Problem 116 - Chemistry The Central Science

Question

Calculate the number of \(\mathrm{H}^{+}(a q)\) ions in 1.0 \(\mathrm{mL}\) of pure water at \(25^{\circ} \mathrm{C} .\)

Step-by-Step Solution

Answer

There are approximately \(6.022 × 10^{13}\) H⁺(aq) ions in 1.0 mL of pure water at 25°C.

Step 1: Determine the ion product of water (Kw)

At 25°C, the ion product of water (Kw) is given as: \[K_w = [H^+][OH^-] = 1.0 × 10^{-14}\] Where [H⁺] is the concentration of H⁺ ions and [OH⁻] is the concentration of OH⁻ ions.

Step 2: Find the concentration of H⁺ ions in pure water

In pure water, the concentrations of H⁺ ions and OH⁻ ions are equal, i.e., [H⁺] = [OH⁻]. We can use this fact along with the Kw value to solve for the concentration of H⁺ ions: \[[H^+] = \sqrt{K_w}\] \[[H^+] = \sqrt{1.0 × 10^{-14}}\] \[[H^+] = 1.0 × 10^{-7} \ \text{M}\] Now we have the concentration of H⁺ ions in the water.

Step 3: Convert the volume of water from mL to Liters

We are given the volume of pure water as 1.0 mL. We need to convert this value to Liters, as the concentration of H⁺ ions is given in moles per liter (M): \[1.0 \ \text{mL} = 1.0 × 10^{-3}\ \text{L}\]

Step 4: Calculate the number of moles of H⁺ ions

Using the given concentration of H⁺ ions [H⁺] and the given volume of water, we can determine the number of moles of H⁺ ions (n): \[n = [H^+] × \text{Volume of water in L}\] \[n = (1.0 × 10^{-7} \ \text{M}) × (1.0 × 10^{-3} \text{L})\] \[n = 1.0 × 10^{-10} \ \text{moles}\]

Step 5: Calculate the number of H⁺ ions

Finally, we can find the number of H⁺ ions using Avogadro's number (6.022 × 10²³ particles per mole): \[\text{Number of H}^+ \text{ ions} = n × \text{Avogadro's number}\] \[\text{Number of H}^+ \text{ ions} = (1.0 × 10^{-10} \ \text{moles}) × (6.022 × 10^{23} \ \text{ions/mol})\] \[\text{Number of H}^+ \text{ ions} = 6.022 × 10^{13} \ \text{ions}\] Thus, there are approximately \(6.022 × 10^{13}\) H⁺(aq) ions in 1.0 mL of pure water at 25°C.

Key Concepts

Water Ionization

Water ionization is an important concept in chemistry, especially when dealing with the properties of pure water. When water undergoes ionization, it splits into hydrogen ions (H⁺) and hydroxide ions (OH⁻). This process can be represented by the following equilibrium equation:
2H₂O ⇌ H₃O⁺ + OH⁻
In pure water at 25°C, this equilibrium creates equal concentrations of H⁺ and OH⁻, which are both 1.0 × 10^{-7} M. This means that, even in its natural state, water contains a small number of ions.
Ionization of water is crucial because it contributes to the pH balance in aqueous solutions, making it a central focus in studies related to acids and bases. The fact that concentrations of both ions are equal in pure water simplifies calculations involved in determining pH and other acid-base phenomena.
- **Equilibrium Equation**: Demonstrates water's ionization into H⁺ and OH⁻. - **Concentration**: In pure water, both ions' concentrations are 1.0 × 10^{-7} M. Understanding water ionization helps in comprehending broader chemical processes, like acid-base reactions and pH calculations.

Avogadro's Number

Avogadro's number is a key concept in chemistry, indicating the number of constituent particles, usually atoms or molecules, that are contained in one mole. The value of Avogadro's number is based on the number of atoms in 12 grams of carbon-12, which is 6.022 × 10^{23} particles.
This number is vital for converting between the number of moles and the number of particles. For instance, when the number of moles of hydrogen ions is known, Avogadro's number allows you to calculate exactly how many ions are present.
- **Definition**: Avogadro's number is 6.022 × 10^{23} particles per mole. - **Conversion Use**: It bridges the macroscopic scale of moles with the microscopic scale of individual ions or molecules. Avogadro's number acts as the link between measurable quantities of substances in chemistry and the vast number of tiny atoms or molecules they contain, greatly facilitating calculations involving chemical reactions and solutions.

Kw (Ion Product of Water)

The ion product of water, denoted as Kw, is a constant at a given temperature that signifies the product of the concentrations of hydrogen ions and hydroxide ions in water. At 25°C, the value of Kw is 1.0 × 10^{-14}.
Kw is essential because it captures the autoionization balance of water:
Kw = [H⁺][OH⁻]
Since the concentration of H⁺ and OH⁻ in pure water is equal, you can deduce their concentrations by calculating the square root of Kw, which results in each being 1.0 × 10^{-7} M under standard conditions.
- **Expression**: Kw = [H⁺][OH⁻] indicating the relationship between ion concentrations. - **Value**: At 25°C, Kw is 1.0 × 10^{-14}. Understanding Kw helps in various calculations in acid-base chemistry, as it allows chemists to determine respective ion concentrations in any aqueous solution, thereby assisting in pH computations and understanding the nature of solutions.