Problem 55 - Precalculus with Limits

Question

Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

Step-by-Step Solution

Answer

The hydrogen ion concentration of apple juice is about 0.000008 times that of drinking water, indicating that the apple juice is significantly more acidic.

Step 1: Determine the pH values

We are given that the pH of apple juice is 2.9 and the pH of drinking water is 8.0. These numbers form the basis of our calculation.

Step 2: Utilizing pH formula

Use the formula for calculating change in concentration which is \(10^{(pH1 - pH2)}\). Since we want to know how many times the concentration of the apple juice is compared to water, we will subtract the pH of water from apple juice i.e., \(10^{(2.9 - 8.0)}\).

Step 3: Calculate the difference

Perform the calculations which gives us \(10^{-5.1}\) . This is a very small number, which indicates that the apple juice is much more acidic than the drinking water.

Step 4: Conversion in standard form

Let's convert this to a typical standard number form. Due to the negative sign, the number calculated is a fraction. The absolute value of the result is the reciprocal of \(10^{5.1}\), which equals approximately to \(0.000008\). This is the factor by which the hydrogen ion concentration of the apple juice is higher than that of the drinking water.

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration is a crucial concept in understanding acidity and basicity. It refers to the number of hydrogen ions present in a solution. The concentration of hydrogen ions in a solution determines its acidity. In a high-acid solution like apple juice, there is a high concentration of hydrogen ions. Conversely, in a less acidic or neutral solution like water, there are fewer hydrogen ions present.

When we talk about hydrogen ion concentration, we are referring to the amount of these ions in one liter of solution. More hydrogen ions mean the solution is more acidic, and fewer mean it’s either neutral or basic. The units typically used to describe hydrogen ion concentration are moles per liter, often denoted as [H⁺].

• High [H⁺] = Strong Acidity
• Low [H⁺] = Weak Acidity or Basic/Neutral

These concentrations can vary greatly, thus requiring a logarithmic scale — the pH scale — for practical use.

Acidity Comparison

Acidity comparison is about determining how acidic one substance is compared to another. This understanding is vital in a wide range of applications, from chemistry to environmental science and beyond. The pH scale, which ranges from 0 to 14, is the most common way to compare acidity levels between substances.

A lower pH means higher acidity. For example, the apple juice in our exercise has a pH of 2.9, indicating high acidity, whereas drinking water with a pH of 8.0 is less acidic. To compare their hydrogen ion concentrations, we need to consider their pH difference. The essence lies in understanding that each unit difference on the pH scale represents a tenfold change in hydrogen ion concentration.

• pH = 7: Neutral (Pure Water)
• pH < 7: Acidic (Apple Juice)
• pH > 7: Basic (Soapy Water)

Hence, a small pH can signify a vastly greater concentration of hydrogen ions, further underscoring the stronger acidity of apple juice compared to drinking water.

pH Formula

The pH formula is essential for quantifying hydrogen ion concentration differences between two substances. The formula is given by:\[pH = -\log_{10}[H^+]\]This mathematical expression helps us convert hydrogen ion concentrations into pH values and vice versa.

In the case of comparing two substances, like apple juice and water, we use the modified formula to determine the concentration change:\[10^{(pH_{\text{water}} - pH_{\text{juice}})}\]This calculation uses the difference between their pH values to give us the concentration factor. For example, in the stated exercise, the factor is calculated as:\[10^{(8.0 - 2.9)} = 10^{5.1}\]But since we are finding how much more concentrated apple juice is compared to water, we use:\[10^{(2.9 - 8.0)} = 10^{-5.1}\]This shows that the concentration of hydrogen ions in apple juice is many times higher than in water, specifically 0.000008 times higher. This formula not only provides a clear comparison but also highlights the significant difference in their acidity levels.