often used to convert between the mass of a substance and the number of molecules or atoms it contains.
In the context of the given exercise, we calculate the moles of hydrochloric acid (HCl) and sodium hydroxide (NaOH). With \(250 \, \text{mL}\) of the solution, the concentration helps us find the number of moles:

• \[ \text{Moles} = \text{Concentration} \times \text{Volume} \]

This measure allows for accurate predictions about how much reactant is needed and how much product will be formed. Moles link the macroscopic world with molecular reality.

solution concentration

Solution concentration is a key concept in chemistry that describes how much solute is present in a specific volume of solvent. It is typically expressed as molarity (M), which is moles of solute per liter of solution. Understanding this helps us predict how substances will interact during reactions.
In the exercise example, the initial concentration of HCl is computed from the pH value. Later, the concentration of NaOH is given. These help in determining the number of moles for both chemicals, key for finding how much remains after they react.

• Initial concentration of HCl: \(1.202 \times 10^{-2} \, \text{M}\)
• Concentration of NaOH: \(0.0105 \, \text{M}\)

This information is crucial in deciding how the final pH is calculated after the reaction.

stoichiometry

Stoichiometry is the calculation method that determines how much reactant or product is involved in a reaction. It's based on the balanced chemical equation, which ensures that matter is conserved. Stoichiometry allows us to understand the quantities needed for a reaction and what you will obtain as a result.
In our case, the chemical equation \( \text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O} \) demonstrates a 1:1 mole ratio. Upon adding the NaOH to the HCl, the stoichiometry tells us how much HCl will react.

• Moles of HCl that reacted: \(2.625 \times 10^{-3} \, \text{mol}\)
• Moles of HCl left over: \((0.380 \times 10^{-3})\)

Understanding stoichiometry ensures the accuracy of predicted outcomes in reactions. It is the backbone for calculating the resulting concentrations and pH.