Problem 90 - Chemistry: Principles and Reactions

Question

A student prepares \(455 \mathrm{~mL}\) of a KOH solution, but neglects to write down the mass of KOH added. His TA suggests that he take the \(\mathrm{pH}\) of the solution. The \(\mathrm{pH}\) is \(13.33 .\) How many grams of KOH were added?

Step-by-Step Solution

Answer

Answer: The student added approximately 5.478 grams of KOH to the solution.

Step 1: Calculate hydroxide (OH-) ion concentration

Since we know that the pOH + pH = 14 and the pH value is given as 13.33, we can calculate the pOH value: pOH = 14 - pH = 14 - 13.33 = 0.67 Now, we can find the hydroxide ion concentration (OH-) using the pOH: (OH-) = 10^(-pOH) = 10^(-0.67) = 0.2149 M

Step 2: Calculate the molarity of the KOH solution

KOH is a strong base, and it dissociates completely in water to form K+ and OH- ions. Since the stoichiometric ratio of KOH to OH- is 1:1, the molarity of KOH is the same as the concentration of OH- ions: [KOH] = 0.2149 M

Step 3: Determine the moles of KOH added

We know the volume of the solution (455 mL) and the molarity of KOH ([KOH] = 0.2149 M). We can calculate the moles of KOH added by multiplying the molarity of KOH by the volume of the solution in liters: moles of KOH = [KOH] * volume = 0.2149 M * (455/1000) L = 0.09768 mol

Step 4: Calculate the mass of KOH

To find the mass of KOH added, multiply the moles of KOH by its molar mass. The molar mass of KOH is approximately 56.11 g/mol: mass of KOH = moles of KOH * molar mass = 0.09768 mol * 56.11 g/mol = 5.478 g Therefore, the student added approximately 5.478 grams of KOH to the solution.

Key Concepts

pH Calculations

The pH scale is a crucial element in understanding acid-base equilibria. It measures the acidity or basicity of a solution. The pH value is derived from the concentration of hydrogen ions (\(\text{H}^+\)) present in solution. In calculating pH, we use the formula:
\[\text{pH} = -\log[\text{H}^+]\]
In the given problem, you calculate the hydroxide concentration indirectly by using the relationship between pH and pOH as follows:

• The sum of pH and pOH always equals 14 (at 25°C), which gives us the equation: \[\text{pH} + \text{pOH} = 14\]
• With the pH given as 13.33, you can find pOH: \[\text{pOH} = 14 - 13.33 = 0.67\]

The pOH provides information on the basicity of the solution, helping determine the hydroxide ion concentration, which is crucial for calculating the amount of a base like KOH.

Molarity Determination

Molarity is a measure of concentration, specifically the number of moles of solute per liter of solution. It's a foundational concept in chemistry used to quantify solutions.
To determine the molarity (\[M\]) of the KOH solution, it is important to first calculate the concentration of hydroxide ions. Given that KOH completely dissociates in water, the concentration of \[\text{OH}^-\] is equal to the molarity of the KOH solution.

• The hydroxide ion concentration from the problem was determined using the formula: \[(\text{OH}^-) = 10^{-\text{pOH}}\] which yields: \[(\text{OH}^-) = 10^{-0.67} = 0.2149\,M\]
• Therefore, the molarity of the KOH solution is also \(0.2149\,M\).

Understanding molarity is crucial for predicting the outcomes in solution reactions, such as determining the concentration of ions or for use in various calculations involving chemical reactions.

Stoichiometry

Stoichiometry is the branch of chemistry that explores the relationships between reactants and products in a chemical reaction. It provides a quantitative framework to calculate the amounts of substances involved.
In this context, stoichiometry helps us understand how much KOH is needed to reach a desired pH in a given volume of water.

• Because KOH dissociates completely in water to produce 1 mole of \[\text{OH}^-\] ions for every mole of KOH, the stoichiometric relationship is 1:1.
• Using the molarity \(0.2149\,M\) and volume \(455\,\text{mL}\) or \(0.455\,\text{L}\), compute moles of KOH:
  \[\text{moles of KOH} = \text{molarity} \times \text{volume} = 0.2149\,\text{M} \times 0.455\,\text{L} = 0.09768\,\text{moles}\]

This stoichiometric information is essential to eventually calculate the mass of KOH.

Hydroxide Ion Concentration

Understanding hydroxide ion concentration is fundamental when dealing with bases. Since KOH is a strong base, it dissociates fully in water, forming potassium ions (\[\text{K}^+\]) and hydroxide ions (\[\text{OH}^-\]).
To calculate the hydroxide ion concentration, the direct relationship with pOH is used.

• Starting with the derived pOH, \[0.67\], we calculate the hydroxide ion concentration as:
  \[(\text{OH}^-) = 10^{-\text{pOH}} = 10^{-0.67}\]
• This results in \[0.2149\,\text{M}\], indicating how basic or alkaline the solution is.

This concentration plays a critical role in the overall acid-base balance in solutions and directly affects their pH and pOH values. Knowing how to find and interpret this is key to solving acid-base problems effectively.