Question

What is the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)in a solution with a \(\mathrm{pH}\) of \(4 ?\)

Short answer

\(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 10^{-4} \text{ M }\)

Step-by-step solution

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

Recognize that pH is the negative logarithm of the hydrogen ion concentration, \(\mathrm{H}_{3}\mathrm{O}^{+}\). Therefore, pH = -log(\[\mathrm{H}_{3}\mathrm{O}^{+}\]).

Calculate the Hydrogen Ion Concentration

Calculate the concentration of hydrogen ions by taking the inverse logarithm (antilog) of the negative pH value. This is done using the formula \(\mathrm{H}_{3}\mathrm{O}^{+} = 10^{-\text{pH}}\).

Applying the pH Value

Substitute the given pH value of 4 into the formula \(\mathrm{H}_{3}\mathrm{O}^{+} = 10^{-4}\), to find the concentration of hydrogen ions.

Key concepts

pH and Hydrogen Ion Concentration

Understanding the relationship between pH and hydrogen ion concentration is fundamental in chemistry, especially in the study of acids and bases. The term 'pH' stands for 'power of hydrogen' and quantifies the level of acidity or basicity of a solution.

The concentration of hydrogen ions, often represented as \[\mathrm{H}_{3}\mathrm{O}^{+}\] or simply \[\mathrm{H}^+\], is pivotal because it directly influences the pH of a solution. A high concentration of hydrogen ions indicates a low pH, which corresponds to an acidic solution, while a low hydrogen ion concentration suggests a high pH, linked to a basic or alkaline solution.

This inverse relationship is expressed in the pH scale, which ranges from 0 to 14, with 7 being neutral. A pH less than 7 denotes an acidic solution, and a pH greater than 7 indicates a basic one. Tools like pH meters or indicators help to measure the pH value, but understanding the mathematical connection allows for precise calculation when these are not available.

Inverse Logarithm

In mathematics, logarithms are used to solve equations in which a number is raised to a power to achieve a certain value, a process that can be reversed using an operation known as the inverse logarithm, or antilogarithm. The concept of inverse logarithm is particularly important in pH calculations where the pH value is given, and the goal is to determine the hydrogen ion concentration.

A logarithm tells us what exponent needed to be used on a base number (usually 10 in pH calculations) to produce a certain value. For instance, if \(10^x = 100\), then the logarithm of 100 to base 10, \(\log_{10}(100)\), is 2. This is because \(10^2 = 100\). In the context of pH, the equation \(\text{pH} = -\log_{10}[\mathrm{H}^+]\) can be reversed using the inverse logarithm to find the concentration of hydrogen ions ({\(\mathrm{H}^+\)}) from a known pH value.

Hydrogen Ion Concentration Calculation

To compute the hydrogen ion concentration from a given pH value, we use the inverse logarithm. From the relationship \(\text{pH} = -\log [\mathrm{H}_{3}\mathrm{O}^+]\), it follows that to reverse the process, we raise 10 to the power of the negative pH value: \([\mathrm{H}_{3}\mathrm{O}^+] = 10^{-\text{pH}}\).

For example, with a pH of 4, the concentration of hydrogen ions \([\mathrm{H}_{3}\mathrm{O}^+]\) would be calculated as \(10^{-4}\) moles per liter. So, we take the antilog of -4, which gives us a hydrogen ion concentration of \(10^{-4}\) M, or 0.0001 moles per liter. This quantitative interpretation provides accuracy in preparing desired pH solutions for various applications, including laboratory experiments and buffer solutions in biological systems.