Problem 125 - College Algebra with Modeling and Visualization

Question

Acid Rain Air pollutants frequently cause acid rain. A measure of the acidity is \(\mathrm{pH}\), which ranges between 1 and \(14 .\) Pure water is neutral and has a \(\mathrm{pH}\) of \(7 .\) Acidic solutions have a \(\mathrm{pH}\) less than 7 , whereas alkaline solutions have a pH greater than \(7 .\) A pH value can be computed by \(\mathrm{pH}=-\log x,\) where \(x\) represents the hydrogen ion concentration in moles per liter. In rural areas of Europe, rainwater typically has \(x=10^{-4.7}\) (a) Find its \(\mathrm{pH}\). (b) Seawater has a \(\mathrm{pH}\) of \(8.2 .\) How many times greater is the hydrogen ion concentration in rainwater from rural Europe than in seawater?

Step-by-Step Solution

Answer

(a) The pH of the rainwater is 4.7. (b) Rainwater has a hydrogen ion concentration that is 3162 times greater than seawater.

Step 1: Calculate the pH of Rainwater

Given the hydrogen ion concentration in rainwater is \(x = 10^{-4.7}\). The formula to calculate pH is \(\mathrm{pH} = -\log x\). Substituting \(x\) into the equation gives: \[\mathrm{pH} = -\log(10^{-4.7}) = 4.7\]Thus, the pH of the rainwater is 4.7.

Step 2: Understand the pH difference and its implication

The pH scale is logarithmic, meaning each unit change corresponds to a tenfold change in hydrogen ion concentration. Seawater has a pH of 8.2, and rainwater has a pH of 4.7. We need to find how many times greater the hydrogen ion concentration in rainwater is compared to seawater.

Step 3: Calculate the Relative Hydrogen Ion Concentration

The hydrogen ion concentration for seawater can be determined using its pH: \[ x_{sea} = 10^{-\mathrm{pH}_{sea}} = 10^{-8.2} \]Since the concentration is logarithmic, the concentration factor difference can be calculated as:\[ 10^{\mathrm{pH}_{sea} - \mathrm{pH}_{rain}} = 10^{8.2 - 4.7} = 10^{3.5} \]Thus, the hydrogen ion concentration in rainwater is \(10^{3.5}\) times greater than in seawater.

Key Concepts

Hydrogen Ion Concentration

The concept of hydrogen ion concentration is foundational when discussing the acidity or alkalinity of a solution.
It is often denoted with the symbol \( x \), which represents the number of moles of hydrogen ions present per liter of solution.
This measure directly influences the pH level of the solution.When the hydrogen ion concentration is high, the solution is more acidic, resulting in a lower pH.
Conversely, a low hydrogen ion concentration leads to a higher pH, making the solution more alkaline.
In the exercise provided, rainwater in rural Europe has a hydrogen ion concentration of \(x = 10^{-4.7} \).

• The concentration \(10^{-4.7} \) indicates that there are very few hydrogen ions, as it is a very small number.
• This results in the rainwater being acidic, as calculated to have a pH of 4.7.

Understanding this concept helps explain why solutions can vary so drastically in their pH values depending on the number of hydrogen ions present.

Logarithmic Scale

The pH scale is a fascinating example of a logarithmic scale in action.
Unlike regular numerical scales, a logarithmic scale means each whole number step represents a tenfold change.
This can be a bit complex at first, but breaking it down helps.### How the Logarithmic Scale WorksEvery time the pH decreases by 1, the hydrogen ion concentration increases by 10 times.
For example, a solution with a pH of 5 has ten times more hydrogen ions than a solution with a pH of 6.

• This is why even small changes in pH values can reflect significant changes in acidity or alkalinity.
• In our exercise, the difference between the pH of seawater (8.2) and rainwater (4.7) means a significant change in hydrogen ion concentration.

Understanding that a logarithmic scale is used for pH calculations helps to comprehend why the concentration of hydrogen ions in the rainwater is \(10^{3.5} \) times greater than in seawater.

Acidic and Alkaline Solutions

Solutions are generally categorized as acidic or alkaline based on their pH values.
The pH scale ranges from 0 to 14, with 7 considered neutral (like pure water).
Knowing whether a solution is acidic or alkaline has many practical applications. ### Characteristics of Acidic Solutions

• pH values less than 7 are labeled as acidic.
• Acidic solutions have higher concentrations of hydrogen ions.
• They often taste sour and can conduct electricity well.

In the given scenario, rainwater with a pH of 4.7 is acidic due to its relatively high hydrogen ion concentration. ### Characteristics of Alkaline Solutions

• pH values greater than 7 are termed alkaline or basic.
• They have a lower concentration of hydrogen ions.
• Alkaline solutions often feel slippery and can neutralize acids.

Seawater, with a pH of 8.2, is a good example of an alkaline solution.
This contrast between acidic rainwater and alkaline seawater demonstrates the diverse nature of solutions and their behaviors.