Problem 27 - Chemistry: The Central Science

Question

If a neutral solution of water, with \(\mathrm{pH}=7.00\), is cooled to \(10^{\circ} \mathrm{C}\), the \(\mathrm{pH}\) 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) \(\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right]\), or (iii) \(\left[\mathrm{H}^{+}\right]<\left[\mathrm{OH}^{-}\right]\)?

Step-by-Step Solution

Answer

The correct statement for the cooled water when the pH rises to 7.27 is (iii) \([\mathrm{H}^+] < [\mathrm{OH}^-]\).

Step 1: Calculate the concentration of H⁺ ions in the cooled water

We are given the pH of water when cooled to \(10^\circ \mathrm{C}\), which is equal to 7.27. Using the pH formula, we can calculate the concentration of H⁺ ions: \[ \mathrm{pH} = -\log_{10} [\mathrm{H}^+] \] Now, we can solve for the concentration of H⁺ ions: \[ [\mathrm{H}^+] = 10^{-\mathrm{pH}} = 10^{-7.27} \]

Step 2: Use the ion product of water to find the concentration of OH⁻ ions

The ion product of water (\(K_\mathrm{w}\)), which is a constant, is given by: \[ K_\mathrm{w} = [\mathrm{H}^+][\mathrm{OH}^-] \] At 25°C, the value of \(K_\mathrm{w}\) is \(1.0 \times 10^{-14}\). Since we are dealing with a neutral solution, the value of \(K_\mathrm{w}\) will not be significantly different at \(10^\circ \mathrm{C}\). Using the ion product of water and the concentration of H⁺ ions, we can now find the concentration of OH⁻ ions: \[ [\mathrm{OH}^-] = \frac{K_\mathrm{w}}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{10^{-7.27}} \]

Step 3: Compare the concentrations of H⁺ and OH⁻ ions

Compare the concentrations of H⁺ and OH⁻ ions to determine which statement is correct: (i)\([\mathrm{H}^+] > [\mathrm{OH}^-]\), (ii)\([\mathrm{H}^+] = [\mathrm{OH}^-]\), or (iii)\([\mathrm{H}^+] < [\mathrm{OH}^-]\) Since we know the values of \([\mathrm{H}^+]\) and \([\mathrm{OH}^-]\), we can determine the correct statement: If \(10^{-7.27} > \frac{1.0 \times 10^{-14}}{10^{-7.27}}\), then statement (i) is correct. Similarly, we can check for the other statements. After comparing the values, we can determine which statement is correct.

Key Concepts

pH Scale

The pH scale is a measure of acidity or alkalinity of a solution. It is a logarithmic scale, meaning that each whole pH value below 7 is ten times more acidic than the next higher value, and each whole pH value above 7 is ten times more alkaline than the next lower value. The scale ranges typically from 0 to 14, with 7 being considered neutral. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic, or alkaline solution. This concept is critical in water chemistry as it determines the nature of the solutes in water.

For the given exercise, the pH of water has risen from 7.00 to 7.27 when cooled, suggesting that the water has become less acidic and more alkaline. Understanding the pH scale helps to make sense of why the concentration of H⁺ ions is changing relative to the OH⁻ ions as the temperature varies.

Ion Product of Water

The ion product of water (denoted as Kw) is the constant product of the concentrations of H⁺ and OH⁻ ions in water. At a specific temperature, this product is constant and it's a fundamental constant of water chemistry. For most purposes, Kw is taken to be approximately 1.0 x 10^-14 at 25°C. The equilibrium represented by this constant shows that as the concentration of hydrogen ions in water increases, the concentration of hydroxide ions decreases, and vice versa, maintaining the product at a constant value. This balance plays a crucial role in understanding the behavior of aqueous solutions and their pH levels.

In our exercise example, we are making use of the ion product to find the concentration of OH⁻ ions by dividing the constant Kw by the calculated concentration of H⁺ ions, which allows us to compare their respective concentrations to ascertain the acidity or alkalinity of the cooled water.

Concentration of H⁺ Ions

The concentration of H⁺ ions in a solution is a direct measure of its acidity. In the pH scale, the pH of a solution is inversely related to the concentration of H⁺ ions; that is, the lower the pH, the higher the concentration of H⁺ ions. This relationship is expressed mathematically by the pH formula, pH = -log[H⁺].

By using this formula, we are able to calculate the concentration of H⁺ ions from the given pH value in our exercise. As the temperature of water decreases, the dynamics of water chemistry change and typically result in a slight decrease in the ionization of water, which can be seen as a change in the concentration of H⁺ ions and consequently, pH.

Concentration of OH⁻ Ions

The concentration of hydroxide ions (OH⁻) is representative of a solution's alkalinity. In water chemistry, OH⁻ ions are important counterparts to H⁺ ions. If the concentration of H⁺ ions governs the acidity of a solution, then the concentration of OH⁻ ions governs its basicity. While pH is the more commonly used measure of acidity or alkalinity, the pOH is sometimes used as a measure of the alkalinity and is related to the pH through the equation pOH = 14 - pH.

In the case presented in our exercise, we calculate the concentration of OH⁻ ions by dividing the ion product constant by the previously determined concentration of H⁺ ions. The resulting concentration will help us to decide the relative concentration balance between the H⁺ ions and OH⁻ ions in the cooled water, which is a crucial step to understanding the solution's final pH.