Animated Solution for Chemistry - Ionic Equilibrium: Assuming that ${\text{Ba(OH)}}_2$ is completely ionised in aqueous solution under the given conditions the concentration of ${\text{H}}_3{\text{O}}^{+}$ ions in $0.005\text{ M}$ aqueous solution of ${\text{Ba(OH)}}_2$ at $298\text{ K}$ is ...... $\times 10^{-12}{\text{ mol L}}^{-1}$. (Nearest integer)

Imagine you are looking at a simple glass of water. To the naked eye, it is just a clear, calm liquid. But at the microscopic level, it is a bustling dance floor of molecules and ions constantly interacting. When we introduce a chemical like Barium Hydroxide into this environment, the dynamics change dramatically. Let's dive into the fascinating world of ionic equilibrium to understand exactly what happens and how we can calculate the precise concentration of the elusive hydronium ions.

Barium Hydroxide

Barium hydroxide, chemically written as ${\text{Ba(OH)}}_2$, is not just any ordinary compound; it is classified as a strong base. In the realm of chemistry, being 'strong' means that it does not hold back. When it enters an aqueous solution, it undergoes complete dissociation. It breaks apart entirely into its constituent ions, leaving no intact molecules behind.

The chemical equation for this process is beautifully simple yet profoundly important:

Notice the stoichiometry here. This is where many students make a critical error. For every single molecule of ${\text{Ba(OH)}}_2$ that dissolves, it releases one Barium ion (${\text{Ba}}^{2+}$) but two Hydroxide ions (${\text{OH}}^{-}$). This $1:2$ ratio is the key to unlocking the entire problem.

The problem states that we have a $0.005\text{ M}$ aqueous solution of ${\text{Ba(OH)}}_2$. Because the dissociation is $100\%$, the concentration of the resulting ions depends directly on this initial value and the stoichiometric coefficients.

Since one mole of the base yields two moles of hydroxide ions, we must multiply the initial concentration by two:

Substituting our given value:

To make our upcoming calculations smoother, it is highly recommended to convert this decimal into scientific notation. Thus, $0.01\text{ M}$ becomes $10^{-2}\text{ M}$. We now have the exact concentration of hydroxide ions dominating our solution.

Autoionization of Water

Now, you might be wondering, "We found the hydroxide ions, but the question asks for hydronium ions (${\text{H}}_3{\text{O}}^{+}$). Where do they come from?"

This is where the magic of water comes into play. Water is not just a passive background solvent; it actively participates in a delicate balancing act known as autoionization. Even in pure water, a tiny fraction of molecules react with each other to form hydronium and hydroxide ions.

At a standard room temperature of $298\text{ K}$, this equilibrium is governed by a strict mathematical rule called the ionic product of water, denoted as $K_w$. The rule states that the product of the concentrations of hydronium and hydroxide ions must always equal a specific constant:

This relationship is a fundamental law of aqueous chemistry. If you increase the amount of ${\text{OH}}^{-}$ (by adding a base like we did), the water will automatically adjust by decreasing the amount of ${\text{H}}_3{\text{O}}^{+}$ to ensure the product remains exactly $10^{-14}$.

We are now in the final stretch. We know the universal constant $K_w$, and we have calculated our specific $[{\text{OH}}^{-}]$. All that is left is to substitute and solve for the unknown $[{\text{H}}_3{\text{O}}^{+}]$.

To isolate the hydronium ion concentration, we divide both sides by $10^{-2}$:

Using the basic rules of exponents (subtracting the denominator's exponent from the numerator's), we get:

The question asks us to express this in the format of $...\times 10^{-12}{\text{ mol L}}^{-1}$. By writing our result as $1\times 10^{-12}\text{ M}$, it is crystal clear that the missing integer is $1$.

Through a logical sequence of understanding strong electrolytes, applying stoichiometry, and leveraging the universal constant of water, we have elegantly arrived at the correct answer. Chemistry is truly just a puzzle waiting to be solved!