Question

If a neutral solution of water, with \(\mathrm{pH}=7.00,\) is cooled to \(10^{\circ} \mathrm{C},\) the pH rises to \(7.27 .\) Which of the following three statements is correct for the cooled water: (i) \(\left[\mathrm{H}^{+}\right]>\left[\mathrm{OH}^{-}\right]\) (ii) \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right], \mathrm{or}\) (iii) \(\left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right] ?\)

Short answer

The correct statement for the cooled water is (iii) \(\left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right]\).

Step-by-step solution

Understand the ion product of water

In a pure water solution, the concentration of hydrogen ions (H+) and hydroxide ions (OH-) are equal, and at 25°C, their product is a constant known as the ion product of water (Kw): \[K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1 \times 10^{-14}\]

Calculate the concentrations of H+ and OH- ions at the initial temperature

For a neutral solution of water with a pH of 7.00, the concentration of hydrogen ions (H+) is: \([H+] = 10^{-\mathrm{pH}} = 10^{-7.00} = 1 \times10^{-7} M\). Since it is a neutral solution, the concentration of hydroxide ions (OH-) is equal to the concentration of hydrogen ions: \([OH^-] = [H^+] = 1 \times 10^{-7} M\).

Calculate the concentrations of H+ and OH- ions in the cooled water

We are given the pH of the cooled water (7.27). To find the concentration of hydrogen ions (H+) in the cooled water, we need to use the same formula we just used: \([H^+] = 10^{-\mathrm{pH}} = 10^{-7.27} M\) We have to find the concentration of OH- ions in the cooled water. To do this, we can use the formula for the ion product of water (Kw): \[K_w = [H^+][OH^-]\] Rearrange the equation to find the concentration of OH- ions: \[ [OH^-] = \frac{K_w}{[H^+]}\]

Compare the concentrations of H+ and OH- ions in the cooled water

Plug in the values for Kw and [H+] that we calculated earlier: \[ [OH^-] = \frac{1 \times 10^{-14}}{10^{-7.27}}\] After calculating the value, we get: \([OH^-] = 1.89 \times 10^{-7} M\) Since \([H^+] = 1.89 \times10^{-7} M<{[OH^-]} = 1\times 10^{-7} M\), statement (iii) is correct: \[ \left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right].\]

Key concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a measure of the amount of hydrogen ions, \(\mathrm{H}^+\), present in a solution. This concept is central to understanding pH, which quantifies the acidity or basicity of a solution. \(\mathrm{H}^+\) ions are crucial because they determine the solution's pH level:

\[ \mathrm{pH} = -\log_{10}([\mathrm{H}^+]) \]

For instance, a neutral solution typically has a pH of 7, meaning the concentration of hydrogen ions is \(1 \times 10^{-7} \, M\).

In both neutral and acidic solutions, the concentration of \(\mathrm{H}^+\) is a decisive factor. If a solution has more \(\mathrm{H}^+\) than \(\mathrm{OH}^-\) (hydroxide ions), it is acidic. If there's less, the solution is basic. Understanding this helps us interpret pH changes when solutions are cooled or heated. Adjustments in temperature can shift the balance of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) ions.

Neutral Solution

A neutral solution is one where the concentration of hydrogen ions \(\left[\mathrm{H}^+\right]\) is equal to the concentration of hydroxide ions \(\left[\mathrm{OH}^-\right]\). This balance results in a pH of 7 at 25°C.

In a neutral solution, the product of these ion concentrations equals the ion product of water:

- \(\left[\mathrm{H}^+\right] = 1 \times 10^{-7} \, M\)
- \(\left[\mathrm{OH}^-\right] = 1 \times 10^{-7} \, M\)

This balance implies that neither acidic nor basic properties dominate. A neutral solution can still vary in its exact ion concentrations due to temperature changes, as seen when water is cooled. Cooling affects the \(\mathrm{pH}\), but as long as \(\left[\mathrm{H}^+\right] = \left[\mathrm{OH}^-\right]\), it remains neutral in its nature.

Ion Product of Water

The ion product of water \(K_w\) is the constant product of the hydrogen ion and hydroxide ion concentrations in pure water. At 25°C, this value is \(1 \times 10^{-14}\). It serves as a benchmark for calculating ion concentrations and understanding solution behavior:

\[ K_w = [\mathrm{H}^+][\mathrm{OH}^-] \]

When water temperature changes, such as cooling, \(K_w\) remains constant, but the individual concentrations of \(\mathrm{H}^+\) and \(\mathrm{OH}^-\) will adjust to maintain this balance. The shift in ion concentrations results in a change in pH, even if the solution stays neutral under different temperatures. This principle explains why the \(\mathrm{pH}\) of water rises when it is cooled, yet remains neutral, because the ion concentrations adapt while keeping the product stable.