Problem 65 - Precalculus

Question

How many times more acidic is a substance with \(\mathrm{pH}=3.2\) than a substance with \(\mathrm{pH}=4.4 ?\) Round your answer to the nearest integer.

Step-by-Step Solution

Answer

The substance with pH 3.2 is 16 times more acidic than the substance with pH 4.4.

Step 1: Understanding pH

The pH scale measures the acidity or basicity of a solution. It is a logarithmic scale, where each integer value change represents a tenfold change in acidity. A lower pH value indicates a stronger acid.

Step 2: Convert pH to Hydrogen Ion Concentration

To find out how many times more acidic one solution is compared to another, we need to convert their pH values to hydrogen ion concentrations. This is done using the formula:\[\text{{[H}^+\text{{]}}} = 10^{-\text{{pH}}}\]For \(\text{pH} = 3.2\), \[\text{{[H}^+\text{{]}}_{3.2}} = 10^{-3.2}\].For \(\text{pH} = 4.4\), \[\text{{[H}^+\text{{]}}_{4.4}} = 10^{-4.4}\].

Step 3: Calculate Hydrogen Ion Concentrations

Calculate the hydrogen ion concentrations for both pH values:\[\text{{[H}^+\text{{]}}_{3.2}} = 10^{-3.2} \approx 6.31 \times 10^{-4}\]\[\text{{[H}^+\text{{]}}_{4.4}} = 10^{-4.4} \approx 3.98 \times 10^{-5}\]

Step 4: Determine Relative Acidity

To find out how many times more acidic the \(\text{pH} = 3.2\) solution is than the \(\text{pH} = 4.4\) solution, divide their hydrogen ion concentrations:\[\frac{\text{{[H}^+\text{{]}}_{3.2}}}{\text{{[H}^+\text{{]}}_{4.4}}} = \frac{6.31 \times 10^{-4}}{3.98 \times 10^{-5}} \approx 15.85\]

Step 5: Round the Result

Round the result to the nearest integer: 15.85 rounded to the nearest integer is 16.

Key Concepts

pH Scale

The pH scale is a measure that tells us how acidic or basic a substance is. Imagine it as a ruler for acidity! This scale ranges from 0 to 14.
If the pH is below 7, the solution is acidic, and if it's above 7, it's basic. Exactly 7 is neutral, like pure water.

What's fascinating about the pH scale is that it's not linear; it's logarithmic. This means each number change on the pH scale represents a tenfold change in acidity. For instance:

• A solution with a pH of 3 is ten times more acidic than one with a pH of 4.
• A pH of 2 is a hundred times more acidic than a pH of 4.

The scale helps us easily quantify the vast differences between the acidity of various solutions.

Hydrogen Ion Concentration

The acidity of a solution relates to its hydrogen ion concentration. More hydrogen ions mean the solution is more acidic. This is represented by \[\text{{[H}}^+\text{{]}}\], symbolizing the concentration of hydrogen ions.
To calculate this from pH, we use the formula:
\[ \text{{[H}}^+\text{{]}} = 10^{-\text{pH}} \]

Here's what's happening under the hood:

• For a lower pH value, there's a higher concentration of hydrogen ions, making the solution more acidic.
• A higher pH indicates fewer hydrogen ions, leading to a more basic (or less acidic) solution.

This is how chemists determine how acidic a sample is quantitatively.

Logarithmic Scale

The term 'logarithmic scale' might sound intimidating, but it's a handy concept when dealing with large and small numbers, like in the pH scale.
A logarithmic scale compresses large ranges of values, which can make large or small numbers more manageable.

Here's why it's perfect for pH:

• The difference between small and large hydrogen ion concentrations can span many orders of magnitude.
• With the logarithmic scale, each step of 1 in pH denotes a 10 times increase or decrease in hydrogen ion concentration.

This means that moving just one unit on the pH scale means a significant change in acidity, thanks to the properties of logarithms. It's no wonder logarithms are key in chemistry for dealing with such diverse quantities!