Problem 64 - A Graphical Approach to College Algebra

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

The \([\mathrm{H}_{3} \mathrm{O}^{+}]\) concentration is approximately \(3.16 \times 10^{-7}\) M.

Step 1: Understand the Relationship Between pH and Hydronium Ion Concentration

The pH of a solution is related to the concentration of hydronium ions, \[\text{pH} = -\log [\mathrm{H}_{3} \mathrm{O}^{+}].\] We need to find the hydronium ion concentration \([\mathrm{H}_{3} \mathrm{O}^{+}]\) given that the pH is 6.5.

Step 2: Apply the Formula to Find \([\mathrm{H}_{3} \mathrm{O}^{+}]\)

To find the hydronium ion concentration, rearrange the formula to solve for \([\mathrm{H}_{3} \mathrm{O}^{+}]\):\[[\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-\text{pH}}.\]

Step 3: Calculate \([\mathrm{H}_{3} \mathrm{O}^{+}]\) for pH 6.5

Substitute the given pH value into the rearranged formula:\[[\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-6.5}.\] Calculate the power of 10 to find the concentration:\[[\mathrm{H}_{3} \mathrm{O}^{+}] \approx 3.16 \times 10^{-7} \text{ M}.\]

Step 4: Convert the Answer

Make sure the calculated hydronium ion concentration is in molarity (M), which it is already in this case. The concentration \([\mathrm{H}_{3} \mathrm{O}^{+}] = 3.16 \times 10^{-7} \text{ M}\) is an appropriate answer for typical scientific reporting.

Key Concepts

Hydronium Ion Concentration

When we talk about hydronium ions, \([\mathrm{H}_{3} \mathrm{O}^{+}]\), we are referring to the form water takes on when it gains a proton (H⁺). This ion is a crucial part of acid-base chemistry. The concentration of hydronium ions in a solution is a direct indicator of the solution's acidity.
In simpler terms, the more hydronium ions present, the more acidic the solution is. This directly influences the pH level, which is a scale used to measure acidity or alkalinity. In practice, finding the concentration of hydronium ions involves calculating based on the solution's pH, using the formula:
\[[\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-\text{pH}}\]
Understanding this relationship helps predict how acidic a solution is based on its hydronium ion concentration. This is especially useful in chemistry since it's a common measurement in analyzing solutions. Grasping the basics of hydronium ions paves the way to understanding more complex acid-base reactions.

Molarity

Molarity is a fundamental concept in chemistry that expresses the concentration of a solution. It is defined as the number of moles of solute present per liter of solution, expressed in moles per liter (M).
Molarity provides a way to convey how much of a substance is dissolved in a given volume of solvent. For solutions involving hydronium ions, the molarity tells us how concentrated the hydronium ions are within the mixture.

• If the molarity of hydronium ions is high, the solution is more acidic.
• If it's lower, the solution is less acidic or more basic.

Understanding molarity is key when preparing solutions and conducting experiments, as it ensures the correct chemical stoichiometry. When calculating hydronium ion concentration using the formula: \([\mathrm{H}_{3} \mathrm{O}^{+}] = 3.16 \times 10^{-7} \text{ M}\), what we are essentially finding is the molarity of hydronium ions in the solution. This concentration value then reflects the solution’s acidity based on the pH level we start with.

Logarithmic Functions

Logarithmic functions are mathematical tools that are used extensively in science to deal with quantities that span many magnitudes. The pH scale is logarithmic by nature, which means that a change in one pH unit reflects a tenfold change in hydronium ion concentration.
The equation \([\text{pH} = -\log [\mathrm{H}_{3} \mathrm{O}^{+}]\) is the core of how pH and hydronium ion concentration are calculated. To reverse this equation and find the concentration from a given pH, we use the inverse, known as the antilogarithm or exponentiation:
\[[\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-\text{pH}}\]

• This conversion allows us to compute precisely how many hydronium ions exist in a solution at a specific pH value.
• For example, a pH of 6.5 implies low acidity, converting to a hydronium ion concentration of \([3.16 \times 10^{-7} \text{ M}\).

Using logarithmic functions in chemistry helps simplify and relate complex concepts that would be otherwise challenging to interpret. It distills vast ranges of ion concentrations into manageable pH numbers that are easy to work with and communicate.