Problem 32 - Algebra 2

Question

The pH of each food is given. Find the concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right] .\) tomato juice, 4.0

Step-by-Step Solution

Answer

The hydrogen ion concentration in tomato juice is \( 10^{-4.0} \) M

Step 1: Write down the given pH

For tomato juice, the pH value is given as 4.0.

Step 2: Use the formula of pH to solve for the hydrogen ion concentration

The formula for pH is \( \text{pH} = -\log[\mathrm{H}^{+}]\). We need to solve this equation for \( [\mathrm{H}^{+}] \). That would give us the formula \( [\mathrm{H}^{+}] = 10^{-\text{pH}}\). Now substitute the pH value of 4.0 into this equation.

Step 3: Calculate the hydrogen ion concentration

The calculation becomes \( [\mathrm{H}^{+}] = 10^{-4.0}\). The result of this calculation will give the hydrogen ion concentration.

Key Concepts

Hydrogen Ion Concentration

Understanding hydrogen ion concentration is key to grasping the concept of pH. The concentration is typically represented as \[[\mathrm{H}^{+}]\]which indicates the number of hydrogen ions present in a solution. This concentration influences the acidity of the solution, and by knowing it, we can determine how acidic or basic a substance is.
For example, in the case of tomato juice with a pH of 4.0, the concentration of hydrogen ions can be calculated using the formula \[[\mathrm{H}^{+}] = 10^{-\text{pH}}\]This means that a lower pH value, which indicates more hydrogen ions, results in a more acidic solution.

• A high hydrogen ion concentration corresponds to low pH levels (acidic solutions).
• A low hydrogen ion concentration corresponds to high pH levels (basic solutions).

By solving the equation for tomato juice, \[[\mathrm{H}^{+}] = 10^{-4.0} = 0.0001 \, \text{M}.\]We find that the hydrogen ion concentration is 0.0001 M, confirming that tomato juice is acidic.

Logarithms

Logarithms are a powerful mathematical tool used to simplify complex calculations, particularly in pH calculation.
When dealing with exponential relationships, as seen with hydrogen ion concentration, logarithms make it easier to express wide-ranging values in a more compact form. The formula for pH uses logarithms: \[\text{pH} = -\log[\mathrm{H}^{+}]\]This formula tells us that the pH is the negative logarithm (base 10) of the hydrogen ion concentration.
To solve for the hydrogen ion concentration from a given pH, we use the inverse operation: exponentiation of base 10. The rearranged formula \[[\mathrm{H}^{+}] = 10^{-\text{pH}}\]allows us to find the concentration by raising 10 to the power of the negative pH.

• Logarithms convert multiplication into addition, making certain calculations faster.
• They help in converting small numbers, like hydrogen ion concentrations, into manageable figures.

In the exercise, \[pH = 4.0\]is used to find \[[\mathrm{H}^{+}] = 10^{-4.0} = 0.0001\]thereby transforming a potentially complex calculation into a straightforward exponential expression.

Acidic Solutions

Acidity is a crucial property of solutions, indicating their potential to donate hydrogen ions.
Solutions with more available \[[\mathrm{H}^{+}]\]ions are termed acidic. A neutral solution (like pure water) has a pH of 7.0, and values lower than 7.0 denote acidic solutions. The pH scale runs from 0 (very acidic) to 14 (very basic), and the acidity increases as the pH decreases.
Tomato juice, with a pH of 4.0, is considered acidic. This is because \[[\mathrm{H}^{+}]\]is much higher than in neutral water. This suggests that compared to neutral solutions, acidic solutions have:

• Lower pH values.
• Higher hydrogen ion concentrations.

Moreover, acidic solutions tend to have a tart taste, and many foods that are naturally acidic, like tomato juice, fall into this category. Understanding the concept of acidity helps in many scientific fields, particularly in chemistry and biology, where the pH can affect chemical reactions and biological processes.