Problem 55 - Chemistry for Allied Health

Question

How can the \(\mathrm{pOH}\) of a solution be determined if its \(\mathrm{pH}\) is known? (Hint: Write a mathematical expression.)

Step-by-Step Solution

Answer

The pOH can be found using the formula: \(\text{pOH} = 14 - \text{pH}\).

Step 1: Understanding the pH and pOH relationship

To solve this problem, it's important to understand the relationship between pH and pOH in aqueous solutions, which is governed by the equation: \[ \text{pH} + \text{pOH} = 14 \]This equation arises from the ion product constant of water (\(K_w\)), which at 25°C is \(1.0 \times 10^{-14}\).

Step 2: Rearrange the pH and pOH equation

Since we are asked to find the pOH when the pH is known, we can rearrange the equation to solve for pOH:\[ \text{pOH} = 14 - \text{pH} \]

Step 3: Determine the pOH with a Known pH

Insert the given pH value into the equation from the previous step to compute the pOH of the solution. For instance, if we know \(\text{pH} = 7\):\[ \text{pOH} = 14 - 7 = 7 \]This example illustrates how you would apply the formula to any given pH value to find the corresponding pOH.

Key Concepts

Ion Product Constant of Water

The ion product constant of water, denoted as \(K_w\), is an important concept in understanding the behavior of aqueous solutions. At 25°C, the value of \(K_w\) is \(1.0 \times 10^{-14}\). This constant arises from the dissociation of water molecules into hydrogen ions \((H^+)\) and hydroxide ions \((OH^-)\). The mathematical expression for this is given by:

• \(K_w = [H^+][OH^-]\)

Here, \([H^+]\) is the concentration of hydrogen ions and \([OH^-]\) is the concentration of hydroxide ions in water. Since pure water is neutral, the concentrations of \(H^+\) and \(OH^-\) are equal, and their product is always constant at the given temperature. Understanding the ion product constant of water is crucial in calculating pH and pOH values in any aqueous solution.

Aqueous Solutions

Aqueous solutions refer to any solution where water is the solvent. Understanding how pH and pOH play a role in these solutions is crucial for evaluating their acidity or alkalinity. In an aqueous solution:

• If the solution has more \(H^+\) ions than \(OH^-\) ions, it is acidic.
• Conversely, if there are more \(OH^-\) ions, the solution is basic (alkaline).
• When \([H^+]=[OH^-]\), the solution is said to be neutral.

Water’s ability to dissociate itself into its ionic components makes it a universal solvent, enabling it to host a vast array of chemical reactions. Knowing how to analyze and manipulate the pH and pOH in these solutions can help predict reaction behaviors and properties of the substances involved.

pH

The term pH is a measure of the acidity or basicity of an aqueous solution. It is defined as the negative logarithm base 10 of the hydrogen ion concentration:

• \(\text{pH} = -\log[H^+]\)

A lower pH represents a higher concentration of \(H^+\) ions, indicating an acidic solution, while a higher pH signifies a basic (or alkaline) solution. The pH scale typically ranges from 0 to 14. Each unit change on the scale corresponds to a tenfold change in hydrogen ion concentration. For example, a solution with a pH of 3 is ten times more acidic than one with a pH of 4. This logarithmic scale helps to conveniently express and compare the wide range of hydrogen ion concentrations.

pOH

pOH is the complementary measure to pH, focusing instead on the hydroxide ion concentration in an aqueous solution:

• \(\text{pOH} = -\log[OH^-]\)

The relationship between pH and pOH is governed by the equation \(\text{pH} + \text{pOH} = 14\) at 25°C. This relation is especially useful because if you know either pH or pOH, you can easily determine the other. In neutral solutions, both pH and pOH are 7, reflecting equal concentrations of hydrogen and hydroxide ions. Understanding pOH provides a complete picture of a solution’s acidity or alkalinity when combined with the pH. This knowledge is critical for various applications, including chemical titrations and buffer solutions in laboratory settings.