/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 The \(p H\) of a solution ranges... [FREE SOLUTION] | 91Ó°ÊÓ

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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.

Short Answer

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

Step by step solution

01

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.
02

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 \(x = 10^{-pH}\).
03

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, \(x = 10^{-2.4}\).
04

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 \(10^{-2.4}\).
05

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 \(0.004\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzce learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x+5 \ln y-6 \ln z $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{125}{y}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \log _{6} x+6 \log _{b} y $$

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