Question

At the freezing point of water \(\left(0^{\circ} \mathrm{C}\right), K_{w}=1.2 \times 10^{-15}\). Calculate \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) for a neutral solution at this temperature.

Short answer

In a neutral solution at the freezing point of water, the concentrations of hydrogen ions and hydroxide ions are approximately equal to \(1.1 \times 10^{-8} \,\mathrm{M}\) each.

Step-by-step solution

Write down the ion product of water expression

The ion product of water expression is given by: \(K_w=\left[\mathrm{H}^{+}\right] \cdot \left[\mathrm{OH}^{-}\right]\)

Set up the equation for a neutral solution

In a neutral solution, the concentration of hydrogen ions is equal to the concentration of hydroxide ions: \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right]\)

Substitute values into the Ion Product of Water expression

Using the given value for \(K_w\) at the freezing point of water and the equation from Step 2, we can set up an equation for the ion concentrations: \(1.2 \times 10^{-15} = \left[\mathrm{H}^{+}\right] \cdot \left[\mathrm{H}^{+}\right]\)

Solve for the hydrogen ion concentration

To find the concentration of hydrogen ions, we can solve the equation above: \(\left[\mathrm{H}^{+}\right]=\sqrt{1.2 \times 10^{-15}}\) \(\left[\mathrm{H}^{+}\right] \approx 1.1 \times 10^{-8} \,\mathrm{M}\)

Determine the hydroxide ion concentration

Since the solution is neutral, the concentration of hydroxide ions is equal to the concentration of hydrogen ions: \(\left[\mathrm{OH}^{-}\right]=\left[\mathrm{H}^{+}\right]\) \(\left[\mathrm{OH}^{-}\right]\approx 1.1 \times 10^{-8}\,\mathrm{M}\)

Write the Final Answer

The concentrations of hydrogen ions and hydroxide ions in a neutral solution at the freezing point of water are approximately equal to \(1.1 \times 10^{-8} \,\mathrm{M}\).

Key concepts

Neutral Solution

In chemistry, a neutral solution is one where the concentration of hydrogen ions \((\text{H}^+)\) is equal to the concentration of hydroxide ions \((\text{OH}^-)\). This balance creates a pH of exactly 7 at room temperature. However, at different temperatures, the neutral point can vary slightly.

At the freezing point of water, the conditions differ from room temperature. Despite this, the concept of neutrality holds true: \[\left[\text{H}^+\right] = \left[\text{OH}^-\right].\] This equality results in a balanced solution, crucial for understanding how water self-ionizes under different thermal conditions.

Hydrogen Ion Concentration

The hydrogen ion concentration \((\left[\text{H}^+\right])\) is an essential part of water chemistry. It determines the acidity of a solution. In pure water, this concentration is typically very low, reflecting water’s status as a weak acid.

At the freezing point of water, the ion product \(K_w = 1.2 \times 10^{-15}\) helps us find \([\text{H}^+]\). Since the solution is neutral, we have: \[\left[\text{H}^+\right] = \sqrt{1.2 \times 10^{-15}} \approx 1.1 \times 10^{-8} \text{M}.\] This highlights that even at low temperatures, hydrogen ions persist due to water's autoprotolysis.

Hydroxide Ion Concentration

Similarly to hydrogen ions, hydroxide ions \((\left[\text{OH}^-\right])\) play a critical role in determining the basicity of a solution. In neutral solutions, like those at 0°C, the concentration of hydroxide ions is directly equal to that of hydrogen ions.

With \(K_w = 1.2 \times 10^{-15}\), and knowing \(\left[\text{H}^+\right] = 1.1 \times 10^{-8} \, \text{M}\), we find: \[\left[\text{OH}^-\right] = 1.1 \times 10^{-8} \, \text{M}.\] This equivalence shows that the ion balance is maintained in neutral water, emphasizing the self-regulating nature of water's chemistry.

Freezing Point of Water

The freezing point of water is the temperature at which it transitions from liquid to solid, precisely 0°C or 32°F at standard atmospheric pressure. At this point, the ion product of water \(K_w\) becomes a crucial factor.

Unlike at room temperature, where \(K_w\) is about \(1.0 \times 10^{-14}\), at 0°C, \(K_w\) decreases to \(1.2 \times 10^{-15}\). This change dictates the concentrations of hydrogen and hydroxide ions in the solution, demonstrating the complex interplay between temperature and ion equilibrium.

Understanding \(K_w\) at the freezing point provides valuable insights into the behavior of aqueous solutions in various thermal conditions.