Problem 31 - Chemistry: The Central Science

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.

Step-by-Step Solution

Answer

In a neutral solution at 0°C, the concentrations of H⁺ and OH⁻ ions are approximately \(1.09 \times 10^{-8}\) M.

Step 1: Write down the ion product constant of water equation for a neutral solution

The ion product constant of water equation is given by: \(K_w = \left[\mathrm{H}^{+}\right]\left[\mathrm{OH}^{-}\right]\) Since we have a neutral solution, the concentrations of H⁺ and OH⁻ ions are equal: \(\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] = x\) Now we can rewrite the equation as: \(K_w = x^2\)

Step 2: Substitute the given value of \(K_w\) and solve for x

We are given the value of \(K_w\) at 0°C: \(K_w = 1.2 \times 10^{-15}\) Now substitute this value into the equation and solve for x: \(1.2 \times 10^{-15} = x^2\) To solve for x, take the square root of both sides: \(x = \sqrt{1.2 \times 10^{-15}}\)

Step 3: Calculate x's numeric value

Now let's calculate the numeric value of x: \(x \approx 1.09 \times 10^{-8}\)

Step 4: Find the concentrations of H⁺ and OH⁻ ions

Since x is equal to the concentrations of H⁺ and OH⁻ ions in the neutral solution, we can write the final values as: \(\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right] \approx 1.09 \times 10^{-8} \, \mathrm{M}\) So, at 0°C, in a neutral solution, the concentrations of H⁺ and OH⁻ ions are approximately \(1.09 \times 10^{-8}\) M.

Key Concepts

Freezing Point

The freezing point is an essential concept when discussing the behavior of substances like water. At the freezing point, which for water is 0 degrees Celsius (32 degrees Fahrenheit), water begins to transition from a liquid to a solid state. This temperature is significant because the properties of water change and this also affects its chemical behavior.

For instance, the ion product constant of water, denoted as \( K_w \), depends on the temperature. At 0°C, \( K_w \) is \( 1.2 \times 10^{-15} \), quite different from its value at 25°C (which is \( 1.0 \times 10^{-14} \)). This implies that water behaves differently at its freezing point compared to higher temperatures.

• Understanding the freezing point helps to predict how substances dissolve or ionize under different thermal conditions.
• The freezing point is fundamental for processes such as crystallization and for understanding weather patterns.

Neutral Solution

A neutral solution is a special state where the concentration of hydrogen ions \( [\mathrm{H}^+] \) equals the concentration of hydroxide ions \( [\mathrm{OH}^-] \). This balance results in a solution that is neither acidic nor basic, typically understood as having a pH of 7 at standard conditions (25°C).

However, this balance changes with temperature. At the freezing point (0°C), the concentrations of ions are determined by the \( K_w \) value at that temperature. In a neutral solution at 0°C, \( K_w = 1.2 \times 10^{-15} \), so both \( [\mathrm{H}^+] \) and \( [\mathrm{OH}^-] \) are approximately \( 1.09 \times 10^{-8} \, \text{M} \). This deviation from the classic \( 1.0 \times 10^{-7} \) M at room temperature is due to temperature's impact on ionization.

• Neutral solutions help to understand varying pH conditions across different temperatures.
• These principles are applicable in fields like biology and environmental science.

Hydrogen Ion Concentration

Hydrogen ion concentration, represented as \( [\mathrm{H}^+] \), is pivotal in determining the acidity of a solution. The concentration level signifies a solution's pH, which reflects its acidic or basic nature.

In the context of a neutral solution at the freezing point of water (0°C), the calculation involves using the ion product constant of water \( K_w \). By solving the equation \( K_w = [\mathrm{H}^+][\mathrm{OH}^-] \), where both concentrations are equal, we find \( [\mathrm{H}^+] = 1.09 \times 10^{-8} \, \text{M} \).

• This value affects how reactions proceed in biochemical and industrial processes under varying thermal conditions.
• The concept is crucial in learning about buffers, neutralization, and water chemistry.