Problem 46 - Precalculus with Calculus Previews

Question

Determine the hydrogen-ion concentration \(\left[\mathrm{H}^{+}\right]\) of a solution with the given \(\mathrm{pH}\). $$ 7.3 $$

Step-by-Step Solution

Answer

The hydrogen-ion concentration is approximately \(5.01 \times 10^{-8}\,\text{mol/L}\).

Step 1: Understanding the pH Formula

The \(\text{pH}\) of a solution is related to its hydrogen-ion concentration \([\mathrm{H}^{+}]\) by the formula: \[\text{pH} = -\log_{10}([\mathrm{H}^{+}])\]. To find \([\mathrm{H}^{+}]\), we will need to rearrange and solve this equation.

Step 2: Rearranging the Formula

To find \([\mathrm{H}^{+}]\), we need to rearrange the formula: \(-\log_{10}([\mathrm{H}^{+}]) = \text{pH}\). This can be transformed into an exponential equation: \([\mathrm{H}^{+}] = 10^{-\text{pH}}\).

Step 3: Substituting the pH Value

Substitute the given \(\text{pH}\) value of 7.3 into the exponential form of the equation: \([\mathrm{H}^{+}] = 10^{-7.3}\).

Step 4: Calculating the Hydrogen-Ion Concentration

Calculate \([\mathrm{H}^{+}]\) using the expression \(10^{-7.3}\). This requires calculating the power of ten with a negative exponent, which involves using a scientific calculator to find \([\mathrm{H}^{+}]\approx 5.01 \times 10^{-8}\,\text{mol/L}\).

Key Concepts

Understanding Hydrogen-Ion Concentration

In chemistry, understanding the hydrogen-ion concentration \([\text{H}^{+}]\) of a solution is essential for grasping many chemical and biological processes. \([\text{H}^{+}]\) is a measure of the abundance of hydrogen ions in a solution, and it directly affects the acidity. \([\text{H}^{+}]\) is usually expressed in moles per liter (mol/L).
Notice that each solution has a certain amount of hydrogen ions freely floating around, which influences its acidic nature.

• If \([\text{H}^{+}]\) is high, the solution is more acidic.
• If \([\text{H}^{+}]\) is low, the solution is more basic or alkaline.

The relationship between \([\text{H}^{+}]\) and pH helps to readily identify the solution’s acidity or basicity level. Since pH is a logarithmic measure, the smaller the \([\text{H}^{+}]\), the higher the pH, resulting in a more basic solution.

Exploring Exponential Equations

Exponential equations are essential in pH calculations. When dealing with pH, we use the formula: \([\text{H}^{+}] = 10^{-\text{pH}}\). Here, the term \(10^{-\text{pH}}\) is critical because it indicates how small fractions of hydrogen ions determine the acidity of a solution.
Understanding exponential equations is vital because they describe how a solution's pH impacts the concentration of hydrogen ions in a massive, exponential manner. Each change of 1 unit in pH reflects a tenfold change in \([\text{H}^{+}]\).
You will often rearrange formulas to convert a logarithmic form to an exponential equation to find unknowns like \([\text{H}^{+}]\). This conversion is crucial for translating pH values back into measurable concentrations in \([\text{H}^{+}]\). Calculators are often used to swiftly compute values due to the nature of exponential growth and decay in these equations.

Logarithmic Relationship Explained

The logarithmic relationship between pH and \([\text{H}^{+}]\) is key to comprehending the scale of measurement in chemistry. Logarithms help us handle numbers that span a broad range, such as hydrogen-ion concentrations.
The pH scale works on a base-10 logarithm, defined as \(\text{pH} = -\log_{10}([\text{H}^{+}])\). This means that each unit decrease in pH represents a tenfold increase in hydrogen-ion concentration.
This logarithmic approach compresses vast differences in \([\text{H}^{+}]\) values into a manageable scale ranging generally from 0 to 14, where:

• A pH of 7 is neutral, marking equal \([\text{H}^{+}]\) and \([\text{OH}^{-}]\) concentrations.
• A pH lower than 7 indicates acidity, while a pH higher than 7 points to alkalinity.

Understanding this scale and logarithmic calculation empowers you to navigate complex chemical concepts with ease.