Problem 59 - Algebra 2 and Trigonometry

Question

The pH (hydrogen potential) measures the acidity or alkalinity of a solution. In general, acids have pH values less than \(7,\) while alkaline solutions (bases) have pH values greater than \(7 .\) Pure water is considered neutral with a \(\mathrm{pH}\) of \(7 .\) The pH of a solution is given by the formula \(\mathrm{pH}=-\log x\) where \(x\) represents the hydronium ion concentration of the solution. Find, to the nearest hundredth, the approximate \(\mathrm{pH}\) of each of the following: a. Blood: \(x=3.98 \times 10^{-8}\) b. Vinegar: \(x=6.4 \times 10^{-3}\) c. A solution with \(x=4.0 \times 10^{-5}\)

Step-by-Step Solution

Answer

Blood pH is approximately 7.40, vinegar pH is 2.19, and the solution's pH is 4.40.

Step 1: Understanding the pH Formula

The pH of a solution is determined by the formula \( \mathrm{pH} = -\log x \), where \( x \) is the hydronium ion concentration. Our task is to find the pH for given concentrations by substituting these values into the formula.

Step 2: Calculate pH for Blood

Substitute \( x = 3.98 \times 10^{-8} \) into the formula: \( \mathrm{pH} = -\log(3.98 \times 10^{-8}) \). Using a calculator, \( \mathrm{pH} \approx -(-7.4006) \approx 7.40 \).

Step 3: Calculate pH for Vinegar

Substitute \( x = 6.4 \times 10^{-3} \) into the formula: \( \mathrm{pH} = -\log(6.4 \times 10^{-3}) \). Using a calculator, \( \mathrm{pH} \approx -(-2.1938) \approx 2.19 \).

Step 4: Calculate pH for the Given Solution

Substitute \( x = 4.0 \times 10^{-5} \) into the formula: \( \mathrm{pH} = -\log(4.0 \times 10^{-5}) \). Using a calculator, \( \mathrm{pH} \approx -(-4.3979) \approx 4.40 \).

Key Concepts

Logarithms in Chemistry

Logarithms play a critical role in chemistry, particularly in calculating pH, which is a measure of acidity or alkalinity. The concept of logarithms can be a bit tricky at first, but they essentially help us simplify very large or small numbers. When dealing with pH, the logarithm base is 10, and we often encounter negative exponents as we're working with small concentrations, like hydronium ions.
In the pH formula, \[pH = -\log x\]where \(x\) is the hydronium ion concentration. This means you take the logarithm base 10 of a number, and then multiply it by -1. This transformation makes it much easier to work with the data. Having a basic understanding of logarithms will significantly simplify your work with many chemical formulas and equations. You'll often find logarithms taking large or small values and presenting them in a more manageable format for interpretation.
Here are some important points:

• Logarithms convert multiplication and division into addition and subtraction.
• A log base 10 of a number is the power to which 10 must be raised to obtain that number.

Hydronium Ion Concentration

Hydronium ion concentration is a crucial aspect of chemistry as it directly influences the pH of a solution. The concentration of hydronium ions, denoted by \([H_3O^+]\), indicates how acidic or basic a solution is. In the pH formula, the hydronium ion concentration is represented as \(x\).
A high concentration of hydronium ions means a low pH value, indicating a more acidic solution. Conversely, a low concentration of these ions suggests a higher pH value, or a more basic (alkaline) solution.
Here's a simplified look at it:

• \([H_3O^+] > \ 1 \times 10^{-7}\) implies acidic solutions.
• \([H_3O^+] < \ 1 \times 10^{-7}\) implies basic solutions.
• \([H_3O^+] = \ 1 \times 10^{-7}\) implies neutral solutions, like pure water.

Understanding how hydronium ion concentration affects pH helps sketch a clearer picture of the solution's characteristics. Many biological and chemical processes rely on specific pH levels, making this knowledge essential for conducting experiments and industrial applications.

Acidity and Alkalinity

Acidity and alkalinity are fundamental concepts that describe a solution's pH level. It's essential to grasp these ideas to understand chemical reactions, biological processes, and everyday substances like food and cleaning agents.
A solution's acidity or alkalinity is based on its hydronium ion concentration. A neutral pH is about 7, typical for pure water. However, solutions can be either:

• Acidic, with a pH less than 7.
• Alkaline (or basic), with a pH greater than 7.

For example, blood has a slightly basic pH of about 7.4, important for its role in bodily functions. Vinegar, on the other hand, is acidic with a pH around 2.4, giving it its characteristic sour taste.
Understanding these concepts is crucial, as they affect everything from the taste of food to the effectiveness of medications and cleaning supplies. Oftentimes, maintaining or adjusting a specific solution's pH level can be vital for certain chemical reactions or biological functions to occur properly. The pH scale not only helps identify the acidity or alkalinity but also aids in the safe and effective handling of various substances.