Question

A solution of hydrogen peroxide is \(30 \%\) by weight \(\mathrm{H}_{2} \mathrm{O}_{2}\). Assuming a density of \(1.11 \mathrm{~g} / \mathrm{cm}^{3}\) and a dissociation constant of \(1.0 \times 10^{-12}\) for \(\mathrm{H}_{2} \mathrm{O}_{2}\), what is the \(\mathrm{pH}\) of the solution?

Short answer

To find the pH of the hydrogen peroxide solution, follow these steps: 1. Calculate the mass of H₂O₂ per unit volume: \(1.11\frac{g}{cm^3} * 0.3\) 2. Calculate the moles of H₂O₂ per unit volume: \(\frac{Mass\:of\:H_{2}O_{2}\:per\:unit\:volume}{Molar\:mass\:of\:H_{2}O_{2}}\) 3. Determine the concentration of H₂O₂ (in mol/L). 4. Calculate the concentration of H⁺ ions using the dissociation constant: [H⁺] = √(K × [H₂O₂]). 5. Calculate the pH using the H⁺ ion concentration: pH = -log₁₀[H⁺].

Step-by-step solution

Calculate the mass of H₂O₂ per unit volume

To determine the mass of H₂O₂ in the solution, we will multiply the density of the solution by the percentage of H₂O₂ by weight: Mass of H₂O₂ per unit volume = (Density) * (Mass percentage of H₂O₂) \(Mass\:of\:H_{2}O_{2}\:per\:unit\:volume = 1.11\frac{g}{cm^3} * 0.3\)

Calculate the moles of H₂O₂ per unit volume

Divide the mass of H₂O₂ found in Step 1 by the molar mass of H₂O₂ to get the moles of H₂O₂ per unit volume. The molar mass of H₂O₂ is 34.01 g/mol. Moles of H₂O₂ per unit volume = \(\frac{Mass\:of\:H_{2}O_{2}\:per\:unit\:volume}{Molar\:mass\:of\:H_{2}O_{2}}\)

Determine the concentration of H₂O₂

The moles of H₂O₂ per unit volume calculated in Step 2 is equivalent to the concentration of H₂O₂ (in mol/L) since 1 cm³ equals 1 mL.

Calculate the concentration of H⁺ ions

Now we will find the concentration of H⁺ ions released by the dissociation of H₂O₂. To do this, we will use the dissociation constant K and the concentration of H₂O₂: \(K = \frac{[H^+][O_2^{2-}]}{[H_2O_2]}\) Since K is very small, we can assume that the dissociation of H₂O₂ is negligible and the concentration of H₂O₂ remains constant. Therefore, the concentration of H⁺ ions can be determined by: [H⁺] = √(K × [H₂O₂])

Calculate the pH

Finally, we will use the concentration of H⁺ ions to calculate the pH of the solution: pH = -log₁₀[H⁺]

Key concepts

Hydrogen Peroxide Solution

Hydrogen peroxide, or \(\mathrm{H}_{2}\mathrm{O}_{2}\), is a chemical compound that appears as a colorless liquid. It is a common disinfectant and is often found in household products. In diluted solutions, it is used safely for various purposes like bleaching and cleaning.
A hydrogen peroxide solution like the one in the exercise is typically expressed in terms of percentage by weight. Here, the solution is \(30\%\) by weight \(\mathrm{H}_{2}\mathrm{O}_{2}\).
This means that in every 100 grams of the solution, 30 grams are hydrogen peroxide, while the remaining 70 grams are composed of water and other components.
The density of the solution is considered to be \(1.11\, \mathrm{g/cm}^3\), which helps in converting the mass percentage to mass per volume, an essential step for calculations.
Knowing these details helps us understand the quantity of \(\mathrm{H}_{2}\mathrm{O}_{2}\) available in any given volume of the solution, forming a basis for further conversion into moles.

Dissociation Constant

The dissociation constant is a critical concept when analyzing the behavior of compounds in solution. Here, it refers to the tendency of hydrogen peroxide to dissociate into its constituent ions.
The dissociation constant is represented as \(K\) and helps us understand the equilibrium between undissociated molecules of \(\mathrm{H}_{2}\mathrm{O}_{2}\) and its ions in solution. In this exercise, the dissociation constant \(K\) is a very small value, \(1.0 \times 10^{-12}\).
This small magnitude essentially suggests that \(\mathrm{H}_{2}\mathrm{O}_{2}\) dissociates only slightly in solution, meaning most of the compound remains as \(\mathrm{H}_{2}\mathrm{O}_{2}\) and only a tiny amount breaks into ions.
For practicality, we can assume the concentration of \(\mathrm{H}_{2}\mathrm{O}_{2}\) does not change significantly due to dissociation, simplifying the calculation of ion concentration.

Molar Mass

Molar mass is the weight of one mole of a substance and is measured in grams per mole. For hydrogen peroxide \(\mathrm{H}_{2}\mathrm{O}_{2}\), the molar mass is calculated by adding the atomic masses of the constituent atoms.
Hydrogen has an atomic mass of approximately 1.01 g/mol, and oxygen's atomic mass is about 16.00 g/mol. Therefore, the molar mass of \(\mathrm{H}_{2}\mathrm{O}_{2}\) is calculated as:

• \(2 \times 1.01\; \text{g/mol} + 2 \times 16.00\; \text{g/mol} = 34.01\; \text{g/mol}\)

This value is crucial when converting between mass and moles, allowing us to find the number of moles of \(\mathrm{H}_{2}\mathrm{O}_{2}\) in a given solution.
Accurate molar mass calculations are essential for many chemical reactions and calculations, as they allow for proper stoichiometric balances in equations.

Concentration of Ions

Calculating ion concentration is pivotal in determining the characteristics of a solution, such as its pH. In this context, the concentration of hydrogen ions \([\mathrm{H}^+]\) is a key factor.
Using the dissociation constant \(K\) and the concentration of \(\mathrm{H}_{2}\mathrm{O}_{2}\), we can find the concentration of \([\mathrm{H}^+]\) using:

• \([\mathrm{H}^+] = \sqrt{K \times [\mathrm{H}_{2}\mathrm{O}_{2}]}\)

This formula assumes that ince the dissociation of \(\mathrm{H}_{2}\mathrm{O}_{2}\) is very slight due to the low \(K\) value, \([\mathrm{H}_{2}\mathrm{O}_{2}]\) remains nearly constant.
The calculated \([\mathrm{H}^+]\) can then be used to determine the pH, a measure of acidity, by applying the formula:

• \(\mathrm{pH} = -\log_{10}[\mathrm{H}^+]\)

This calculation provides insight into the solution's acidic properties, which are essential for understanding how it reacts in different environments.