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 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 is (iii): \(\left[\mathrm{H}^{+}\right] < \left[\mathrm{OH}^{-}\right]\) for the cooled water at 10ºC with a pH of 7.27.

Step 1: Understanding the relationship between pH, pOH, \(\mathrm{H}^{+}\), and \(\mathrm{OH}^{-}\) concentrations

The pH of a solution is the measure of its acidity. The pH is calculated using the following formula: pH = -log \(\left[\mathrm{H}^{+}\right]\) Additionally, there's a relationship between pH and pOH: pH + pOH = 14 where pOH is the measure of a solution's basicity and is calculated using the following formula: pOH = -log \(\left[\mathrm{OH}^{-}\right]\) We will need this relationship to compare the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in the cooled water.

Step 2: Calculating the pOH of the cooled water at 10ºC

We are given the pH of the cooled water, which is 7.27. We can use the relationship between pH and pOH to find the pOH of the cooled water: pOH = 14 - pH pOH = 14 - 7.27 = 6.73

Step 3: Calculating the \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) concentrations of the cooled water

Next, we need to calculate the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) using the formulas for pH and pOH: \(\left[\mathrm{H}^{+}\right] = 10^{-\mathrm{pH}}\) \(\left[\mathrm{OH}^{-}\right] = 10^{-\mathrm{pOH}}\) When we plug in the values of pH and pOH we found earlier, we get: \(\left[\mathrm{H}^{+}\right] = 10^{-7.27} \approx 5.36 \times 10^{-8} \mathrm{M}\) \(\left[\mathrm{OH}^{-}\right] = 10^{-6.73} \approx 1.87 \times 10^{-7} \mathrm{M}\)

Step 4: Comparing the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) to determine the correct statement

Now that we have the concentrations of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\), we can compare them and find the correct option between (i), (ii), or (iii): (i) \(\left[\mathrm{H}^{+}\right] > \left[\mathrm{OH}^{-}\right]\): \(5.36 \times 10^{-8} \mathrm{M} > 1.87 \times 10^{-7} \mathrm{M} \Rightarrow\) False (ii) \(\left[\mathrm{H}^{+}\right] = \left[\mathrm{OH}^{-}\right]\): \(5.36 \times 10^{-8} \mathrm{M} = 1.87 \times 10^{-7} \mathrm{M} \Rightarrow\) False (iii) \(\left[\mathrm{H}^{+}\right] < \left[\mathrm{OH}^{-}\right]\): \(5.36 \times 10^{-8} \mathrm{M} < 1.87 \times 10^{-7} \mathrm{M} \Rightarrow\) True Hence, the correct statement is (iii): \(\left[\mathrm{H}^{+}\right] < \left[\mathrm{OH}^{-}\right]\) for the cooled water at 10ºC with a pH of 7.27.

Key Concepts

Acid-Base Equilibrium

In chemistry, the acid-base equilibrium is crucial for understanding how acids and bases interact in a solution. It typically involves the transfer of protons (H⁺) between molecules. Water (H₂O) can self-ionize to form H⁺ and OH^- ions, and the concentration of these ions defines the acidity or basicity of the solution. When we say a solution is neutral, it means that the concentrations of H⁺ and OH^- ions are equal, a condition typically observed in pure water at 25°C. However, temperature can affect this equilibrium, leading to changes in pH even without adding any acid or base.

For a solution to remain neutral at a given temperature, the product of the concentrations of H⁺ and OH^-, known as the ion product of water, must be constant. This concept helps us interpret the pH level changes with temperature variations, such as in the exercise's scenario, where cooling water leads to a pH increase.

Hydrogen Ion Concentration

The hydrogen ion concentration in a solution, represented as [H⁺], is a measure of the number of free protons present. The more H⁺ ions there are, the more acidic the solution is. This concentration plays a pivotal role in determining the pH value. High [H⁺] corresponds to low pH (acidic conditions), while low [H⁺] corresponds to high pH (basic conditions).

The pH scale, which runs typically from 0 to 14, allows us to quantify the level of acidity or basicity in a miraculous way that avoids unwieldy scientific notation. The scale is logarithmic, meaning a one-unit change in pH corresponds to a tenfold change in [H⁺]. So when comparing the pH of two solutions, it is important to remember that a small change in pH actually indicates a large change in the actual hydrogen ion concentration.

Hydroxide Ion Concentration

Conversely to hydrogen ions, hydroxide ions (OH^-) indicate the basicity of a solution. The hydroxide ion concentration, [OH^-], can be calculated from the pOH, which is the negative logarithm of the [OH^-] value. The relationship between the concentrations of [H⁺] and [OH^-] ions is such that when one increases, the other decreases to maintain the constant ion product of water. Hence, a higher hydroxide ion concentration means a lower hydrogen ion concentration, leading to a basic or alkaline solution. The assessment of [OH^-] is just as important as [H⁺] in understanding the solution's overall chemical properties.

pH Calculation

The pH calculation is a clear example of the logarithmic relationship between the concentration of H⁺ ions in a solution and the pH value. By using the formula pH = -log[H⁺], we convert the often small concentration numbers into a more manageable scale. Additionally, the relationship pH + pOH = 14 serves as a foundation for understanding the interdependence of acid and base properties: when we know the pH, we can easily find the pOH and vice versa.

In the exercise example, observing the pH increase when the water is cooled demonstrates how the balance of [H⁺] and [OH^-] shifts without adding any substances—just by changing the temperature. The cooled water at 10°C with a pH of 7.27 has a lesser concentration of H⁺ ions compared to OH^- ions, leading to a slightly basic nature, which could intrigue students and stimulate a comprehensive understanding of these nuanced concepts.