Problem 29 - Algebra and Trigonometry

Question

The pH reading of a sample of each substance is given. Calculate the hydrogen ion concentration of the substance. (a) Vinegar: \(\mathrm{pH}=3.0\) (b) Milk: \(\mathrm{pH}=6.5\)

Step-by-Step Solution

Answer

Vinegar: \([\mathrm{H}^+] = 1.0 \times 10^{-3}\, \text{M}\); Milk: \([\mathrm{H}^+] = 3.16 \times 10^{-7}\, \text{M}\)."

Step 1: Understanding the Relationship

To find the hydrogen ion concentration from pH, remember the formula \[ \mathrm{pH} = -\log_{10}[\mathrm{H}^+] \] where \([\mathrm{H}^+]\) is the hydrogen ion concentration. We need to find \([\mathrm{H}^+]\) by rearranging the formula.

Step 2: Rearranging the Formula

To isolate \([\mathrm{H}^+]\), use the equation \[ [\mathrm{H}^+] = 10^{-\mathrm{pH}} \] This allows you to find the concentration of hydrogen ions when given the pH.

Step 3: Calculating for Vinegar

Substitute the pH of vinegar (3.0) into the formula: \[ [\mathrm{H}^+] = 10^{-3.0} \] Calculate to find \[ [\mathrm{H}^+] = 1.0 \times 10^{-3} \text{ M} \]

Step 4: Calculating for Milk

Substitute the pH of milk (6.5) into the formula: \[ [\mathrm{H}^+] = 10^{-6.5} \] Calculate to find \[ [\mathrm{H}^+] = 3.16 \times 10^{-7} \text{ M} \]

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a key concept when understanding pH and how acidic or basic a solution is. It is denoted by \([\mathrm{H}^+]\). The more hydrogen ions present in a solution, the more acidic the solution is.
On the pH scale, which ranges from 0 to 14, a lower pH value indicates a higher hydrogen ion concentration.To calculate the hydrogen ion concentration from a given pH, we use the formula:

• \([\mathrm{H}^+] = 10^{-\mathrm{pH}}\)

This formula is derived from the definition of pH itself. In practical terms, if you know the pH of a substance, you can quickly compute the concentration of hydrogen ions using this relationship.
For instance, vinegar with a pH of 3.0 has a hydrogen ion concentration of \(10^{-3.0}\), or \((1.0\times10^{-3})\) M, making it acidic.

Acid-Base Chemistry

Acid-base chemistry is fundamental in understanding the pH scale and reactions associated with acids and bases.In simple terms, an acid is a substance that increases the \([\mathrm{H}^+]\)when dissolved in water, while a base decreases \([\mathrm{H}^+]\)levels.Some common properties of acids include:

• Tasting sour, like citrus or vinegar.
• Reacting with metals to produce hydrogen gas.
• Turning blue litmus paper red.

Milk, with a pH of 6.5, is slightly acidic compared to a neutral pH of 7. Its hydrogen ion concentration is \(3.16 \times 10^{-7}\) M, indicating fewer hydrogen ions than vinegar. On a molecular level, understanding how compounds affect hydrogen ion concentration is crucial for predicting chemical behavior in solutions.
Acid-base reactions are central to numerous biological processes and industrial applications.

Logarithms

Logarithms are a mathematical concept used extensively in pH calculations.The logarithm is essentially the inverse operation of exponentiation, and for pH, it refers to base 10 logarithms which are shown as \(-\log_{10}[\mathrm{H}^+]\).Here are some key points about logarithms in the context of pH:

• Logarithms can transform multiplicative relationships into additive ones.
• In acid-base chemistry, logarithms help simplify the calculation of hydrogen ion concentrations from pH values.
• A small change in pH implies a tenfold change in hydrogen ion concentration since it is a logarithmic scale.

The power of using logarithms in pH calculations lies in its ability to handle wide ranges of concentrations easily and succinctly. For example, it allows us to express the concentration of hydrogen ions in a manageable number format, even if those concentrations vary widely.