Problem 68 - Chemistry An Atoms First Approach

Question

A student dissolves 0.0100 mole of an unknown weak base in \(100.0 \mathrm{mL}\) water and titrates the solution with \(0.100 \mathrm{M} \mathrm{HNO}_{3}\) After \(40.0 \mathrm{mL}\) of \(0.100 \mathrm{M} \mathrm{HNO}_{3}\) was added, the \(\mathrm{pH}\) of the resulting solution was \(8.00 .\) Calculate the \(K_{\mathrm{b}}\) value for the weak base.

Step-by-Step Solution

Answer

The \(K_b\) value for the weak base is \(6.67 \times 10^{-8}\).

Step 1: Calculate the moles of the weak base and HNO3

To start, let's first determine the moles of the weak base and the moles of HNO3 added. We are given that there are 0.0100 moles of weak base, and that the concentration of HNO3 is 0.100 M. For HNO3, we can calculate the moles using the volume of the solution and its concentration: Moles of HNO3 = Volume × Concentration Moles of HNO3 = 40.0 mL × 0.100 M = 0.0040 moles (since 1L = 1000mL, 40mL = 0.040L)

Step 2: Calculate the moles of the weak base and its conjugate acid after titration

During the titration, the weak base will react with HNO3, and the resulting solution will contain the conjugate acid of the weak base. Since 0.0100 moles of the base react with 0.0040 moles of HNO3, we can calculate the moles of each species after the reaction: Moles of weak base after titration = Initial moles of weak base - moles of HNO3 reacted = 0.0100 - 0.0040 = 0.0060 moles Moles of conjugate acid = Moles of HNO3 reacted = 0.0040 moles

Step 3: Calculate the concentration of OH- ions using the pH

After titration, the pH of the resulting solution is given as 8.00. We can use this to calculate the pOH and then find the concentration of OH- ions: pOH = 14 - pH pOH = 14 - 8.00 = 6.00 Now we can calculate the concentration of OH- ions using the pOH: [OH-] = 10^(-pOH) = 10^(-6.00) = 1.00 x 10^(-6) M

Step 4: Calculate the concentrations of the weak base and conjugate acid

Now we need to determine the concentrations of the weak base and its conjugate acid. We have a total volume of 140.0 mL after mixing (100.0 mL weak base solution + 40.0 mL HNO3 solution). Concentration of weak base = moles of weak base / total volume = 0.0060 moles / 0.140 L = 0.04286 M Concentration of conjugate acid = moles of conjugate acid / total volume = 0.0040 moles / 0.140 L = 0.02857 M

Step 5: Calculate the Kb value for the weak base

Now, we can use the concentrations of weak base, conjugate acid, and OH- ions to determine the Kb value for the weak base. The reaction can be expressed as: Weak base + H2O ⇌ Conjugate Acid + OH- The equilibrium expression for this reaction is: Kb = ([Conjugate Acid][OH-]) / [Weak base] Kb = (0.02857 M × 1.00 × 10^(-6) M) / (0.04286 M) Kb = 6.67 × 10^(-8) The Kb value for the weak base is 6.67 × 10^(-8).

Key Concepts

Kb value calculation

In the realm of chemistry, particularly when dealing with acid-base reactions, calculating the equilibrium constant for bases, known as the 'base ionization constant' or 'Kb', is crucial for understanding the strength of a base and its behavior in solution. To find the Kb value, you must first understand that it denotes the equilibrium concentration of products over reactants for the ionization of a weak base in water.

As outlined in the exercise, once the moles of the base and titrant acid (HNO3) are known, and after considering the titration's stoichiometry, we can calculate the remaining concentration of the weak base and its conjugate acid. Using the pH provided, we determine the pOH, from which we calculate the hydroxide ion concentration [OH-]. With these concentrations, the Kb is simply the product of the concentrations of the conjugate acid and the OH- ions divided by the concentration of the weak base. It’s worth noting that larger Kb values correspond to stronger bases, since they ionize more extensively to form OH- ions in solution.

acid-base titration

Acid-base titration is a fundamental technique used in chemistry to determine the concentration of an unknown acid or base. In this analytical procedure, a known volume and concentration of an acid is carefully added to a base, or vice versa, until the reaction reaches its end point, which is often indicated by a color change due to an indicator or by measuring the pH.

During a weak base titration with a strong acid, like in the exercise, the weak base (B) reacts with the strong acid (HNO3) to form its conjugate acid (BH+). The titration aims to reach the equivalence point, where moles of acid are stoichiometrically equivalent to the moles of base. However, the exercise problem was concerned with the pH at a specific point of the titration, before reaching this equivalence point. This information, combined with careful calculations and knowledge of the reaction's stoichiometry, allows chemists to deduce the concentration of all species in solution and, ultimately, the base's Kb value.

pH and pOH calculations

Understanding the pH and pOH scales is essential in chemistry for describing the acidity or basicity of a solution. pH stands for 'power of hydrogen' and measures the hydrogen ion (H+) concentration in a solution, while pOH does the same for hydroxide ions (OH-). The pH scale typically runs from 0 (very acidic) to 14 (very basic), with 7 being neutral.

To relate pH and pOH, we use the equation: pH + pOH = 14. In the context of weak base titration, determining the pOH from the given pH allows us to calculate the concentration of OH- using the formula [OH-] = 10^(-pOH). Understanding these calculations is not only important for finding the concentration of ions in solution but also for progressing to further necessary calculations like determining the Kb of a base, as they provide a starting point for the mathematical journey towards comprehensive acid-base equilibrium analysis.