Problem 28 - Chemistry Matter and Change

Question

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

Step-by-Step Solution

Answer

a. pH ≈ 9.52, pOH ≈ 4.48; b. pH ≈ 2.02, pOH ≈ 11.98.

Step 1: Calculating pOH from given [OH⁻]

To find the pOH, use the formula: \( \mathrm{pOH} = -\log_{10}( [\mathrm{OH}^-] ) \). For \([\mathrm{OH}^-] = 0.000033 \, M\), calculate:\[pOH = -\log_{10}(0.000033) \approx 4.48\]

Step 2: Calculating pH from pOH

Use the relation between pH and pOH for water at 298 K: \( \mathrm{pH} + \mathrm{pOH} = 14 \). Solve for pH:\[\mathrm{pH} = 14 - \mathrm{pOH} = 14 - 4.48 \approx 9.52 \]

Step 3: Calculating pH from given [H⁺]

To find the pH, use the formula: \( \mathrm{pH} = -\log_{10}( [\mathrm{H}^+] ) \). For \([\mathrm{H}^+] = 0.0095 \, M\), calculate:\[pH = -\log_{10}(0.0095) \approx 2.02\]

Step 4: Calculating pOH from pH

Use the relation between pH and pOH for water at 298 K: \( \mathrm{pH} + \mathrm{pOH} = 14 \). Solve for pOH:\[\mathrm{pOH} = 14 - \mathrm{pH} = 14 - 2.02 \approx 11.98 \]

Key Concepts

Ionic Concentration

When dealing with solutions, understanding ionic concentration is crucial. This refers to the amount of ions that are present within a given volume of solution. For instance, if we have \([\text{OH}^-]\) with a concentration of 0.000033 M, this tells us there are 0.000033 moles of hydroxide ions per liter of solution. These concentrations are fundamental in determining other properties of the solution, such as pH and pOH.
Ionic concentration plays a central role in chemistry and biology since it affects reactions and processes. In aqueous solutions, ions are formed when compounds dissociate in water, influencing the acidity or basicity of the solution. Therefore, knowing the ionic concentration allows us to calculate and predict other chemical properties. This sets the stage for pH and pOH calculations, which are deeply rooted in these concentrations.

Logarithmic Calculations

Logarithmic calculations are essential in determining the pH and pOH of solutions. The pH scale, which measures how acidic or basic a solution is, is logarithmic. This means each step on the pH scale represents a tenfold change in concentration.
To calculate pH from the concentration of hydrogen ions in the solution, the formula is \[ \text{pH} = -\log_{10}([\text{H}^+]) \] This calculation utilizes the base-10 logarithm, translating the concentration of ions into a more manageable number for human interpretation. Similarly, pOH is calculated from the concentration of hydroxide ions using \[ \text{pOH} = -\log_{10}([\text{OH}^-]) \]
These logarithmic calculations convert tiny numbers (like 0.0095 M) into clear, understandable figures (such as a pH of 2.02). This method provides an intuitive understanding of a solution's acidity or basicity, which can be crucial in scientific and industrial contexts.

Aqueous Solutions

Aqueous solutions are those where water acts as the solvent. Water's polarity allows it to dissolve a wide range of substances, making it an excellent medium for chemical reactions, especially acid-base reactions. In these solutions, substances like acids and bases dissociate into their respective ions.
For example, hydrochloric acid ( HCl ) dissociates into H⁺ and Cl⁻ ions in water, making it an acidic solution. Conversely, a base such as sodium hydroxide ( NaOH ) separates into OH⁻ and Na⁺ ions, increasing the solution’s basicity.
The unique properties of water, like its high dielectric constant, facilitate the easy dissociation into ions necessary for conducting electricity and driving chemical reactions. Thus, aqueous solutions provide an accessible environment for studying ionic concentration, pH, and other chemical phenomena.

Acid-Base Chemistry

Acid-base chemistry revolves around the balance of hydrogen ions (H⁺) and hydroxide ions (OH⁻) in a solution. An acid increases the concentration of H⁺ ions, while a base increases that of OH⁻ ions.
The pH value is a measure of the concentration of H⁺ ions, while pOH relates to OH⁻. Their relationship is expressed by the equation \(\text{pH} + \text{pOH} = 14\) at 298 K, which is known as the water dissociation constant at room temperature.This equilibrium relationship helps in determining the acidity or basicity of a solution when given either the concentration of H⁺ or OH⁻. The pH scale ranges from 0 to 14, with 7 being neutral—indicating equal concentrations of H⁺ and OH⁻. Values below 7 indicate acidity, while those above 7 indicate alkalinity.

• Acids contribute to lowering the pH by increasing H⁺ concentration.
• Bases contribute to raising the pH by increasing OH⁻ concentration.

This basic understanding of acid-base chemistry is pivotal for industries like pharmaceuticals, agriculture, and environmental science, where controlling the acidity or basicity can influence product effectiveness and safety.