Problem 28 - Chemistry Matter and Change

Question

Calculate the pH and pOH of aqueous solutions with the following concentration at 298 \(\mathrm{K}\) . a. \(\left[\mathrm{OH}^{-}\right]=0.000033 M\) b. \(\left[\mathrm{H}^{+}\right]=0.0095 M\)

Step-by-Step Solution

Answer

For part a, we have pOH = -log(0.000033) = 4.48. Therefore, pH = 14 - pOH = 14 - 4.48 = 9.52. For part b, we have pH = -log(0.0095) = 2.02. Therefore, pOH = 14 - pH = 14 - 2.02 = 11.98.

Step 1: Definition of pH and pOH

pH is the negative logarithm of the concentration of hydrogen ions [H⁺] and pOH is the negative logarithm of the concentration of hydroxide ions [OH⁻]. The following formulas are used to calculate pH and pOH: pH = -log\([\mathrm{H}^{+}]\) pOH = -log\([\mathrm{OH}^{-}]\)

Step 2: Relationship between pH, pOH, and Kw

The ion product constant for water, Kw, relates the concentrations of H⁺ and OH⁻. At 298 K, the value of Kw is 1.0 x 10⁻¹⁴. The relationship between pH, pOH, and Kw is given by the following equation: pH + pOH = 14

Step 1: Calculate pH and pOH for part a.

In part a, we are given the concentration of OH⁻ as 0.000033 M. Therefore, we can calculate pOH by using the formula for pOH: pOH = -log\(\left[\mathrm{OH}^{-}\right]\) = -log\(0.000033\) Now, we can calculate pH using the relationship between pH, pOH, and Kw: pH = 14 - pOH

Step 2: Calculate pH and pOH for part b.

In part b, we are given the concentration of H⁺ as 0.0095 M. Therefore, we can calculate pH using the formula for pH: pH = -log\(\left[\mathrm{H}^{+}\right]\) = -log\(0.0095\) Now, we can calculate pOH using the relationship between pH, pOH, and Kw: pOH = 14 - pH After calculating the pH and pOH values for both parts a and b, we will be able to present the final results.

Key Concepts

pOH calculation

Calculating pOH involves understanding the relationship between the concentration of hydroxide ions \(\text{OH}^-\) and logarithms. The procedure is quite straightforward. Start by identifying the concentration of hydroxide ions provided. In our example for part a, this concentration is \( [\text{OH}^-] = 0.000033 \, \text{M} \). The formula used to find the pOH is: \[\text{pOH} = -\log ([\text{OH}^-])\]The negative logarithm of the hydroxide ion concentration gives us the pOH value. Calculating the logarithm involves using a calculator or logarithm table since it helps in converting the concentration to a more manageable number, specifically for comparison or further calculations. Once you've calculated the pOH, you have one piece of the puzzle. This pOH value is essential in the broader picture of understanding the solution’s acidity or alkalinity, often used in conjunction with pH.

Kw (ion product constant of water)

The ion product constant for water, represented as \( K_w \), is a central concept in understanding the equilibrium of water's ionization. At room temperature, 298 K, \( K_w \) is equal to **1.0 x 10\(^{-14}\)**. This value comes from the equilibrium expression involving the concentrations of hydrogen ions \( [\text{H}^+] \) and hydroxide ions \( [\text{OH}^-] \). The equilibrium expression is:\[K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14}\]This relationship is vital as it forms the basis of the pH scale and allows us to calculate the pH or pOH when one is known. For example, with \( [\text{H}^+] \) determined, you can find \( [\text{OH}^-] \) by rearranging the expression: \[[\text{OH}^-] = \frac{K_w}{[\text{H}^+]}\]Knowing \( K_w \) allows chemists to understand not just isolated ions but the balance of acidity and basicity in the entire solution. Additionally, the relationship between pH and pOH comes from this fundamental constant, establishing that:\[\text{pH} + \text{pOH} = 14\]This guideline makes it possible to swap easily between calculating pH and pOH, a crucial ability in chemistry.

hydrogen ion concentration

Understanding hydrogen ion concentration \( [\text{H}^+] \) is crucial as it directly influences the pH of a solution. In part b of the problem, the hydrogen ion concentration provided is \( [\text{H}^+] = 0.0095 \, \text{M} \). The pH calculation involves taking the negative logarithm of this concentration:\[\text{pH} = -\log ([\text{H}^+])\]This calculation translates the concentration of hydrogen ions into a more manageable number on the pH scale, which ranges from 0 to 14. Lower pH values correspond to more acidic solutions, while higher values indicate basic solutions. pH not only indicates acidity but also guides chemical reactivity, biological activity, and industrial applications. Also, when you know the pH, calculating the pOH becomes simple thanks to the relationship \( \text{pH} + \text{pOH} = 14 \), aiding in exploring the solution’s full characteristics without calculating \( [\text{OH}^-] \) directly.Hydrogen ion concentrations are necessary for diverse fields beyond chemistry, influencing medicine, biology, and even environmental science, showing the wide-reaching importance of this concept.