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 0°C with \(K_w = 1.2 \times 10^{-15}\), both the hydrogen ion concentration and the hydroxide ion concentration are approximately \(3.5 \times 10^{-8} M\).

Step-by-step solution

Write the Ion Product of Water Expression

For a neutral solution at 0°C, the ion product of water, \(K_w\), is given as \(1.2 \times 10^{-15}\). We can write the expression for the ion product of water as follows: \[K_w = \left[\mathrm{H}^{+}\right]^2\] Here, \(K_w\) is the ion product constant for water. The concentrations of hydrogen ions and hydroxide ions are equal in a neutral solution, since the number of hydrogen ions and the number of hydroxide ions are equal.

Solve for the Hydrogen Ion Concentration

Plug in the given value for \(K_w\) and solve for \(\left[\mathrm{H}^{+}\right]\): \(1.2\times 10^{-15} = \left[\mathrm{H}^{+}\right]^2\) To find the hydrogen ion concentration, take the square root of both sides: \(\left[\mathrm{H}^{+}\right] = \sqrt{1.2\times 10^{-15}}\)

Calculate the Hydrogen Ion Concentration

Evaluate the square root: \(\left[\mathrm{H}^{+}\right] \approx 3.5 \times 10^{-8} M\)

Find the Hydroxide Ion Concentration

Since the solution is neutral, the hydroxide ion concentration equals the hydrogen ion concentration: \(\left[\mathrm{OH}^{-}\right] = \left[\mathrm{H}^{+}\right]\) Thus, \(\left[\mathrm{OH}^{-}\right] \approx 3.5 \times 10^{-8} M\). In conclusion, for a neutral solution at 0°C, both the hydrogen ion concentration and the hydroxide ion concentration are approximately \(3.5 \times 10^{-8} M\).

Key concepts

Neutral Solution

In chemistry, a neutral solution is one where the concentrations of hydrogen ions (\(\left[\mathrm{H}^{+}\right]\)) and hydroxide ions (\(\left[\mathrm{OH}^{-}\right]\)) are equal.This balance of ions results in a solution that is neither acidic nor basic.At different temperatures, the ion product of water, \(K_w\), varies, which affects the exact concentration of these ions in neutral solutions.
For example, at 0°C, which is the freezing point of water, the value of \(K_w\) is \(1.2 \times 10^{-15}\).This is different from the value at 25°C (\(1.0 \times 10^{-14}\)), which is more commonly used in room temperature conditions. Understanding how \(K_w\) changes with temperature helps predict the behavior of ions in a solution as conditions vary.
In practice, when dealing with neutral solutions at different temperatures, always remember:

• \([\mathrm{H}^{+}] = [\mathrm{OH}^{-}]\)
• The product \([\mathrm{H}^{+}][\mathrm{OH}^{-}] = K_w\)

This knowledge helps in calculating ion concentrations within neutral solutions.

Hydrogen Ion Concentration

The hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) is a measure of the acidity of a solution.In a neutral solution at any given temperature, \(\left[\mathrm{H}^{+}\right]\) is equal to the hydroxide ion concentration.To determine this concentration at a specific temperature, such as 0°C, we use the ion product of water, \(K_w\).
Given the equation \(K_w = \left[\mathrm{H}^{+}\right]^2\) for a neutral solution, solving for \(\left[\mathrm{H}^{+}\right]\) involves taking the square root of \(K_w\).For instance, at 0°C, \(K_w = 1.2\times 10^{-15}\), so we find:\[\left[\mathrm{H}^{+}\right] = \sqrt{1.2\times 10^{-15}}\]Evaluating this expression gives \(\left[\mathrm{H}^{+}\right] \approx 3.5 \times 10^{-8} \) M.This calculation reveals that the concentration of hydrogen ions in a neutral solution varies with temperature,underlining the importance of knowing \(K_w\) for precise acidity measurement.

Hydroxide Ion Concentration

The concentration of hydroxide ions \(\left[\mathrm{OH}^{-}\right]\) in a neutral solution is equal to that of hydrogen ions \(\left[\mathrm{H}^{+}\right]\).This balance is crucial for maintaining neutrality within a solution.When \(\left[\mathrm{H}^{+}\right]\) is known, \(\left[\mathrm{OH}^{-}\right]\) can be directly determined in neutral solutions. For example, at 0°C where \(K_w = 1.2 \times 10^{-15}\), if \(\left[\mathrm{H}^{+}\right] = 3.5 \times 10^{-8}\) M:
Then, by definition of neutrality:\[\left[\mathrm{OH}^{-}\right] = \left[\mathrm{H}^{+}\right]\]Thus, \(\left[\mathrm{OH}^{-}\right] \approx 3.5 \times 10^{-8} \) M.This equality simplifies calculations, allowing us to focus on either ion and infer the other's concentration instantly.It highlights how neutrality in solutions is defined not by the exact value of ion concentrations but by their equality at a given temperature.Such understanding assists with predicting solution behavior as it transitions between neutral, acidic, or basic states.