Question

A 20.0 -mL sample of 0.200 $\mathrm{M}$ HBr solution is titrated with 0.200 $\mathrm{M}\mathrm{NaOH}$ solution. Calculate the $\mathrm{pH}$ of the solution after the following volumes of base have been added: (a) 15.0 $\mathrm{mL}$ $(\mathbf{b})19.9\mathrm{mL},(\mathbf{c})20.0\mathrm{mL},(\mathbf{d})20.1\mathrm{mL},(\mathbf{e})35.0\mathrm{mL}$

Short answer

For each volume of NaOH added: (a) 15.0 mL: pH ≈ 1.14 (b) 19.9 mL: pH ≈ 1.99 (c) 20.0 mL: pH ≈ 7.00 (d) 20.1 mL: pH ≈ 12.04 (e) 35.0 mL: pH ≈ 12.87

Step-by-step solution

Calculate initial moles of HBr and NaOH

To calculate the initial moles of each substance, use the formula: Moles = molarity × volume For HBr, moles = $0.200\frac{mol}{L}\times 20.0mL$ Now, calculate the moles of NaOH for each given volume. (a) 15.0 mL: moles = $0.200\frac{mol}{L}\times 15.0mL$ (b) 19.9 mL: moles = $0.200\frac{mol}{L}\times 19.9mL$ (c) 20.0 mL: moles = $0.200\frac{mol}{L}\times 20.0mL$ (d) 20.1 mL: moles = $0.200\frac{mol}{L}\times 20.1mL$ (e) 35.0 mL: moles = $0.200\frac{mol}{L}\times 35.0mL$

Determine the limiting reactant

The limiting reactant is the one that will be completely used up in the reaction. In this case, the limiting reactant will be the one with fewer moles. For each lettered part of the problem, compare the moles of $\text{HBr}$ and $\text{NaOH}$ and determine which is the limiting reactant.

Calculate the final concentrations of HBr and NaOH

Once the limiting reactant is determined, subtract the moles of the limiting reactant from both reactants to find the final moles of each substance. Then, divide the final moles by the total volume of the solution to get the final concentration. Do this for each lettered part of the problem.

Determine the pH of the solution

The pH of a solution is given by: pH = $-\log [{\text{H}}^{+}]$ For each lettered part of the problem, calculate the ${\text{H}}^{+}$ concentration from the final concentrations of $\text{HBr}$ and $\text{NaOH}$, and then use the formula above to calculate the pH of the solution.

Key concepts

Acid-Base Titration

Acid-base titration is a laboratory method used to determine the concentration of an acid or a base in a solution. This process involves the gradual addition of a known solution, called the titrant, to the unknown solution until the chemical reaction between the acid and base is complete, which is often indicated by a color change with the use of an indicator or by reaching a pH that characterizes the equivalence point. In the case of our exercise, we are titrating a strong acid, hydrobromic acid (HBr), with a strong base, sodium hydroxide (NaOH). This reaction proceeds until all the HBr has reacted with NaOH to form water and sodium bromide (NaBr), which is a neutral salt.

Understanding the stoichiometry of the acid-base reaction is crucial for predicting the outcome at various stages of the titration. For example, when an equal molarity of HBr and NaOH are mixed in identical volumes, a complete neutralization occurs, which means the pH of the solution will be neutral (pH = 7 at 25°C). However, if we add less NaOH than HBr, the solution remains acidic, and conversely, if we add more NaOH than HBr, the solution becomes basic.

Limiting Reactant

In any chemical reaction, the limiting reactant is the one that is completely consumed first, preventing any further reaction from occurring. This concept is crucial in titration exercises, as it helps to identify the point at which all of one reactant has been used up in the reaction. In the exercise given, we compare the moles of HBr and NaOH to determine the limiting reactant at different stages of the titration. For instance, before the equivalence point, HBr is in excess and NaOH is the limiting reactant. At the equivalence point, the moles of HBr and NaOH are equal and thus, neither is in excess. If we were to go past the equivalence point, NaOH would be in excess and HBr would then be the limiting reactant. Knowing which reactant is limiting enables us to calculate the concentration of reactants or products at various stages of the titration process.

For efficient problem-solving, it is crucial to perform a mole-to-mole comparison based on the balanced chemical equation and the mole ratios. Understanding the concept of the limiting reactant is also key to predicting the pH of a solution in a titration.

Molarity

Molarity, often denoted by the symbol 'M', is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution. In the context of our titration exercise, we calculate initial molarities of HBr and NaOH to find out how many moles of each are present before the titration begins. Calculating molarity is a foundational step that allows us to understand the strength of the solutions and assists in predicting the course of the reaction.

To calculate the molarity, you use the formula: $M=\frac{moles}{liters}$. This calculation leads us to determine how many moles of acid and base we initially started with and subsequently, through titration, how the concentration changes. A fundamental grasp of molarity is necessary for students to properly engage with exercises involving chemical reactions, especially titrations where the concentrations of reactants are integral to determining the point of neutralization and the pH of the final solution.

pH Calculation

The pH scale is a measure of the acidity or basicity of an aqueous solution. It is calculated as the negative logarithm (base 10) of the hydrogen ion concentration (\text{H}^+). The pH scale typically ranges from 0 to 14, with 7 being neutral, values less than 7 indicating acidity, and values greater than 7 indicating basicity. In our exercise, after identifying the limiting reactant and calculating the final moles of HBr and NaOH, we derive the concentration of \text{H}^+ to calculate the pH at various stages of the titration.

For strong acids and bases, like HBr and NaOH, the concentration of \text{H}^+ or \text{OH}^- ions in the solution can be directly derived from their molarity after determining any excess reagent post-reaction. The ability to calculate pH is not only key to solving titration problems but also for understanding the behavior of solutions in a wide range of chemical and biological contexts. Students can use the pH formula, $pH=-\log [{\text{H}}^{+}]$, to quickly ascertain the nature of the solution at various points in an acid-base titration.