Question

Calculate the \(\mathrm{pH}\) of each solution. (a) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=2.4 \times 10^{-10} \mathrm{M}\) (b) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=7.6 \times 10^{-2} \mathrm{M}\) (c) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=9.2 \times 10^{-13} \mathrm{M}\) (d) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=3.4 \times 10^{-5} \mathrm{M}\)

Short answer

The pH values are (a) 9.62, (b) 1.12, (c) 12.04, and (d) 4.47.

Step-by-step solution

- Understanding pH calculation

The pH of a solution is calculated using the formula: \(\text{pH} = -\log_{10}[\text{H}_3\text{O}^+]\). The concentration of hydronium ions \([\text{H}_3\text{O}^+]\) is given, so we will apply the logarithm to these values to find the pH.

- Calculate pH for (a)

Substitute \([\text{H}_3\text{O}^+]=2.4 \times 10^{-10}\) into the formula: \(\text{pH} = -\log_{10}(2.4 \times 10^{-10})\).

- Calculate pH for (b)

Substitute \([\text{H}_3\text{O}^+]=7.6 \times 10^{-2}\) into the formula: \(\text{pH} = -\log_{10}(7.6 \times 10^{-2})\).

- Calculate pH for (c)

Substitute \([\text{H}_3\text{O}^+]=9.2 \times 10^{-13}\) into the formula: \(\text{pH} = -\log_{10}(9.2 \times 10^{-13})\).

- Calculate pH for (d)

Substitute \([\text{H}_3\text{O}^+]=3.4 \times 10^{-5}\) into the formula: \(\text{pH} = -\log_{10}(3.4 \times 10^{-5})\).

Key concepts

Hydronium Ion Concentration

Understanding the role of hydronium ion concentration in pH calculation is fundamental in chemistry, particularly in the field of acid-base reactions. The hydronium ion, represented as \( \text{H}_3\text{O}^+ \), is the form of a proton \( \text{H}^+ \) combined with a water molecule \( \text{H}_2\text{O} \). The concentration of hydronium ions in a solution determines its acidity or basicity.

A high concentration of hydronium ions indicates an acidic solution, while a low concentration points to a basic or alkaline solution. The concentration is often expressed in molarity (M), which is the number of moles of hydronium ions per liter of solution. To grasp the significance of the hydronium ion concentration, consider that a neutral solution at room temperature, like pure water, has a concentration of \(1.0 \times 10^{-7} M\).

In the given exercise, the task is to calculate the pH based on different given concentrations of hydronium ions. It's crucial for students to recognize that small changes in hydronium ion concentration can result in a significant change in pH, reflecting a logarithmic relationship which we will explore in the next section.

Logarithmic Functions in Chemistry

Logarithmic functions are critical in chemistry for conveying relationships that span many orders of magnitude, like the concentration of hydronium ions when calculating pH. The pH scale itself is logarithmic, meaning a one-unit change in pH corresponds to a tenfold change in hydronium ion concentration.

The formula for pH \( \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \) encompasses the use of a base-10 logarithm to indicate the inverse log relationship between pH and hydronium ion concentration. When students use this logarithmic function, they can convert a cumbersome exponential expression into an easily understandable number that represents the acidity or basicity of a solution.

For example, if you were given a hydronium ion concentration of \(2.4 \times 10^{-10} M\), the corresponding pH would be calculated using the negative logarithm of this number. It's important for students to be familiar with how to operate logarithms, especially when dealing with negative exponents, as these can be common pitfalls when dealing with acid-base chemistry calculations.

Acid-Base Chemistry

Acid-base chemistry is a cornerstone of chemical understanding, encompassing the study of pH, and the behavior of acids and bases. It explains not only the chemical properties and reactions of acids and bases but also their roles in biological and environmental systems.

Acids are substances that increase the concentration of hydronium ions when dissolved in water, whereas bases decrease this concentration. The pH scale, ranging from 0 to 14, serves as a measure of the acidity or basicity of a solution; a pH less than 7 indicates an acidic solution, pH equal to 7 indicates a neutral solution, and pH greater than 7 indicates a basic or alkaline solution.

When students calculate pH from a given hydronium ion concentration, they are essentially determining where the solution lies on the pH scale. By understanding the concepts of acids and bases, and how to calculate pH, students can predict the behavior of substances in a chemical reaction or environment, which is an essential skill in many scientific fields.