Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\). Drinking water, 6.5

Step-by-Step Solution

Answer

\([\mathrm{H}_3\mathrm{O}^+] = 3.16 \times 10^{-7} \text{ M}\).

Step 1: Understanding pH Definition

The pH of a solution is related to the concentration of hydronium ions \( [\mathrm{H}_3\mathrm{O}^+] \) by the formula \[ pH = -\log_{10} \,([\mathrm{H}_3\mathrm{O}^+]). \] Here, a pH of 6.5 is given.

Step 2: Reorganizing the Formula

To find \( [\mathrm{H}_3\mathrm{O}^+] \) given the pH, we need to rearrange the formula: \[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}}. \] Substituting for pH, this becomes \[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-6.5}. \]

Step 3: Calculating Hydronium Ion Concentration

Now we calculate \( 10^{-6.5} \). This is equal to \[ 3.16 \times 10^{-7} \] (approximately).

Step 4: Expressing the Result

We express our final answer in scientific notation, showing the hydronium ion concentration as \( 3.16 \times 10^{-7} \, \text{M} \).

Key Concepts

Hydronium Ion Concentration

The concentration of hydronium ions (\( [\mathrm{H}_3\mathrm{O}^+] \)) is a key factor in determining the acidity of a solution. This measurement directly influences the pH value. When we talk about hydronium ions, they form when water (\( \mathrm{H}_2\mathrm{O} \)) accepts an extra hydrogen ion (\( \mathrm{H}^+ \)), creating \( \mathrm{H}_3\mathrm{O}^+ \).

• Higher concentrations of hydronium ions result in a lower pH, indicating acidity.
• Lower concentrations lead to a higher pH, signifying alkalinity.

Given the formula \[ pH = -\log_{10}( [\mathrm{H}_3\mathrm{O}^+] ) \]we can assess how the concentration and pH are related. When we know the pH value, we can rearrange this formula to find \( [\mathrm{H}_3\mathrm{O}^+] \) using:\[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} \]In our case, substituting \( pH = 6.5 \), we calculate \( [\mathrm{H}_3\mathrm{O}^+] \) for substances like drinking water. Understanding this relationship is vital in chemistry and environmental science.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a simplified form, making calculations more manageable. This notation involves writing numbers as a product of a coefficient (between 1 and 10) and a power of 10.For example, the number \( 3.16 \times 10^{-7} \)is expressed in scientific notation. Here, \( 3.16 \) is the coefficient and \( 10^{-7} \) is the power of 10. This format is especially useful in science and engineering, where such values are frequent.

• The positive exponent indicates the number is large and is moved to the right of the decimal.
• The negative exponent, like \(-7\), shows the number's decimal point is moved to the left, signifying a small value.

In our calculation of hydronium ion concentration for water with a pH of 6.5, the result \( 3.16 \times 10^{-7} \) highlights how scientific notation helps in expressing these small measurements effectively.

Exponential Function

An exponential function is crucial in the context of pH calculations. It involves a mathematical expression where a constant base, like 10, is raised to a variable exponent. This type of function grows very fast and is used to convert linear data, such as the pH scale, to exponential.In the hydronium ion concentration formula:\[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} \]the exponential function comes into play. Here,

• The base is 10, which is standard for pH calculations.
• The exponent is the negative pH value, which changes according to the substance's pH.

This results in transforming pH values into the ion concentrations, reflecting the acidity or basic character of a solution. Thanks to the exponential nature of this relationship, small changes in pH can lead to significant changes in \([\mathrm{H}_3\mathrm{O}^+] \). Thus, deeply understanding exponential functions is essential for accurate pH and concentration calculations.