Question

A \(0.400-M\) solution of ammonia was titrated with hydrochloric acid to the equivalence point, where the total volume was 1.50 times the original volume. At what pH does the equivalence point occur?

Short answer

The pH at the equivalence point of the titration between a $0.400-M$ ammonia solution and hydrochloric acid is approximately 4.93.

Step-by-step solution

Calculate moles of ammonia

We are given the concentration of ammonia, \(0.400\,M\). In order to find the moles of ammonia, we need to know the volume of the solution. We're told the total volume at the equivalence point was 1.50 times the original volume. Since the volume of ammonia and hydrochloric acid combined is 1.50 times the original volume, the volume of the ammonia must be half of the total volume. Let's denote the volume of ammonia as \(V_{NH_3}\), then we have: \(2 \times V_{NH_3} = 1.50 V_{NH_3}\) Solving for \(V_{NH_3}\), we get: \(V_{NH_3} = 0.75 V_{NH_3}\) Now, we can find the moles of ammonia using the provided concentration: moles \(NH_3 = 0.400\,M \times 0.75 V_{NH_3}\)

Calculate moles of hydrochloric acid

At the equivalence point, the moles of ammonia and hydrochloric acid are equal. Therefore, the moles of hydrochloric acid are the same as the moles of ammonia. So: moles \(HCl = 0.400\,M \times 0.75 V_{NH_3}\)

Calculate the pH at the equivalence point

Since ammonia is a weak base and hydrochloric acid is a strong acid, the product of the reaction is the conjugate acid of ammonia, which is ammonium chloride (\(NH_4Cl\)). \(NH_3 + HCl \rightarrow NH_4Cl\) At the equivalence point, all the ammonia has reacted with the hydrochloric acid to form ammonium ions (\(NH_4^+\)). The equilibrium reaction for the ammonium ion is: \(NH_4^+ \rightleftharpoons NH_3 + H^+\) To calculate the pH at the equivalence point, we need to find the equilibrium concentration of \(H^+\) ions. We can do this using the equilibrium constant expression for the reaction: \(K_a = \frac{[NH_3][H^+]}{[NH_4^+]}\) We are given the initial concentration of ammonia (0.400 M), and we know that it has reacted with an equal amount of hydrochloric acid. Therefore, the initial concentration of ammonium ion is also 0.400 M. Since the reaction is a 1:1 ratio, the final concentration of \(NH_3\) and \(H^+\) at equilibrium will be the same. Thus, we can simplify the \(K_a\) expression as: \(K_a = \frac{x^2}{0.400\,M-x}\) The \(K_a\) value for ammonia is approximately \(5.56 \times 10^{-10}\). By substituting this value into the equation above and solving for x, we can find the concentration of \(H^+\) ions at equilibrium.

Solve for the equilibrium concentration of \(H^+\) ions

\(5.56 \times 10^{-10} = \frac{x^2}{0.400\,M-x}\) Solve for x (concentration of \(H^+\)) using either a quadratic equation or an approximation method. For simplicity, we can approximate that x is much smaller than 0.400 M so the equation becomes: \(5.56 \times 10^{-10} \approx \frac{x^2}{0.400\,M}\) Solve for x: \(x \approx 1.18 \times 10^{-5}\,M\)

Calculate the pH at the equivalence point

Now that we have the equilibrium concentration of \(H^+\) ions, we can use the pH formula to find the pH at the equivalence point: pH = -log\([H^+]\) pH = -log\((1.18 \times 10^{-5}\,M)\) pH ≈ 4.93 At the equivalence point, the pH is approximately 4.93.

Key concepts

Ammonia Titration

Ammonia titration involves the process of gradually adding a solution of hydrochloric acid (HCl) to a solution of ammonia ( NH_3 ) to reach the equivalence point. The equivalence point is the juncture where the amount of added acid is stoichiometrically equal to the amount of base present in the solution.
This reaction between ammonia and hydrochloric acid yields ammonium chloride ( NH_4Cl ), changing the pH of the solution.
This setup illustrates a titration of a weak base with a strong acid. Understanding the complete titration process involves knowing how ammonia, a weak base, reacts and what products form as a result.
During this process, the pH of the solution shifts from its initial basic value towards a more acidic range as the titrant (HCl) is added.

Weak Base Strong Acid Titration

In a titration involving a weak base and a strong acid, such as ammonia titrated with hydrochloric acid, the reaction dynamics are crucial to comprehend. When the strong acid is added to the weak base, it leads to the formation of a conjugate acid.
The general reaction scheme for this process is:

• The weak base ( NH_3 ) reacts with the strong acid ( HCl ).
• The result is the formation of the conjugate acid ( NH_4Cl ).

This type of titration usually doesn't result in a neutral pH at the equivalence point, as one might expect with strong acid-strong base titrations. Instead, the resulting solution tends to be more acidic due to the weak base's conjugate acid in the solution.
One must consider the behavior of the added strong acid and how it completely reacts with the weak base to alter the pH of the solution during the titration process.

Ammonium Ion Equilibrium

Once ammonia (NH_3) reacts with hydrochloric acid, it forms ammonium ions (NH_4^+). This transformation highlights the ammonium ion equilibrium that's pivotal to understand in this context.
The ammonium ion exists in equilibrium with ammonia and hydrogen ions (H^+) as per the equation: \[NH_4^+ \rightleftharpoons NH_3 + H^+ \]This equilibrium is characterized by its own equilibrium constant, known as the acid dissociation constant (K_a).
At the equivalence point, these ammonium ions significantly affect the solution's pH level. Understanding how to calculate the concentrations of these ions helps in determining the pH, which is influenced by this equilibrium.This set of reactions and the behavior of ammonium ions is vital in analyzing the acidity of the solution post-titration.

pH Calculation

pH calculation at the equivalence point involves determining the concentration of hydrogen ions (H^+) in the solution. In this scenario, the solution is acidic because it contains ammonium ions.
To find the pH, one must first calculate the concentration of the hydrogen ions resulted from the equilibrium:\[K_a = \frac{[NH_3][H^+]}{[NH_4^+]} \]The value of K_a for ammonia is essential here to solve for [H^+]. In this instance, we consider the initial concentrations and equilibrium shifts to approximate [H^+] using simplified balance equations, especially when working under assumptions.
After finding [H^+], the pH can be calculated using the formula:\[pH = -\log([H^+])\]This method provides the pH at the equivalence point, which reflects the acidic nature due to the presence of the conjugate acid.

Weak Base Equilibrium Reaction

The weak base equilibrium reaction plays a crucial role in the entire ammonia titration process. When ammonia reacts with hydrochloric acid, the equilibrium that the NH_4^+ ions establish is part of this equilibrium process.
For a weak base, such as ammonia, the equilibrium reaction is dictated by its tendency to release hydroxide ions (OH^-) when dissolved in water and subsequently neutralize hydrogen ions added from the acid.

• The weak base equilibrium can be represented with the following equation: \[ NH_3 + H_2O \rightleftharpoons NH_4^+ + OH^- \]

This reaction signifies that even after forming the conjugate acid, NH_4^+, little OH^- remains, thus imparting the acidic character seen at the equivalence point.
The equilibrium principles are essential in understanding how the weak base transforms during titration and how its resultant products dictate the final pH of the solution.
This basic understanding of weak base behavior and equilibrium reactions enriches the conceptual grasp of such titrations.