Question

In which of the following cases is the approximation that the equilibrium concentration of \(\mathrm{H}^{+}(a q)\) is small relative to the initial concentration of HA likely to be most valid: (a) initial \([\mathrm{HA}]=0.100 \mathrm{M}\) and \(K_{a}=1.0 \times 10^{-6}\), (b) initial \([\mathrm{HA}]=0.100 \mathrm{M}\) and \(K_{a}=1.0 \times 10^{-4}\), (c) initial \([\mathrm{HA}]=0.100 \mathrm{M}\) and \(K_{a}=1.0 \times 10^{-3} ?[\) Section \(16.6]\)

Short answer

The approximation is most valid in Case (a) where [HA] = 0.100 M and \(K_a = 1.0 \times 10^{-6}\), as the degree of ionization (x) is much smaller than the initial concentration of HA.

Step-by-step solution

Write the Reaction, Equilibrium Expression and Approximation Condition

First, let's write the ionization reaction of the weak acid, HA: HA (aq) ⇌ H⁺ (aq) + A⁻ (aq) The equilibrium constant expression for the reaction is given by: \(K_a = \dfrac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]}\) Here, the approximation condition we need to check is that the equilibrium concentration of H⁺ is much smaller than the initial concentration of HA. In other words, we want: \([\mathrm{H}^{+}] \ll [\mathrm{HA}]_0\)

Calculate the Degree of Ionization for all 3 cases

Let's introduce a variable x to represent the degree of ionization of the weak acid. At equilibrium, the concentrations are: [H⁺] = [A⁻] = x, and [HA] = [HA]₀ - x. We can rewrite the equilibrium constant expression: \(K_a = \dfrac{x^2}{[\mathrm{HA}]_0 - x}\) Now, we can plug in the given initial concentrations and Ka values from each case and calculate x for each case. Case (a): \([\mathrm{HA}]_0 = 0.100\) M and \(K_a = 1.0 \times 10^{-6}\) \(10^{-6} = \dfrac{x^2}{0.100 - x}\) Case (b): \([\mathrm{HA}]_0 = 0.100\) M and \(K_a = 1.0 \times 10^{-4}\) \(10^{-4} = \dfrac{x^2}{0.100 - x}\) Case (c): \([\mathrm{HA}]_0 = 0.100\) M and \(K_a = 1.0 \times 10^{-3}\) \(10^{-3} = \dfrac{x^2}{0.100 - x}\)

Evaluate the Approximation Condition

Now, we can use the quadratic formula to calculate x for each case. The more important part is to check the approximation condition: If the degree of ionization (x) is much smaller than the initial concentration ([HA]₀), the approximation is considered more valid. Case (a): \(x_a \approx 3.16 \times 10^{-4}\) Case (b): \(x_b \approx 0.010\) Case (c): \(x_c \approx 0.031\) By comparing the calculated x values with the initial concentrations, we can find out for which case the approximation is most valid: Case (a): \(x_a \ll [\mathrm{HA}]_0\) (approximation is valid) Case (b): \(x_b \approx [\mathrm{HA}]_0/10\) (approximation is somewhat valid) Case (c): \(x_c \approx [\mathrm{HA}]_0/3\) (approximation is not valid)

Conclusion

The approximation that the equilibrium concentration of \(\mathrm{H}^{+}(a q)\) is small relative to the initial concentration of HA is most valid in Case (a), where [HA] = 0.100 M and \(K_a = 1.0 \times 10^{-6}.\)

Key concepts

Weak Acids

Weak acids only partially ionize in aqueous solutions, which means they don't release all of their hydrogen ions into the solution. Instead, a significant fraction of the acid remains in its undissociated form. This behavior is contrasted with strong acids, which completely ionize in solution.

Because weak acids do not ionize entirely, they establish an equilibrium between the ionized and non-ionized forms. For instance, when a weak acid, HA, is dissolved in water, it partially dissociates into \( ext{H}^+\) ions and \( ext{A}^-\) ions. This partial dissociation can be expressed by the chemical reaction:

• HA (aq) ⇌ H⁺ (aq) + A⁻ (aq)

Understanding the equilibrium state of weak acids helps predict their behavior in different chemical environments, including pH changes and buffer solutions. Much of this revolves around balancing the degree of ionization alongside the initial concentration of the acid.

Ionization

Ionization refers to the process where a molecule or compound breaks apart into ions, often in a solution. For weak acids, ionization is a partial process that results from the dynamic balance between the dissociated and undissociated molecules.

In the case of weak acids like HA, the equation \( ext{HA (aq)} ⇌ ext{H}^+ ext{(aq)} + ext{A}^- ext{(aq)}\) provides a simple representation of how ionization takes place. The amount of the acid that ionizes can be expressed in terms of ionization equilibrium.

The 'x' in the equilibrium equation \( ext{K}_a = \frac{x^2}{[ ext{HA}]_0 - x}\) denotes the degree of ionization. In practical scenarios, the value of 'x' is usually determined by the magnitude of the equilibrium constant \( ext{K}_a\) and the initial concentration of the acid. For many weak acids, this is much less than its initial concentration, validating the approximation method used.

Equilibrium Constant

The equilibrium constant (\(K_a\) for weak acids) quantifies the extent of ionization of an acid in a solution at equilibrium. It is an important numerical value that provides insight into how much of the acid dissociates in the solution.

Mathematically it is expressed as:

• \(K_a = \frac{[ ext{H}^+][ ext{A}^-]}{[ ext{HA}]}\)

When \(K_a\) is small, like \(1.0 imes 10^{-6}\), it implies that the acid is weak, and only a small fraction ionizes. A larger value, such as \(1.0 imes 10^{-3}\), indicates stronger ionization, though still considered a weak acid compared to strong acids like hydrochloric acid that have much higher equilibrium constants.

The value of \(K_a\) allows chemists to predict and calculate the concentrations of the different species in an equilibrium scenario. This understanding is crucial for applications ranging from calculating the pH of solutions to designing buffer systems.

Approximation Method

The approximation method simplifies complex equilibrium calculations by assuming that the change in concentration of the weak acid due to ionization is negligible. This is most valid when the equilibrium constant is much smaller and the initial concentration of the acid is relatively large.

In the problem setup, the approximation states that \([ ext{H}^+] \ll [ ext{HA}]_0\). This allows us to simplify the algebra of the equilibrium expressions. By using the approximation that \([ ext{HA}]_0 - x \approx [ ext{HA}]_0\), calculations become more manageable.

This technique is highly beneficial:

• For saving time on calculations
• For aligning with the chemical intuition of weak acids having minor ionization
• For focusing on large scale trends without getting lost in meticulous algebra

Approximations are vital in chemistry for making quick, initial assessments, especially when dealing with weak acids, and should always be confirmed using more precise methods if needed.