Problem 51 - Precalculus with Calculus Previews

Question

Determine how many more times acidic the first substance is compared to the second substance. acidic rain, \(\mathrm{pH}=3.8 ;\) clean rain, \(\mathrm{pH}=5.6\)

Step-by-Step Solution

Answer

Acidic rain is about 63 times more acidic than clean rain.

Step 1: Understanding pH and Acidity

The pH scale measures how acidic or basic a substance is on a scale of 0 to 14, with lower values indicating higher acidity. The scale is logarithmic, meaning that each whole pH value represents a tenfold difference in acidity.

Step 2: Calculating the Hydrogen Ion Concentrations

First, calculate the hydrogen ion concentrations from the pH values: For acidic rain (\text{pH} = 3.8), the concentration is \([H^+] = 10^{-3.8}\). For clean rain (\text{pH} = 5.6), the concentration is \([H^+] = 10^{-5.6}\).

Step 3: Determine the Ratio of Acidity

To find out how many times more acidic the acidic rain is compared to clean rain, divide the hydrogen ion concentration of acidic rain by that of clean rain: \( \frac{10^{-3.8}}{10^{-5.6}} = 10^{(-3.8 + 5.6)} = 10^{1.8} \).

Step 4: Calculating the Result

Compute \(10^{1.8}\) using a calculator. This is approximately equal to 63.1.

Key Concepts

Acidity Calculation

Calculating acidity is an essential part of understanding chemical properties in substances. Acidity is often expressed as hydrogen ion concentration, \([H^+]\), in solutions. To find how much more acidic one substance is compared to another, it involves a comparison of their \([H^+]\) values.

• A lower pH value indicates higher acidity, meaning a stronger concentration of \([H^+]\) ions.
• When comparing substances like acidic rain (pH 3.8) and clean rain (pH 5.6), it's crucial to convert pH values into hydrogen ion concentrations first.

By calculating the ratio of \([H^+]\), one can determine how much more acidic one substance is. In this case, acidic rain is many times more acidic than clean rain, demonstrated through the calculation. This ratio gives a clear, quantitative understanding of their acidity levels.

Hydrogen Ion Concentration

The hydrogen ion concentration is a key indicator of a solution's acidity. It tells us how many hydrogen ions are present in a given amount of the substance. This concentration is derived from the pH value of the substance.

• The formula to calculate \([H^+]\) from pH is \([H^+] = 10^{-\text{pH}}\).
• For example, for a substance with a pH of 3.8, the hydrogen ion concentration is \([H^+] = 10^{-3.8}\), and for a pH of 5.6, it is \([H^+] = 10^{-5.6}\).

This calculation highlights the exponential nature of acidity. Remember, while the pH scale seems linear, each unit change represents a tenfold difference in \([H^+]\). Therefore, small pH difference equates to significant acidity difference.

Logarithmic Scale

The pH scale is a classic example of a logarithmic scale. Unlike linear scales, a logarithmic scale means each step represents a multiplication of the previous step. Here, each pH unit change represents a tenfold change in hydrogen ion concentration.

• For instance, moving from a pH of 5 to 4 means the solution is 10 times more acidic.
• This tenfold rule is due to the way pH is calculated: \([H^+] = 10^{-\text{pH}}\).

The logarithmic nature of the pH scale means that acidity calculations are non-intuitive. Two digital units on this scale represent a 100 times difference in terms of hydrogen ion concentration. This emphasizes the importance of understanding logarithmic calculations in chemistry when assessing acidity levels.