Problem 70 - 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 \(69-70\) The figure shows very acidic rain in the northeast United States. What is the hydrogen ion concentration of rainfall with a pH of \(4.2 ?\) Express the answer as a power of \(10,\) and then round to the nearest hundredthousandth.

Step-by-Step Solution

Answer

The hydrogen ion concentration of the rainfall, rounded to the nearest hundredthousandth, is \(0.00000630957\) moles per litre.

Step 1: Writing Down the Given Information and Requirements

The pH value of the rain is given as \(4.2\). The formula for pH is \(pH = -log(x)\), where \(x\) is the hydrogen ion concentration. The task is to find the value of \(x\).

Step 2: Applying the Formula and Solving for x

Substitute the pH value into the formula: \(4.2 = -log(x)\). Rearranging for \(x\), we get: \(x = 10^{-pH}\). Substitute \(pH = 4.2\) into this equation to get \(x\).

Step 3: Calculating the Value of x

Calculate \(x = 10^{-4.2}\). The value should be expressed as a power of 10, and then rounded to the nearest hundredthousandth.

Step 4: Final Answer

After calculating, the concentration would be \(x = 0.00000630957\) when rounded to the nearest hundredthousandth.

Key Concepts

Understanding Acidic Solution pH

The pH level of a solution is a crucial measure that indicates whether a substance is acidic, neutral, or basic. For acidic solutions, the pH value is below 7. This seemingly simple scale actually represents a complex relationship between the concentration of hydrogen ions and the property of acidity. With the pH scale being logarithmic, each whole number change represents a tenfold increase or decrease in acidity. For example, a solution with a pH of 4 is ten times more acidic than one with a pH of 5.

This important facet of acidity is particularly relevant when discussing environmental phenomena such as acid rain, which can have profound effects on ecosystems and man-made structures. Normal rainwater is slightly acidic with a pH of about 5.6 due to dissolved carbon dioxide forming carbonic acid, but when pollutants are present, rain can become much more acidic.

Hydrogen Ion Concentration in Solutions

When we talk about hydrogen ion concentration, we refer to the amount of hydrogen ions present in a solution. The higher the concentration of hydrogen ions, the more acidic the solution becomes. Conversely, a lower hydrogen ion concentration typically indicates a more basic or alkaline solution. The concentration is measured in moles per liter (mol/L), giving scientists and educators a quantifiable way to express the acidity or basicity of a solution.

Understanding the relationship between pH and hydrogen ion concentration is vital in fields like chemistry, biology, environmental science, and even medicine, as many processes are pH-dependent. One can determine the effect substances will have on each other or on biological systems by knowing their respective pH levels and corresponding hydrogen ion concentrations.

Logarithmic Nature of pH Scale

The pH scale is not a linear scale - it's a logarithmic one. What this means is that the pH of a solution is inversely proportional to the concentration of hydrogen ions in the solution. The logarithmic scale presents a much more manageable way to represent the concentration, which can vary greatly. The formula \(\text{pH} = -\log [H^+]\) showcases this relationship.

On the logarithmic pH scale, every single unit change in pH equals a tenfold change in hydrogen ion concentration. This is crucial for students to grasp because a small numerical change in pH can represent a significant change in chemical properties. For example, a change from pH 4 to pH 3 indicates a tenfold increase in acidity, not just an incremental step.

pH Calculation Exercises

To solidify your understanding of pH and hydrogen ion concentration, practicing pH calculation exercises is invaluable. Such exercises often ask you to calculate either the pH from a given hydrogen ion concentration or vice versa. By becoming proficient in these calculations, you deepen your comprehension of the intimate relationship between pH value, acidity, and basicity. Carrying out these calculations will generally involve using the logarithmic formula \(\text{pH} = -\log [H^+]\) and the inverse \( [H^+] = 10^{-\text{pH}} \).

When working through these exercises, paying close attention to significant figures and rounding rules is crucial. This attention to detail is key in scientific accuracy and communication. Additionally, exercises like these serve as practice not just in chemistry but in understanding how logarithms work, further boosting your math skills.