Calculating \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) and \(\left[\mathrm{HO}^{-}\right]\) using \(\mathrm{K} \mathrm{W}\) 1\. A solution at \(25^{\circ} \mathrm{C}\), is known to have a hydronium ion concentration of \(4.5 \times 10^{-5} \mathrm{M}\); what is the concentration of hydroxide ion in this solution? 2\. A solution at \(25^{\circ} \mathrm{C}\), is known to have a hydroxide ion concentration of \(7.5 \times 10^{-2} \mathrm{M}\); what is the concentration of hydronium ion in this solution? 3\. A solution is known to have a hydronium ion concentration of \(9.5 \times 10^{-8} \mathrm{M}\); what is the concentration of hydroxide ion in this solution?

Understanding the Ion-Product Constant for Water, known as \(K_W\), is crucial for anyone studying chemistry, particularly when examining acid-base reactions. At a certain temperature, this constant represents the equilibrium concentration of water molecules dissociated into hydronium (\([H_3O^+]\)) and hydroxide (\([OH^-]\)) ions.

At the significant temperature of \(25^\circ\mathrm{C}\), \(K_W\) is valued at \(1.0 \times 10^{-14}\). This value is derived from the principle that water is in a dynamic equilibrium where water molecules dissociate into ions and ions recombine to form water molecules at equal rates. When the concentrations of \([H_3O^+]\) and \([OH^-]\) are multiplied, the result remains constant at \(K_W\), provided that the temperature is stable.

This relationship is key in determining the properties of solutions, understanding the pH scale, and assessing the acid or base nature of a substance. In an ideal neutral solution, the concentrations of \([H_3O^+]\) and \([OH^-]\) are equivalent, which also explains why pure water has a neutral pH of 7.

The hydronium ion concentration in a solution, represented as \([H_3O^+]\), is an important parameter in the field of acid-base chemistry. It indicates the concentration of hydrogen ions present in a solution, which directly affects the solution's acidity. A higher \([H_3O^+]\) corresponds to a more acidic solution, while a lower concentration suggests a more alkaline solution.

To find the \([H_3O^+]\) in a solution when the \([OH^-]\) is known, we use the \(K_W\) value and the expression \(K_W = [H_3O^+][OH^-]\). Algebraic manipulation allows us to isolate \([H_3O^+]\) and calculate its value. For instance, if a solution at \(25^\circ\mathrm{C}\) has an \([OH^-]\) of \(7.5 \times 10^{-2} \mathrm{M}\), the \([H_3O^+]\) can be calculated by dividing \(K_W\) by the known \([OH^-]\), thus helping to classify the solution as acidic or basic.

Likewise, the hydroxide ion concentration, denoted as \([OH^-]\), gives us an insight into the alkalinity of a solution. It is the counterpart to the hydronium ion concentration and an increase in \([OH^-]\) means a decrease in the solution's acidity, making it more basic or alkaline.

To find the \([OH^-]\) when \([H_3O^+]\) is given, one simply needs to rearrange the equilibrium expression for water. For example, if the hydronium ion concentration is \(4.5 \times 10^{-5} \mathrm{M}\), the hydroxide ion concentration can be found by dividing \(K_W\) by this value of \([H_3O^+]\). This is an essential method in understanding the full composition of a solution and predicting its behavior in acid-base reactions.

The concept of Acid-Base Equilibrium is central to understanding the balance between acidic and basic ions in a solution. When an acid donates a proton (H+), it increases the \([H_3O^+]\), whereas a base accepts a proton, increasing the \([OH^-]\). Equilibrium is the state where the rate of proton donation by acids equals the rate of proton acceptance by bases.

At equilibrium in pure water or a neutral solution, the concentrations of the hydronium and hydroxide ions are equal. However, in a solution where an acid or base has been added, the equilibrium shifts to re-establish the balance, leading to changes in ion concentrations until a new equilibrium is achieved. Understanding this dynamic is fundamental for predicting the direction of the reaction and calculating the pH of various substances in solution.