Problem 116 - Chemistry: The Central Science in SI Units, Global Edition

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 \(6.022 \times 10^{13}\) \(\mathrm{H}^{+}(a q)\) ions in \(1.0 \mathrm{~mL}\) of pure water at \(25^{\circ} \mathrm{C}\).

Step 1: Identify the ion product constant of water.

At \(25^{\circ} \mathrm{C}\), the ion product constant of water, \(K_{w}\), is given by: \[K_{w} = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14}\] The concentration of \(\mathrm{H}^{+}(a q)\) and \(\mathrm{OH}^{-}(a q)\) ions is equal in pure water.

Step 2: Calculate the \(\mathrm{H}^{+}\) concentration.

Since the concentration of \(\mathrm{H}^{+}(a q)\) and \(\mathrm{OH}^{-}(a q)\) ions is equal in pure water, we can write the equation \[[\mathrm{H}^{+}]^2 = K_{w}\] Now, calculate the concentration of \(\mathrm{H}^{+}(a q)\) ions: \[[\mathrm{H}^{+}] = \sqrt{K_{w}} = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \, \text{mol/L}\]

Step 3: Convert volume to liters.

Since we're given the volume in milliliters, we need to convert it to liters before proceeding. \[1.0 \, \text{mL} = \frac{1.0}{1000} \, \text{L} = 1.0 \times 10^{-3} \, \text{L}\]

Step 4: Calculate the number of \(\mathrm{H}^{+}(a q)\) ions.

To find the number of \(\mathrm{H}^{+}(a q)\) ions in \(1.0 \mathrm{~mL}\) of pure water, we simply multiply the concentration of \(\mathrm{H}^{+}(a q)\) ions with the volume of water in liters: \[\text{Number of } \mathrm{H}^{+}\text{ ions} = [\text{H}^{+}](V) = (1.0 \times 10^{-7} \, \text{mol/L})(1.0 \times 10^{-3}\, \text{L}) = 1.0 \times 10^{-10}\, \text{mol}\]

Step 5: Convert moles to ions.

Finally, we will convert the moles of \(\mathrm{H}^{+}(a q)\) ions to the actual number of ions using Avogadro's number: \[\text{Number of }\mathrm{H}^{+}\text{ ions} = 1.0 \times 10^{-10} \, \text{mol} \times \frac{6.022 \times 10^{23} \, \text{ions}}{1\, \text{mol}}=6.022\times 10^{13}\, \text{ions}\] So, there are approximately \(6.022 \times 10^{13}\) \(\mathrm{H}^{+}(a q)\) ions in \(1.0 \mathrm{~mL}\) of pure water at \(25^{\circ} \mathrm{C}\).

Key Concepts

Ion Product Constant

The ion product constant of water, commonly expressed as \( K_w \), is a fundamental property of water that measures the equilibrium between hydrogen ions \( \text{H}^+ \) and hydroxide ions \( \text{OH}^- \) at a given temperature. At \( 25^\circ \text{C} \), this equilibrium is described by the equation:

• \( K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \)

Both the hydrogen and hydroxide ion concentrations are equal in pure water, meaning the concentration of each ion is \( \sqrt{K_w} \).
This value of \( K_w \) remains constant under standard conditions and is crucial for determining the pH of neutral water, which is 7, as both ion concentrations are \( 1.0 \times 10^{-7} \text{ mol/L} \).
Understanding \( K_w \) helps in comprehending how changes in temperature and other conditions could influence ion concentration in solutions.

Avogadro's Number

Avogadro's number, also known as Avogadro's constant, is a key concept in chemistry representing the number of constituent particles (usually atoms or molecules) in one mole of a substance. This number is expressed as:

• \( 6.022 \times 10^{23} \) particles/mol

It provides a bridge between the macroscopic scale (moles) and the microscopic scale (individual ions or molecules).
This constant is immensely helpful in calculating the number of ions in a given volume of water once the molar concentration is known.
For example, when you find \( 1.0 \times 10^{-10} \text{ mol} \) of \( \text{H}^+ \) ions in 1 mL of water, multiplying by Avogadro's number gives the actual count of ions, which is approximately \( 6.022 \times 10^{13} \) ions.
Thus, Avogadro’s number allows chemists to convert between the microscopic realm of ions and observable quantities.

Molar Concentration

Molar concentration or molarity is a measure of the concentration of a solute in a solution. It quantifies the number of moles of a solute per liter of solution and is denoted as \( \text{mol/L} \). In the context of pure water, the challenge is often to find the concentration of hydrogen ions \( \text{H}^+ \) or hydroxide ions \( \text{OH}^- \).

• For pure water at \( 25^\circ \text{C} \), the \( \text{H}^+ \) concentration is \( 1.0 \times 10^{-7} \text{ mol/L} \).

Once the molar concentration is determined, it can be used to find the number of moles in a specific volume.
For instance, multiplying the \( \text{H}^+ \) concentration by the volume in liters gives the moles of \( \text{H}^+ \), which can then be converted to the number of ions using Avogadro's number. Molarity is a central concept in quantifying solutes in chemistry.

Pure Water

Pure water is a unique substance characterized by its neutrality and equilibrium, where the concentrations of \( \text{H}^+ \) and \( \text{OH}^- \) ions are equal. In pure water:

• The concentrations of \( \text{H}^+ \) and \( \text{OH}^- \) are \( 1.0 \times 10^{-7} \text{ mol/L} \) at \( 25^\circ \text{C} \).
• The pH is 7, indicating neutrality.

This state is true at the ion product constant \( K_w = 1.0 \times 10^{-14} \) and is used as a baseline measurement in pH calculations.
The concept of pure water, with its balanced ionic concentration, serves as an essential reference point in chemistry for understanding the properties of solutions and how they differ when solutes are added.
Even though pure water is often considered as a baseline, its properties like conductivity arise due to the presence of these minimal but significant concentrations of ions.