Two drops \((0.1 \mathrm{~mL})\) of \(1.0 \mathrm{M} \mathrm{HCl}\) are added to water to make \(1.0 \mathrm{~L}\) of solution. What is the \(\mathrm{pH}\) of this solution if the \(\mathrm{HCl}\) is \(100 \%\) ionized?

When hydrochloric acid (HCl) is added to water, it undergoes 100% ionization. This means each HCl molecule breaks apart into its constituent ions:
\( \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \)
HCl is a strong acid which will always ionize completely in water, producing hydronium ions (\text{H}^+) and chloride ions (\text{Cl}^-). The amount of \text{H}^+ ions equals the initial amount of HCl molecules added, making the calculation straightforward.
Therefore, to determine the \text{H}^+ ion concentration, you simply need to know the initial concentration of HCl.

Acid-base chemistry revolves around the concept of hydrogen ions (H\text{+}) and hydroxide ions (OH\text{-}). In the context of strong acids like HCl, acids are substances that donate hydrogen ions in a solution.
According to the Bronsted-Lowry theory, these hydrogen ions are then accepted by water molecules to form hydronium ions (\text{H}\(_3\)\text{O}^+), but for simplicity, we often refer to them just as H\text{+}.
A strong acid like HCl dissociates completely, making it very efficient at increasing the \text{H}^+ concentration in a solution. This complete ionization is key when calculating the solution's pH since the amount of H\text{+} directly impacts the pH value.

To solve for the concentration of a solution, we use the formula:
\[ \text{concentration (M)} = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
In the given exercise, we calculated the concentration of HCl after dilution. We started by finding the moles of HCl added using the provided concentration and volume.
For HCl, the moles added were found using: \[ \text{moles of HCl} = \text{concentration (1.0 M)} \times \text{volume (0.1 mL or 0.1} \times 10^{-3} \text{ L)} = 1.0 \times 10^{-4} \text{ moles}} \]
Then, we calculated the concentration in the final solution: \[ \text{Final concentration} = \frac{\text{moles of HCl}}{\text{total volume of solution (1.0 L)}} = 1.0 \times 10^{-4} \text{ M} \]

Understanding the concept of moles is crucial in chemistry, particularly in solution chemistry. A mole is a unit that measures the quantity of substance; one mole equals Avogadro's number (approximately 6.022 \times 10^{23}) of particles (atoms, molecules, etc.).
In the context of solutions, knowing the number of moles helps us understand how concentrated a solution is.
For example, in the given problem, we used the number of moles of HCl to determine how much of it is present in the solution. This information was crucial for calculating the final concentration and thereby the pH of the solution.
Using the relationship: \( \text{moles} = \text{concentration} \times \text{volume} \), we can convert between these values as needed to solve various types of problems in chemistry. Thus, the mole concept is foundational for several types of calculations, including finding pH.