Question

At \(40 .^{\circ} \mathrm{C}\) the value of \(K_{\mathrm{w}}\) is \(2.92 \times 10^{-14}\) a. Calculate the \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) in pure water at \(40 .^{\circ} \mathrm{C} .\) b. What is the pH of pure water at \(40 .^{\circ} \mathrm{C} ?\) c. If the hydroxide ion concentration in a solution is 0.10\(M\) what is the pH at \(40 .^{\circ} \mathrm{C} ?\)

Short answer

a. The concentrations of H+ and OH- ions in pure water at 40°C are approximately \(1.71 \times 10^{-7}\,M\). b. The pH of pure water at 40°C is approximately 6.77. c. The pH of the solution when the OH- concentration is 0.10 M at 40°C is approximately 12.53.

Step-by-step solution

Write the ion product expression for water

The ion product of water is given by the following expression: \(K_w = [H^+][OH^-]\).

Use the given Kw value to find the H+ and OH- concentrations

We are given a value of \(K_w = 2.92 \times 10^{-14}\) at 40°C. Since pure water is neutral, it has equal concentrations of H+ and OH- ions. Let the concentrations of both ions be x. So, we have: \[K_w = x^2\] Now, substitute the given Kw value and solve for x: \[(2.92 \times 10^{-14}) = x^2\] To find x, take the square root of both sides: \[x = \sqrt{2.92 \times 10^{-14}}\]

Calculate x, the concentrations of H+ and OH- ions

By calculating the square root, we find x: \[x \approx 1.71 \times 10^{-7}\] Thus, the concentrations of H+ and OH- ions in pure water at 40°C are approximately \(1.71 \times 10^{-7}\,M\). b. To find the pH of pure water at 40°C:

Use the pH formula and the H+ concentration

The pH formula is given by: \[pH = -\log[H^+]\] Now, substitute the H+ concentration we calculated in part a: \[pH = -\log(1.71 \times 10^{-7})\]

Calculate the pH

By calculating the logarithm of the value, we find the pH: \[pH \approx 6.77\] Thus, the pH of pure water at 40°C is approximately 6.77. c. To find the pH when the OH- concentration is 0.10 M at 40°C:

Use the given OH- concentration and the Kw value to find the H+ concentration

We are given the OH- concentration is 0.10 M. We can use the expression for the ion product of water to find the H+ concentration: \[K_w = [H^+][OH^-]\] Substitute the given values and solve for the H+ concentration: \[(2.92 \times 10^{-14}) = [H^+] (0.10)\] Divide both sides by 0.10 to find the H+ concentration: \[[H^+] = \frac{2.92 \times 10^{-14}}{0.10} \approx 2.92 \times 10^{-13}\]

Calculate the pH using the H+ concentration

Now, we can use the pH formula to find the pH at 40°C when the OH- concentration is 0.10 M: \[pH = -\log(2.92 \times 10^{-13})\] By calculating the logarithm, we find the pH: \[pH \approx 12.53\] Thus, the pH of the solution when the OH- concentration is 0.10 M at 40°C is approximately 12.53.

Key concepts

pH calculation

The pH of a solution is a measure of its acidity or basicity. It is calculated using the formula: \[pH = -\log[H^+]\] where \([H^+]\) is the hydrogen ion concentration in the solution. This formula is derived from the logarithmic scale of the hydrogen ion concentration.
For pure water, which is neutral, the concentration of hydrogen ions \([H^+]\) is equal to that of hydroxide ions \([OH^-]\), leading to a balanced pH. Typically, at 25°C, the pH of pure water is 7. However, this changes with temperature. - For instance, at 40°C, when calculating the pH of pure water, we use the specific \(K_w\) value for that temperature. - At 40°C, with \(K_w = 2.92 \times 10^{-14}\), the pH of pure water was determined to be approximately 6.77, indicating slight acidity compared to the standard 7. By understanding how to compute the pH using the hydrogen ion concentration, you can determine how acidic or basic a solution is.

hydrogen ion concentration

Hydrogen ion concentration, denoted as \([H^+]\), is a vital component in understanding the acidity of a solution. It represents the number of hydrogen ions in a specific volume of solution. The higher the \([H^+]\), the more acidic the solution is.In chemical equilibria, particularly with water, the hydrogen ion concentration is linked with the hydroxide ion concentration \([OH^-]\) through the ion product of water, \(K_w\): \[K_w = [H^+][OH^-]\] At 40°C, this equilibrium gives us \(K_w = 2.92 \times 10^{-14}\). In pure water, \([H^+]\) and \([OH^-]\) are equal, allowing us to calculate their concentration by solving \(K_w = x^2\) to find: - \(x = \sqrt{2.92 \times 10^{-14}} = 1.71 \times 10^{-7} M\). Understanding the hydrogen ion concentration is crucial for calculating the pH and determining a solution's properties. It dictates the acidity level and helps in adjusting conditions for various chemical reactions.

hydroxide ion concentration

Hydroxide ion concentration \([OH^-]\) is a critical factor when discussing basic solutions. It measures the quantity of hydroxide ions in a solution. Much like \([H^+]\) affects acidity, \([OH^-]\) influences basicity.Both \([H^+]\) and \([OH^-]\) are linked through the ion product of water, \(K_w\). For instance, in an alkaline solution with a given hydroxide ion concentration (e.g., 0.10 M), you can determine the corresponding hydrogen ion concentration using: \[K_w = [H^+][OH^-]\] With \(K_w = 2.92 \times 10^{-14}\) at 40°C: - Using \([OH^-] = 0.10 M\), calculate \([H^+]\) as \([H^+] = \frac{2.92 \times 10^{-14}}{0.10} = 2.92 \times 10^{-13} M\). By knowing the hydroxide ion concentration, a simple re-arrangement of the ion product equation can inform you about the hydrogen ion concentration and, subsequently, the pH of your solution. This calculation is essential for adjusting conditions in processes requiring specific pH levels and understanding the solution's nature.