Problem 29 - Algebra 2

Question

The pH of each food is given. Find the concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right] .\) cider vinegar, 3.1

Step-by-Step Solution

Answer

The concentration of hydrogen ions in cider vinegar with a pH of 3.1 is approximately \(7.94 \times 10^{-4}\) moles per litre.

Step 1: Understand the pH and H+ Ion Concentration Relationship

The pH is a scale used to specify the acidity or basicity of an aqueous solution. It is defined as the negative logarithm (base 10) of the activity of dissolved hydrogen ions (H+). The formula is given as: \(\mathrm{pH} = -\log [\mathrm{H}^{+}]\) where \([\mathrm{H}^{+}]\) is the concentration of hydrogen ions.

Step 2: Rearrange the Formula

To find the concentration of hydrogen ions given a pH, rearrange the pH formula to solve for \([\mathrm{H}^{+}]\). This gives: \([\mathrm{H}^{+}] = 10^{-\text{pH}}\)

Step 3: Plug in Known pH

The problem states that the pH of cider vinegar is 3.1. Use this known pH and put it into the rearranged formula to find the concentration of hydrogen ions: \([\mathrm{H}^{+}] = 10^{-3.1}\)

Step 4: Calculate the Concentration

Now, calculate the value to find the concentration of hydrogen ions in the cider vinegar. Use a scientific calculator or a mathematical software, which can handle exponentiation of numbers, to evaluate the expression: \([\mathrm{H}^{+}] = 10^{-3.1}\)

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is an essential aspect in determining how acidic or basic (alkaline) a solution is. In chemical terms, the concentration of hydrogen ions is represented as \([\mathrm{H}^+]\). This concentration is crucial because it directly influences the pH level of a solution.

• More hydrogen ions mean a higher acidity and a lower pH.
• Fewer hydrogen ions signify a higher basicity and a higher pH.

The hydrogen ion concentration is measured in moles per liter (mol/L). When dealing with acidity, small changes in \([\mathrm{H}^+]\) can result in significant shifts in pH, which is why even minor differences in hydrogen ion concentration need to be accounted for in calculations. Understanding this concept helps in various scientific fields, such as chemistry, biology, and environmental science, where pH plays a critical role.

Logarithmic Functions

Logarithmic functions are mathematical functions that help us understand relationships where values change exponentially. In the context of pH and hydrogen ion concentration, the logarithmic relationship serves as a bridge between a solution's pH and its \([\mathrm{H}^+]\). The pH of a solution is calculated using the formula:\[\mathrm{pH} = -\log ([\mathrm{H}^+])\]

This formula shows that pH is the negative logarithm (base 10) of the hydrogen ion concentration. This is why:

• A decrease in one pH unit represents a tenfold increase in hydrogen ion concentration.
• Conversely, an increase in one pH unit signifies a tenfold decrease in \([\mathrm{H}^+]\).

By converting exponential relationships to linear ones, logarithmic functions make it easier to work with large or small changes in data. This simplification is beneficial in scientific calculations because it allows easier interpretation of the pH scale and its wide spectrum of values.

Acidity and Basicity

Acidity and basicity determine a solution's chemical nature and are measured primarily through the pH scale, which ranges from 0 to 14. Here's how it breaks down:

• A pH less than 7 is acidic.
• A pH greater than 7 is basic (or alkaline).
• A pH of exactly 7 is neutral, like in pure water.

Acidic solutions have a higher concentration of hydrogen ions and examples include lemon juice or vinegar. Basic solutions have fewer hydrogen ions and examples include baking soda and soapy water.

Understanding acidity and basicity is key in many daily tasks, such as cooking, gardening, and medicine. In the exercise, cider vinegar's pH of 3.1 means it's quite acidic. This acidity comes from the presence of acetic acid in the vinegar. Grasping the concept of pH in relation to hydrogen ion concentration enables students to better understand how various substances affect chemical balance and reactivity.