Question

Find the \(\mathrm{pH}\) of a solution in which \(\left[\mathrm{H}^{+}\right]=6.38 \times 10^{-6}\) mole/liter.

Short answer

The pH of the solution is approximately 5.20, calculated using the formula pH = -log10(\(6.38 \times 10^{-6}\)) with the given concentration of H⁺ ions.

Step-by-step solution

Identify the given information

We are given the concentration of the H⁺ ions in the solution, which is: \(\left[\mathrm{H}^{+}\right]=6.38 \times 10^{-6}\) mole/liter.

Recall the formula for calculating pH

The formula for calculating the pH of a solution is: pH = -log10([H⁺]) Where [H⁺] is the concentration of hydrogen ions in the solution.

Insert the given [H⁺] value into the formula and calculate the pH

Replace the value of [H⁺] in the pH formula with the given value: pH = -log10(6.38 × 10⁻⁶)

Calculate the logarithm and find the pH of the solution

Now, we will calculate the logarithm and find the pH: pH = -log10(6.38 × 10⁻⁶) ≈ 5.20 Hence, the pH of the solution is approximately 5.20.

Key concepts

The pH Scale

The pH scale is a measure of the acidity or alkalinity of an aqueous solution. It ranges from 0 to 14, with 7 being neutral. Solutions with a pH less than 7 are considered acidic, while those with a pH greater than 7 are considered basic or alkaline.

Understanding this scale is crucial for chemists, biologists, and students alike, as it helps predict the behavior of substances and their reactions in different environments. For instance, stomach acid has a low pH around 2, making it very acidic, suitable for digesting food; on the other hand, household bleach has a high pH around 12, making it very basic, ideal for disinfection and cleaning.

It's important to recognize that the pH scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic than the next higher value. For example, a solution with a pH of 4 is ten times more acidic than a solution with a pH of 5 and a hundred times more acidic than a solution with a pH of 6. This logarithmic nature allows the scale to encompass a wide range of hydrogen ion concentrations in a compact scale.

Hydrogen Ion Concentration

Hydrogen ion concentration, denoted by \( \left[\mathrm{H}^{+}\right] \), is a key factor in determining the acidity of a solution. It represents the molarity of hydrogen ions present in a liter of solution, which are essentially free protons. When acids dissolve in water, they release hydrogen ions, increasing the solution's hydrogen ion concentration and, consequently, its acidity.

Understanding Concentration Units

It is expressed in moles per liter (M/L or mol/L), a standard unit of concentration in chemistry. For example, if a solution is reported as having \( \left[\mathrm{H}^{+}\right]=1 \times 10^{-3} \) mol/L, it means there are 0.001 moles of hydrogen ions in each liter of solution, indicating an acidic environment.

Various factors can affect hydrogen ion concentration, such as the nature of the acid or base, its strength, dilution, and temperature. As these concentrations can vary greatly, from very small values in basic solutions to larger values in acidic solutions, it's essential to utilize logarithmic calculations for an easier and more manageable representation, which leads to the concept of pH.

Logarithm in pH Calculation

The logarithmic relationship in pH calculation is what allows us to convert the potentially unwieldy hydrogen ion concentration into a more convenient number on the pH scale. Recall the pH definition as a negative logarithm:

\[\mathrm{pH} = -\log_{10}\left(\left[\mathrm{H}^{+}\right]\right)\]
Here, the log function refers to the base-10 logarithm, right at the heart of what makes the pH scale so useful. The negative sign indicates that a higher concentration of hydrogen ions (more acidic) will result in a lower pH value.

Simplification of the pH Calculation

The use of logarithms transforms the multiplication and division of concentrations into addition and subtraction of pH values, simplifying the mathematics involved in acid-base chemistry. This also illustrates why the pH scale is logarithmic: as the concentration of hydrogen ions increases by an order of magnitude, the pH value decreases by one unit.

For students and practitioners, mastering logarithmic calculations is essential for accurate pH computation. With a calculator, finding the logarithm of a number like in the given exercise is straightforward, thus making the determination of pH a relatively simple task once the concentration is known.