Question

Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)for each solution. (a) \(\mathrm{pH}=8.55\) (b) \(\mathrm{pH}=11.23\) (c) \(\mathrm{pH}=2.87\) (d) \(\mathrm{pH}=1.22\)

Short answer

\([\mathrm{H}_{3} \mathrm{O}^{+}]\) is \(2.82 \times 10^{-9}\) M for (a), \(5.88 \times 10^{-12}\) M for (b), \(1.35 \times 10^{-3}\) M for (c), and \(6.03 \times 10^{-2}\) M for (d).

Step-by-step solution

- Understanding the Relationship between pH and \(\mathrm{H}_{3} \(\mathrm{O}^{+}\)\)

pH is the negative logarithm (base 10) of the hydronium ion concentration \(\mathrm{H}_{3} \mathrm{O}^{+}\). The formula is \(\mathrm{pH} = -\log(\mathrm{H}_{3} \mathrm{O}^{+})\). To find the concentration of hydronium ions, the inverse of this process is used: \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-\mathrm{pH}}\).

- Calculate \(\mathrm{H}_{3} \mathrm{O}^{+}\) for \(\mathrm{pH}=8.55\)

Use the formula \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-\mathrm{pH}}\) to find the concentration of hydronium ions. \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-8.55}\).

- Calculate \(\mathrm{H}_{3} \mathrm{O}^{+}\) for \(\mathrm{pH}=11.23\)

Again apply the formula \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-\mathrm{pH}}\). \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-11.23}\).

- Calculate \(\mathrm{H}_{3} \mathrm{O}^{+}\) for \(\mathrm{pH}=2.87\)

Use the formula \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-\mathrm{pH}}\). \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-2.87}\).

- Calculate \(\mathrm{H}_{3} \mathrm{O}^{+}\) for \(\mathrm{pH}=1.22\)

Employ the formula one last time: \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-\mathrm{pH}}\). \(\mathrm{H}_{3} \mathrm{O}^{+} = 10^{-1.22}\).

Key concepts

pH Scale

The pH scale is a numerical representation of acidity or basicity in a solution. It ranges from 0 to 14, with 7 being neutral, values below 7 indicating acidity, and values above 7 indicating basicity. Pure water, with a pH of exactly 7, is considered neutral because it has equal concentrations of hydronium ions ((H_3O^+)) and hydroxide ions ((OH^-)).

When you see a pH value, it is essentially a measure of the activity of hydronium ions in the solution. What's fascinating is that each unit of change in the pH scale represents a tenfold change in ion concentration. This means that a solution with a pH of 6 has ten times more (H_3O^+) ions than a solution with a pH of 7.

Logarithmic Relationships

Understanding the pH scale requires grasping logarithmic relationships because pH is calculated using the logarithm of the hydronium ion concentration. A logarithm is simply a way to express power or exponents, making huge or minuscule numbers more manageable.

In the case of the pH scale, we use the base 10 logarithm. Here's a key point: the pH is the negative logarithm of the hydronium ion concentration. Why negative? To keep the pH numbers positive since we usually deal with (H_3O^+) concentrations less than 1—which would result in negative log values. This logarithmic relationship means that a small change in pH corresponds to a large change in ion concentration.

Hydronium Ions

Hydronium ions, denoted as (H_3O^+), play a central role in acid and base chemistry. They are formed when a hydrogen ion ((H^+)) associates with a water molecule ((H_2O)). The concentration of these ions determines the acidity of a solution.

Knowing the hydronium ion concentration allows chemists to predict the behavior of substances in different environments—such as how they react, dissolve, or interact with other compounds. For example, highly acidic solutions with high concentrations of (H_3O^+) can lead to corrosion, while basic solutions can neutralize excess acidity.

Acid and Base Chemistry

In the realm of acid and base chemistry, pH and (H_3O^+) concentration are indispensable in predicting the strength and behavior of acids and bases. Acids are substances that donate hydrogen ions to the solution, increasing the (H_3O^+) concentration, while bases accept hydrogen ions, thus decreasing it.

Strong acids completely dissociate in water, providing a higher concentration of (H_3O^+) and thus a lower pH. On the other hand, strong bases fully dissociate to give (OH^-) ions that then react with (H_3O^+) to form water, leading to a lower concentration of hydronium ions and a higher pH. It's this balance of (H_3O^+) and (OH^-) that governs a substance's acidity or basicity and its reactions in various scenarios.