Calculate the \(\left[\mathrm{H}^{+}\right], \mathrm{pH},\) and \(\mathrm{pOH}\) in each of the following solutions in which the hydroxide ion concentrations are (a) \(0.0068 M\) (c) \(1.6 \times 10^{-8} \mathrm{M}\) (b) \(6.4 \times 10^{-5} M\) (d) \(8.2 \times 10^{-12} M\)

Hydrogen ion concentration, often expressed as \( [H^+] \), is a measurement of the number of hydrogen ions per unit volume in a solution. It's a fundamental aspect of understanding acid-base chemistry, as the concentration of these ions determines the acidity of a solution. A high concentration indicates an acidic solution, while a low concentration suggests a basic solution.

The relationship between hydrogen ions and hydroxide ions is critical to maintaining the pH balance in various chemical reactions and biological systems. By using the constant \( K_w = 1.0 \times 10^{-14} \) at 25°C, one can find the \( [H^+] \) if the \( [OH^-] \) is known, using the formula \( [H^+] = \frac{K_w}{[OH^-]} \). This relation illustrates the inverse proportionality between the concentrations of hydrogen and hydroxide ions in aqueous solutions.

Conversely, hydroxide ion concentration, expressed as \( [OH^-] \), is the measure of the concentration of hydroxide ions. This measurement provides insight into the basicity of a solution. In the context of the given exercise, students are starting with known \( [OH^-] \) values and must calculate the corresponding \( [H^+] \) values.

Understanding hydroxide ion concentration is also important in various real-life applications such as water treatment, environmental monitoring, and the food industry, where pH control is crucial. To convert hydroxide ion concentration to \( pOH \), the negative logarithm is taken ( \( pOH = -\log([OH^-]) \) ). This provides a way to quickly assess the basicity of a solution in a more digestible form, rather than dealing with small decimal numbers.

Acid-base equilibrium refers to the state of balance between acid and base concentrations in a solution. It's an extension of the relationship between the hydrogen and hydroxide ion concentrations. The product of these concentrations at equilibrium equals the autoionization constant of water, \( K_w \), which can be affected by factors such as temperature.

Any shift in this equilibrium, like adding an acid or a base, will lead to a corresponding shift in the concentration of \( [H^+] \) and \( [OH^-] \) to maintain the constant value of \( K_w \). This concept is essential when analyzing buffer solutions, understanding enzymatic activity in biology, or predicting the behavior of chemical reactions in different media.

The logarithmic scale in chemistry is a way to express wide-ranging quantities in a more manageable form. This is particularly useful when dealing with concentrations of ions in acid-base chemistry. pH and pOH are perfect examples of logarithmic scales that simplify the expression of hydrogen and hydroxide ion concentrations.

The formulae \( pH = -\log([H^+]) \) and \( pOH = -\log([OH^-]) \) highlight the use of this scale. A pH or pOH value changes by one unit corresponds to a tenfold change in ion concentration, which demonstrates the scale’s compressive nature. This log scale allows chemists and students alike to quickly comprehend and compare the relative acidity or basicity in chemical solutions.