Question

Is it possible for a solution to have \(\left[\mathrm{H}^{+}\right]=0.002 \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=5.2 \times 10^{-6} M\) at 25 C \(?\) Explain.

Short answer

No, it is not possible for a solution to have \(\left[\mathrm{H}^{+}\right]=0.002 \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=5.2 \times 10^{-6} \mathrm{M}\) at 25°C. This is because the ion product of the given solution \(K_{solution} = 1.04 \times 10^{-8}\) does not equal the required ion product constant value of \(K_w = 1.0 \times 10^{-14}\) at 25°C.

Step-by-step solution

Finding the ion product of water at 25°C

At 25°C, the ion product of water (Kw) is a constant value which can be found using the following expression: \[K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}]\] At 25°C, the value of Kw is well known and equal to: \[K_w = 1.0 \times 10^{-14}\] Now, let's test if the given ion concentrations of the solution meet the required value for Kw.

Calculate the ion product of the given solution

We are given the concentrations of both hydronium and hydroxide ions in the solution. \[\left[\mathrm{H}^{+}\right] = 0.002 \mathrm{M}\] \[\left[\mathrm{OH}^{-}\right] = 5.2 \times 10^{-6} \mathrm{M}\] Calculate the ion product of this solution using Kw expression: \[K_{solution} = [\mathrm{H}^{+}][\mathrm{OH}^{-}]\] Plug in the given concentrations: \[K_{solution} = (0.002 \mathrm{M})(5.2 \times 10^{-6} \mathrm{M})\]

Compare K_solution with Kw

Now, multiply the concentrations of both ions: \[K_{solution} = 1.04 \times 10^{-8}\] Since the ion product for the given solution \(K_{solution} \neq K_w\), the given concentrations of hydronium and hydroxide ions are not possible to maintain at 25°C, as it does not meet the required ion product constant value of \(1.0 \times 10^{-14}\). In conclusion, it is not possible for a solution to have \(\left[\mathrm{H}^{+}\right]=0.002 \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=5.2 \times 10^{-6} \mathrm{M}\) at 25°C.

Key concepts

Ion Product of Water

The ion product of water, denoted as \( K_w \), is a crucial concept in understanding the characteristics of aqueous solutions. At 25°C, this value is constant and equal to \( 1.0 \times 10^{-14} \). This constant represents the product of the molar concentrations of hydrogen ions \( [\mathrm{H}^+] \) and hydroxide ions \( [\mathrm{OH}^-] \) in pure water. This equilibrium expression shows the self-ionization of water: \[ \mathrm{H_2O} (l) \rightleftharpoons \mathrm{H^+} (aq) + \mathrm{OH^-} (aq) \] This equation implies that any water-based solution, regardless of its pH level, will balance these ions in a way that their product always equals \( K_w \). Thus, if the concentration of one increases, the other must decrease to maintain this product. Understanding this relationship helps determine the acidic or basic nature of a solution and is foundational in calculations involving pH and pOH.

pH and pOH

pH and pOH are numerical scales used to specify the acidity or basicity of an aqueous solution. These values are inversely related to the concentrations of hydronium and hydroxide ions. The pH is calculated using the formula: \[ \text{pH} = -\log [\mathrm{H}^+] \] While pOH is determined through: \[ \text{pOH} = -\log [\mathrm{OH}^-] \] For any aqueous solution at 25°C, the sum of pH and pOH is always 14: \[ \text{pH} + \text{pOH} = 14 \] This relationship is derived from the constant \( K_w \) and provides a simple method to find one value if the other is known.
Knowing the pH provides insights into the acidity of a solution:

• A pH less than 7 indicates an acidic solution.
• A pH of 7 is neutral.
• A pH greater than 7 indicates a basic or alkaline solution.

Understanding pH and pOH can help gauge the reactivity and behavior of chemical substances in solutions.

Concentration Calculations

In chemistry, calculating the concentration of ions is fundamental for analyzing solutions. Concentration is often expressed in molarity (M), which is the number of moles of a solute per liter of solution. When given concentrations of \([\mathrm{H}^+]\) and \([\mathrm{OH}^- ]\), we can determine if they adhere to expected norms at a specified temperature, often 25°C in standard conditions.
To perform these calculations:

• Identify the known concentration values.
• Utilize the ion product of water formula \( [\mathrm{H}^+]\times [\mathrm{OH}^- ]= K_{w} \).
• Input the given concentrations to confirm if \( K_{solution} = K_{w} \).

If \( K_{solution} \) doesn't equal the expected \( K_{w} \), it suggests that such a concentration combination is not feasible under the given conditions. Effective concentration calculations are crucial for predicting how solutions behave and for designing chemical reactions accurately.