Suppose \(1.00\) mole of \(\mathrm{HCl}\) is dissolved in enough water to give \(500.0 \mathrm{~mL}\) of solution. (a) What is the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration? (Hint: When 1 mole of \(\mathrm{HCl}\) dissociates, we get 1 mole of \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions because the acid is strong and monoprotic. Your answer should be in moles of \(\mathrm{H}_{3} \mathrm{O}^{+}\) per liter of solution.) (b) What is the \(\mathrm{OH}^{-}\) concentration? (Hint: Use the \(K_{\mathrm{w}}\) relationship.)

Understanding molarity is fundamental to describing the concentration of a solute in a solution. Molarity (M) is measured as the number of moles of a solute per liter of solution. In simpler terms, it tells us how much of a substance is dissolved in a given volume of liquid.

To calculate molarity, we use the formula:
\[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
This equation allows us to determine how concentrated a solution is, which is crucial when conducting experiments or when a specific concentration is needed for a reaction to occur. In the given exercise, once we know the volume of the solution and the amount of solute in moles, we can easily find the molarity.

Strong acids, like hydrochloric acid (HCl), are known for their ability to completely dissociate in water. This means that when HCl is added to water, it will release its hydrogen ions, forming hydronium ions \( \mathrm{H}_3\mathrm{O}^+ \) in a one-to-one ratio. Due to this full dissociation, the concentration of hydronium ions in the solution is directly related to the original concentration of the strong acid.

For a strong monoprotic acid like HCl, which releases only one hydrogen ion per molecule, the resulting molarity of \( \mathrm{H}_3\mathrm{O}^+ \) ions will be the same as the molarity of the dissolved HCl, exemplified in our exercise. Fully understanding the behavior of strong acids helps in predicting the outcomes when they are added to different solutions.

The measurement of acidity or basicity of a solution can be expressed using the pH scale, where pH stands for the 'power of hydrogen'. The pH is defined by the negative logarithm of the hydronium ion concentration:
\[ \text{pH} = -\log [\mathrm{H}_3\mathrm{O}^+] \]
pOH, on the other hand, refers to the power of hydroxide ion concentration and is given by:
\[ \text{pOH} = -\log [\mathrm{OH}^-] \]
The pH and pOH scales are related, and together they must always add up to 14 at 25°C, which leads us to the equation \( \text{pH} + \text{pOH} = 14 \). Understanding pH and pOH is essential when dealing with acid-base reactions, buffer solutions, and determining the acidity or basicity of a substance.

Water itself partially dissociates into hydronium \( \mathrm{H}_3\mathrm{O}^+ \) and hydroxide \( \mathrm{OH}^- \) ions. The extent to which this happens at a given temperature is defined by the ion product constant of water (\( K_{\mathrm{w}} \)). At 25°C, the equilibrium constant value is \( 1.00 \times 10^{-14} \).

This constant is vital in calculations involving acid-base equilibrium as it connects the concentrations of \( \mathrm{H}_3\mathrm{O}^+ \) and \( \mathrm{OH}^- \) ions:
\[ K_{\mathrm{w}} = [\mathrm{H}_3\mathrm{O}^+] [\mathrm{OH}^-] \]
Knowing the concentration of one ion allows us to find the other, as shown in the exercise when we calculate the hydroxide ion concentration using the known hydronium ion concentration. The concept of \( K_{\mathrm{w}} \) plays an integral role in predicting the behavior of acids and bases in solution and is a cornerstone of acid-base chemistry.