Problem 32 - Algebra and Trigonometry

Question

The pH readings for wines vary from 2.8 to 3.8. Find the corresponding range of hydrogen ion concentrations.

Step-by-Step Solution

Answer

The hydrogen ion concentration ranges from \(1.58 \times 10^{-4}\) to \(1.58 \times 10^{-3}\) moles per liter.

Step 1: Understand the pH Formula

The pH of a solution is defined as the negative logarithm (base 10) of its hydrogen ion concentration, i.e., \( pH = -\log[H^+] \). This means the hydrogen ion concentration can be found using the formula \( [H^+] = 10^{-pH} \).

Step 2: Calculate the Hydrogen Ion Concentration for pH 2.8

Use the formula \( [H^+] = 10^{-pH} \) and substitute \( pH = 2.8 \). This gives \([H^+] = 10^{-2.8}\). Calculate this to get approximately \( [H^+] = 1.58 \times 10^{-3} \) moles per liter.

Step 3: Calculate the Hydrogen Ion Concentration for pH 3.8

Similarly, substitute \( pH = 3.8 \) into the formula \([H^+] = 10^{-3.8}\). Calculate this to get approximately \( [H^+] = 1.58 \times 10^{-4} \) moles per liter.

Step 4: Determine the Range of Hydrogen Ion Concentrations

With the concentrations calculated in Steps 2 and 3, the range of hydrogen ion concentrations is from \( 1.58 \times 10^{-4} \) to \( 1.58 \times 10^{-3} \) moles per liter.

Key Concepts

pH formula

The pH scale is a way to measure how acidic or alkaline a solution is. It ranges from 0 to 14, where values less than 7 indicate acidity, a value of 7 is neutral, and values greater than 7 indicate alkalinity. The pH formula is:

• It's defined mathematically as the negative logarithm (base 10) of the hydrogen ion concentration: \( pH = -\log[H^+] \).

• This means that, as the hydrogen ion concentration increases, the pH value decreases, signifying a higher acidity level.

Knowing how to use the pH formula is crucial for chemists and scientists, especially when determining the acidity of solutions like wines. This formula helps to reverse calculate the hydrogen ion concentration from a given pH value by rearranging it to \( [H^+] = 10^{-pH} \). This rearranged formula is essential when looking to understand the actual concentration of hydrogen ions in a solution.

logarithms

Logarithms are a key mathematical concept used in the pH formula. They are the inverse operation to exponentiation, just like subtraction is to addition. Understanding logarithms can make complex multiplications more manageable:

• A logarithm answers the question: "To what exponent must the base be raised to get that number?"

• In the pH context, we use base 10 logarithms. For example, \( \log_{10}(100) = 2 \) because \( 10^2 = 100 \).

When you see the formula \( pH = -\log[H^+] \), it implies that you need to determine the power or exponent to which 10 must be raised to get the hydrogen ion concentration. This is why calculating pH can help us understand the acidity level directly.

hydrogen ion concentration

Hydrogen ion concentration \([H^+]\) in a solution determines its acidity. More hydrogen ions mean the solution is more acidic. It is quantified in moles per liter and can have a significant range:

• The concentration is crucial for calculating the pH of a solution, using \([H^+] = 10^{-pH} \).

• Lower concentrations of \([H^+]\) imply less acidity and a higher pH value, while higher concentrations imply more acidity and a lower pH value.

This concept is valuable in various fields, including chemistry and environmental science, where precise acidity measurements are critical for research and industry practices.

acidity

Acidity refers to the level of acid present in a substance or solution, often measured by its pH level:

• A low pH value indicates high acidity, meaning more hydrogen ions are present.

• High acidity corresponds to solutions like lemon juice or vinegar, and in this context, wines with a pH range of 2.8 to 3.8 display varied acidity levels.

Acidity impacts taste, preservation, and chemical behavior. Understanding acidity is not only vital for winemakers but also in food science and other industries where maintaining specific acidity levels is crucial.

mole calculation

Mole calculations help us quantify the number of entities, like atoms or ions, in a given volume. In chemistry, a mole represents Avogadro's number \(6.022 \times 10^{23}\) of things:

• For calculating hydrogen ion concentration, knowing the number of moles per liter is essential, which is expressed as molarity (moles/L).

• The mole concept helps translate between microscopic quantities of ions and macroscopic measurements that are more tangible.

Using mole calculations in the context of pH and hydrogen ion concentration enables scientists to measure and understand the behavior of acidic or alkaline solutions in practical and precise terms.