Problem 71 - Precalculus : Building Concepts and Connections

Question

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Find the \(\mathrm{pH}\) of a solution with \(\left[\mathrm{H}^{+}\right]=6 \times 10^{-8}\).

Step-by-Step Solution

Answer

The pH of the solution is approximately 7.22

Step 1: Understanding the given values

The exercise provides the concentration of Hydrogen ions represented as \(\left[\mathrm{H}^{+}\right] = 6 \times 10^{-8}\) M.

Step 2: Applying the formula for pH

The pH of a solution is calculated using the formula: pH = -log \([H^+]\). Here, the negative logarithm base 10 of the Hydrogen ion concentration \([H^+]\) is taken.

Step 3: Substituting the values into the formula

Substituting the given [H+] value into the formula, we obtain: pH = -log \((6 \times 10^{-8})\).

Step 4: Calculating the pH

Use the properties of logarithms to simplify the expression. Logarithm of a product can be written as the sum of the logarithms. Therefore -log(a*b) = -log(a) - log(b). Applying this to simplify the expression, we get: pH = -(\(log 6 + log 10^{-8})\), which simplifies to pH = \(-0.78 - (-8)\), leading to the final result: pH = 7.22.

Step 5: Conclusion statement

The pH of the solution with Hydrogen ion concentration of \(6 \times 10^{-8}\) M is approximately 7.22.

Key Concepts

Logarithms

Logarithms are a mathematical concept used to simplify complex calculations, especially those that involve multiplication and division. They are the inverse operation of exponentiation. In the context of pH calculations, logarithms allow us to transform the hydrogen ion concentration, a value that can sometimes be very small, into a more manageable and interpretable number on the pH scale. The formula for calculating pH is based on the base-10 logarithm of the hydrogen ion concentration:

• pH = -log ([H⁺])

This means we take the log base 10 of the hydrogen ion concentration and then multiply it by -1 to get the pH. For instance, if [H⁺] = 6 x 10^-8, we compute the pH as follows:

• pH = -( log(6) + log(10^-8))

Understanding how to manipulate logarithms can simplify the process dramatically, as in this case, where we are dealing with negative exponents.

Hydrogen Ion Concentration

The hydrogen ion concentration, often denoted by [H⁺], reflects the activity of hydrogen ions in a solution. It is a crucial measure in chemistry as it helps us understand a solution's acidity or basicity. Typically, hydrogen ion concentrations can range over many orders of magnitude, making it challenging to compare directly. Hence, the use of the pH scale.
The concentration is usually expressed in molarity (M), and it's placed in the pH formula to derive the pH value. For instance, knowing [H⁺] = 6 x 10^-8 M allows us to calculate the solution's pH using the logarithmic pH formula:

• pH = - log([H⁺])

The negative logarithm translates the wide range of hydrogen ion concentrations into a more condensed and useful scale, helping scientists and students alike evaluate and compare the acidity of different solutions easily.

Acid-Base Chemistry

Acid-base chemistry covers a significant domain within chemistry focused on solutions' acidity and basicity, commonly quantified via the pH scale. This field examines how acids and bases interact, neutralize each other, and affect various chemical reactions.
The balanced nature of water makes it a neutral reference point on the pH scale, which ranges from 0 to 14.

• Acids have pH values less than 7
• Bases have pH values greater than 7
• Neutral solutions like pure water have a pH of approximately 7

Acids typically release hydrogen ions (H⁺) into solutions, increasing the hydrogen ion concentration. This results in a lower pH. Bases, on the other hand, reduce the hydrogen ion concentration either by accepting H⁺ ions or releasing hydroxide (OH^-) ions. Understanding these interactions and how they impact pH is fundamental in predicting and controlling the outcome of chemical reactions. For example, a solution with a [H⁺] of 6 x 10^-8 M and a calculated pH of 7.22 would indicate a solution very close to neutral, aligning with what is expected in acid-base chemistry.