Problem 162 - Chemistry

Question

The \(\mathrm{pH}\) of \(1.0 \times 10^{-8} \mathrm{M}\) hydrochloric acid is not \(8.00\). The correct \(\mathrm{pH}\) can be calculated by considering the relationship between the molarities of the three principal ions in the solution \(\left(\mathrm{H}^{+}, \mathrm{Cl}^{-},\right.\), and \(\mathrm{OH}^{-}\) ). These molarities can be calculated from algebraic equations that can be derived from the considerations given below. a. The solution is electrically neutral. b. The hydrochloric acid can be assumed to be \(100 \%\) ionized. c. The product of the molarities of the hydronium ions and the hydroxide ions must equal \(K_{\mathrm{w}}\). Calculate the \(\mathrm{pH}\) of a \(1.0 \times 10^{-8} \mathrm{HCl}\) solution.

Step-by-Step Solution

Answer

The pH of the 1.0 x 10^-8 M hydrochloric acid solution is approximately 5.995.

Step 1: Write down the ionization equation for HCl and autoionization of water

HCl ionizes in water as follows: HCl (aq) → H+ (aq) + Cl- (aq) Water (H2O) autoionizes as follows: H2O (l) ↔ H+ (aq) + OH- (aq) The equilibrium constant for this reaction is represented as Kw.

Step 2: Determine the initial concentrations of ions

We know that the concentration of HCl is 1.0 x 10^-8 M and it is 100% ionized. So the initial concentration of H⁺ ions from HCl is: [H⁺]HCl = 1.0 x 10^-8 M

Step 3: Let x represent the concentration of the additional H⁺ ions from water's autoionization

Let x be the concentration of H⁺ ions that are generated from the autoionization of water: [H⁺]H2O = x Now, the total concentration of H⁺ ions will be: [H⁺]total = [H⁺]HCl + [H⁺]H2O = 1.0 x 10^-8 + x

Step 4: Write the equation for electrical neutrality and the dissociation constant for water (Kw)

The solution should be electrically neutral, and therefore: [H⁺]total = [OH⁻] (1) We know that the autoionization of water has the equilibrium constant, Kw: Kw = [H⁺]H2O * [OH⁻] (2) Kw for water is 1 x 10^-14 at room temperature.

Step 5: Solve the equations for x (concentration of H⁺ ions from water's autoionization) and [OH⁻]

From equation (1), we can write [OH⁻] in terms of [H⁺]total: [OH⁻] = [H⁺]total Substitute this in the equation for Kw (2) and solve for x: Kw = [H⁺]H2O * [H⁺]total 1 x 10^-14 = x * (1.0 x 10^-8 + x) Taking into account that x is very small compared to 1.0 x 10^-8, we can simplify this equation to: 1 x 10^-14 ≈ x * 1.0 x 10^-8 Solving for x, we get: x ≈ 1 x 10^-6 Now, we have the concentration of H⁺ ions from water's autoionization.

Step 6: Calculate the total concentration of H⁺ ions and pH

Next, we'll find the total concentration of H⁺ ions in the solution using the value of x: [H⁺]total = 1.0 x 10^-8 + x ≈ 1.0 x 10^-8 + 1 x 10^-6 Therefore, [H⁺]total ≈ 1.01 x 10^-6 M Now, we can find the pH of the solution using the formula: pH = -log10[H⁺] pH = -log10(1.01 x 10^-6) Calculate the pH: pH ≈ 5.995

Step 7: Final Answer

The pH of the 1.0 x 10^-8 M hydrochloric acid solution is approximately 5.995.

Key Concepts

Hydrochloric Acid Ionization

When hydrochloric acid (HCl) dissolves in water, it ionizes, or splits into its component ions. This process can be represented by the simple equation:
HCl (aq) → H⁺ (aq) + Cl⁻ (aq)
In dilute solutions, such as the example with a concentration of 1.0 x 10⁻⁸ M, HCl can be assumed to dissociate completely. This assumption leads us to conclude that every molecule of HCl provides one hydrogen ion (H⁺) and one chloride ion (Cl⁻) to the solution. As a result, the same concentration of chloride ions will also exist in the solution. Understanding this ionization process is crucial for calculating the pH of an HCl solution and addressing the exercise improvement advice, we ensure that the concept of essentially complete ionization is highlighted for clarity.

Autoionization of Water

Water itself slightly ionizes in a process known as autoionization or self-ionization, which has the equation:
H₂O (l) ↔ H⁺ (aq) + OH⁻ (aq)
This process is essential when considering very dilute solutions of strong acids or bases, or even pure water. The equilibrium constant for this reaction is the ion product of water, Kw. In such solutions, the contributions of H⁺ and OH⁻ ions from water's autoionization can significantly affect pH calculations, a point not to be overlooked in our educational approach.

Concentration of Ions

The concentration of ions in a solution refers to the number of ions of a particular type (e.g., H⁺, OH⁻, Cl⁻) present in a unit volume of the solution. It is typically expressed in molarity (moles per liter, M). In the given exercise, the initial concentration of H⁺ ions from HCl is known to be 1.0 x 10⁻⁸ M. However, due to water's autoionization, additional H⁺ ions are present, and their concentration must be determined to calculate the actual pH accurately—thus adhering to easy-to-understand content, we emphasize this consideration to improve students' comprehension.

Electrical Neutrality

In any aqueous solution, there is a fundamental principle of electrical neutrality: the sum of the positive charges must equal the sum of the negative charges. When calculating ion concentrations, this principle must be satisfied. As such, the total concentration of positive ions (particularly H⁺ ions) must equal the total concentration of negative ions (OH⁻ ions and any other anions present). For the exercise at hand, electrical neutrality helps set up an important relationship between the concentrations of hydronium and hydroxide ions, key to solving for pH.

Dissociation Constant (Kw)

The dissociation constant for water, Kw, is the product of the concentrations of hydrogen ions and hydroxide ions in pure water. At room temperature (25°C), Kw is always 1 x 10⁻¹⁴. This constant is crucial for understanding the relationship between H⁺ and OH⁻ ion concentrations. When solving for the additional H⁺ ions from water's autoionization, Kw is used to calculate the potential increase in H⁺ ions, which, in turn, affects pH calculation. This explanation is in line with the imperative to provide content that ensures a student can easily grasp the intricacies of pH calculations involving Kw.