Suppose \(0.0100\) mole of \(\mathrm{Ba}(\mathrm{OH})_{2}\) is dissolved in enough water to give \(500.0 \mathrm{~mL}\) of solution. (a) What is the \(\mathrm{OH}^{-}\) concentration? (Hint: Remember that every time \(1 \mathrm{~mole}\) of \(\mathrm{Ba}(\mathrm{OH})_{2}\) dissociates, we get 2 moles of \(\mathrm{OH}^{-}\) ions.) (b) What is the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration? (Hint: Use the \(K_{w}\) relationship.)

When a compound like barium hydroxide, or Ba(OH)2, is added to water, it readily separates into ions in a process called dissociation. This is a characteristic behavior of many ionic compounds when they dissolve in water. For Ba(OH)2, dissociation leads to the formation of barium ions (Ba2+) and hydroxide ions (OH-).

It's important to remember that Ba(OH)2 is a strong base, meaning it dissociates completely in aqueous solution. This complete dissociation is crucial for determining the concentration of OH- ions produced. For every single mole of Ba(OH)2, two moles of OH- ions are released into the solution due to its chemical formula indicating two hydroxide groups per molecule of barium hydroxide. Therefore, if one starts with 0.0100 moles of Ba(OH)2, the resulting solution will contain 0.0200 moles of OH- ions.

Once Ba(OH)2 dissociates in water, the resulting OH- ion concentration can be determined by considering the stoichiometry of the dissociation process. As mentioned earlier, because Ba(OH)2 provides two hydroxide ions for every one molecule, we can simply multiply the molarity of the dissolved Ba(OH)2 by two.

Therefore, a 0.0200 M Ba(OH)2 solution, which is found by dividing the moles of Ba(OH)2 by the volume of the solution in liters, will yield a hydroxide ion concentration ([OH-]) of 0.0400 M. To put it simply, if the starting concentration of Ba(OH)2 in the solution is known, the concentration of OH- ions will always be double that value.

Understanding the Balance between OH- and H3O+ Ions

In any aqueous solution at 25°C, the concentrations of OH- and H3O+ ions are inversely related, governed by the constant value of the ion product for water, Kw. Essentially, in a solution where the concentration of OH- ions is known to be high, as is the case with our Ba(OH)2 example, the concentration of H3O+ ions must be low to maintain the balance dictated by Kw.

Through a simple algebraic manipulation, we can find the concentration of H3O+ ions by dividing Kw by the concentration of OH- ions. As such, with a calculated [OH-] concentration of 0.0400 M, the corresponding [H3O+] would be incredibly low, specifically 2.5 x 10^{-16} M. This relationship is critical for understanding the acidic or basic nature of a solution.

Defining Kw

The constant Kw, known as the ion product of water, is the product of the concentrations of H3O+ ions and OH- ions in a solution. At room temperature (25°C), its value universally remains at 1.0 x 10^{-14}. This fundamental concept in chemistry signifies the self-ionization of water, where water molecules occasionally dissociate into H3O+ and OH- ions even without the addition of an acid or base.

Significance of Kw

The value of Kw becomes particularly useful when deducing the concentration of one ion type from the other in a solution. For instance, in the exercise, knowing the concentration of OH- ions directly aids in calculating the concentration of H3O+ ions. This constant Kw is a pivotal tool in understanding a wide range of chemical phenomena, from the behavior of acids and bases to the workings of various biological systems.