Question

Calculate the \(\left[\mathrm{OH}^{-}\right], \mathrm{pH},\) and \(\mathrm{pOH}\) for each of the following solutions in which the \(\mathrm{H}^{+}\) concentrations are (a) \(3.5 \times 10^{-7} M\) (c) \(2.5 \times 10^{-11} M\) (b) \(0.0017 M\) (d) \(7.9 \times 10^{-2} M\)

Short answer

For solution (a) \(\left[\mathrm{OH}^{-}\right] = 2.86 \times 10^{-8} M\), \(\mathrm{pH} = 6.46\), and \(\mathrm{pOH} = 7.54\). Similar calculations for (c), (b), and (d) respectively will yield their respective concentrations and pH values.

Step-by-step solution

Calculate \(\left[\mathrm{OH}^{-}\right]\) for solution (a)

Use the water dissociation constant \(K_w = 1.0 \times 10^{-14}\) at 25°C to find the hydroxide ion concentration \(\left[\mathrm{OH}^{-}\right]\): \[\left[\mathrm{OH}^{-}\right] = \frac{K_w}{\left[\mathrm{H}^{+}\right]} = \frac{1.0 \times 10^{-14}}{3.5 \times 10^{-7}}.\] Calculate this value to find \(\left[\mathrm{OH}^{-}\right]\) for solution (a).

Calculate \(\mathrm{pOH}\) and \(\mathrm{pH}\) for solution (a)

Use the formulas \(\mathrm{pOH} = -\log(\left[\mathrm{OH}^{-}\right])\) and \(\mathrm{pH} = 14 - \mathrm{pOH}\) to find the pH and pOH of the solution. Calculate \(\mathrm{pOH}\) by taking the negative logarithm of the \(\left[\mathrm{OH}^{-}\right]\) found in Step 1. Then, subtract the \(\mathrm{pOH}\) from 14 to find the pH.

Calculate \(\left[\mathrm{OH}^{-}\right]\), \(\mathrm{pOH}\), and \(\mathrm{pH}\) for solution (c)

Repeat Steps 1 and 2 for solution (c) using \(\left[\mathrm{H}^{+}\right] = 2.5 \times 10^{-11} M\). Calculate the hydroxide ion concentration, then the pOH, and finally the pH using the procedures described above.

Calculate \(\left[\mathrm{OH}^{-}\right]\), \(\mathrm{pOH}\), and \(\mathrm{pH}\) for solution (b)

Repeat Steps 1 and 2 for solution (b) using \(\left[\mathrm{H}^{+}\right] = 0.0017 M\). Calculate the hydroxide ion concentration, then the pOH, and finally the pH using the procedures described above.

Calculate \(\left[\mathrm{OH}^{-}\right]\), \(\mathrm{pOH}\), and \(\mathrm{pH}\) for solution (d)

Repeat Steps 1 and 2 for solution (d) using \(\left[\mathrm{H}^{+}\right] = 7.9 \times 10^{-2} M\). Calculate the hydroxide ion concentration, then the pOH, and finally the pH using the procedures described above.

Key concepts

Hydroxide Ion Concentration

Understanding the hydroxide ion concentration is crucial in the realm of acid-base chemistry. It represents the amount of hydroxide ions (OH-) present in a solution. The concentration of these ions directly influences the solution's acidity or alkalinity.

In the provided exercise, calculating the hydroxide ion concentration involves using the water dissociation constant (Kw), which at 25°C is 1.0 \times 10^{-14}. The formula \[\left[\mathrm{OH}^-\right] = \frac{K_w}{\left[\mathrm{H}^+\right]}\] connects the concentration of hydrogen ions (H+) to that of hydroxide ions. A higher H+ concentration means a lower OH- concentration and vice versa. It is this inverse relationship that helps maintain the balance of pH in water and aqueous solutions.

Water Dissociation Constant

The water dissociation constant, denoted as Kw, is a fundamental concept in acid-base chemistry. It characterizes the level at which water molecules dissociate into hydrogen and hydroxide ions, and its value at 25°C is universally accepted to be 1.0 \times 10^{-14}.

This constant holds a pivotal role because it is the product of the concentrations of H+ and OH- ions that are always present in a given water-based solution, represented by the equilibrium expression: \[K_w = [\mathrm{H}^+][\mathrm{OH}^-]\] When you know the concentration of one of these ions, you can easily calculate the concentration of the other, allowing for further calculations such as pH and pOH.

pH Scale

The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It ranges typically from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, while a pH greater than 7 indicates a base. The pH is calculated based on the concentration of hydrogen ions using the formula: \[\mathrm{pH} = -\log([\mathrm{H}^+])\] Conversely, pOH measures the alkalinity of a solution and is found using the concentration of hydroxide ions. The pH scale is vital because it affects various chemical reactions, biological systems, and environmental conditions. A deviation in pH from a biological system's normal range can have significant consequences.

Acid-Base Chemistry

At the heart of acid-base chemistry is the interaction between acids and bases. Acids are substances that increase the concentration of hydrogen ions (H+) in a solution, while bases do the opposite by increasing the concentration of hydroxide ions (OH-). Their strengths are quantified by their ability to dissociate in water, creating an equilibrium that is fundamental to various scientific and industrial processes.

An important aspect of acid-base chemistry is the ability to manipulate and measure pH. Understanding how different substances act as acids or bases and how they affect pH is vital in many fields of science, from biochemistry to environmental science. It's a complex balance, but by knowing the concentrations of H+ and OH- ions, we can determine the nature and the strength of any acid or base.