Problem 69 - College Algebra

Question

The \(p H\) of a solution ranges from 0 to \(14 .\) An acid solution has a pH less than 7. Pure water is neutral and has a pH of 7. Normal, unpolluted rain has a p \(H\) of about \(5.6 .\) The pH of a solution is given by $$ \mathrm{pH}=-\log x $$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve Exercises \(69-70\) An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? Express the answer as a power of \(10,\) and then round to the nearest thousandth.

Step-by-Step Solution

Answer

The concentration of hydrogen ions in the solution is approximately \(0.004\).

Step 1: Understand the pH formula

Analyze the pH formula. The pH of a solution is given by \(-\log x\), where \(x\) is the concentration of the hydrogen ions in moles per litre.

Step 2: Transform the formula

To find the hydrogen ion concentration (\(x\)), the formula needs to be transformed. The formula \(-\log x = pH\) can be written as \(x = 10^{-pH}\).

Step 3: Substitute the given pH value into the formula

Substitute the value \(2.4\) (given as the pH of the most acidic rainfall) into the formula. Therefore, \(x = 10^{-2.4}\).

Step 4: Calculate the Hydrogen ion concentration

Now, calculate the value of \(x\). To keep the answer as a power of 10, the hydrogen ion concentration is \(10^{-2.4}\).

Step 5: Round the answer to the nearest thousandth

Calculate \(10^{-2.4}\) and round the answer to the nearest thousandth. The concentration of hydrogen ions in the solution is approximately \(0.004\).

Key Concepts

acid rain

Acid rain is a form of precipitation that includes acidic components. It is caused by the atmosphere's reaction with pollutants, such as sulfur dioxide and nitrogen oxide. These gases are released into the sky from burning fossil fuels. When these pollutants mix with water vapor in the air, they form sulfuric and nitric acids, resulting in acid precipitation.
Acid rain can have a pH as low as 2.4, making it highly acidic and harmful to the environment.

• It can damage forests by leaching essential nutrients from the soil.
• Lakes and streams can become uninhabitable for fish and other wildlife due to increased acidity.
• Monuments and buildings can erode faster due to acid's corrosive nature on certain materials.

Understanding acid rain helps us appreciate the importance of reducing air pollution to protect our ecosystems and infrastructure.

hydrogen ion concentration

Hydrogen ion concentration is crucial in measuring the acidity or basicity of a solution. It is expressed as the number of hydrogen ions present per liter of solution. The more hydrogen ions there are, the more acidic the solution is.
For example, in the pH calculation, the concentration of hydrogen ions is represented by the variable \(x\). This relates directly to the pH value of the solution according to the formula:

• \(\text{pH} = -\log x\)

This formula means that if you know the pH of a solution, you can determine its hydrogen ion concentration. For instance, a pH of 2.4, as cited with the most acidic rainfall, corresponds to a high hydrogen ion concentration, specifically \(x = 10^{-2.4}\).
Calculating this gives approximately \(0.004\), indicating that the solution is quite acidic. Understanding hydrogen ion concentration helps interpret the impacts of such conditions, including their potential harm to biological and environmental systems.

logarithms

Logarithms are a mathematical tool used to simplify complex calculations, especially with very large or very small numbers. In pH calculations, logarithms are vital because they enable conversion between the pH and hydrogen ion concentration in manageable terms.

• The term \("\log"\) in mathematics refers to the exponent that a base number, often 10, must be raised to produce a given number.
• The formula \(\text{pH} = -\log x\) indicates a relationship between pH and hydrogen ion concentration by using logarithms to transform the scale.

Logarithms help in making data comparisons easier in scenarios involving significant variations, such as differences in hydrogen ion concentration levels. They are particularly useful in chemistry and environmental science because they provide a simplified way to express how concentrated a substance is in terms of its pH and hydrogen ion concentration. When using logarithms for pH, you can easily move from one expression level to another, such as from pH to original concentrations, making them a valuable and often-used calculation method in various scientific disciplines.