Problem 49 - Precalculus with Calculus Previews

Question

Determine how many more times acidic the first substance is compared to the second substance. lemon juice, \(\mathrm{pH}=2.3\); vinegar, \(\mathrm{pH}=3.3\)

Step-by-Step Solution

Answer

Lemon juice is 10 times more acidic than vinegar.

Step 1: Understand the pH Scale

The pH scale measures the acidity of a solution. It is logarithmic, meaning that each whole pH value below 7 (neutral) is ten times more acidic than the next higher value. Therefore, a solution with a lower pH is more acidic.

Step 2: Calculate the Hydrogen Ion Concentration

To determine the concentration of hydrogen ions \(\[H^{+}\]\), use the formula \(\[H^{+}\] = 10^{-\mathrm{pH}}\). For lemon juice, \(\mathrm{pH}=2.3\), so \(\[H^{+}\] = 10^{-2.3}\). For vinegar, \(\mathrm{pH}=3.3\), so \(\[H^{+}\] = 10^{-3.3}\).

Step 3: Perform the Calculations

Calculate the hydrogen ion concentration for each substance: 1. Lemon juice: \[H^{+} = 10^{-2.3} \approx 5.01 \times 10^{-3}\]2. Vinegar: \[H^{+} = 10^{-3.3} \approx 5.01 \times 10^{-4}\]

Step 4: Determine the Factor of Acidity

To find out how many times more acidic lemon juice is than vinegar, divide the hydrogen ion concentration of lemon juice by that of vinegar: \[\text{Factor} = \frac{5.01 \times 10^{-3}}{5.01 \times 10^{-4}} = 10\]

Step 5: Conclude

Since the factor of acidity is 10, lemon juice is 10 times more acidic than vinegar.

Key Concepts

Logarithmic Scale and the pH Scale

The pH scale is a powerful tool to measure the acidity or basicity of a solution. It ranges from 0 to 14, where a value of 7 is neutral, like pure water. This scale is not linear; it is logarithmic. This means each whole number change in pH reflects a tenfold change in acidity. For instance, a pH of 2 is ten times more acidic than a pH of 3. This is because a logarithmic scale manages large ranges of values by interpreting them as powers of ten.

By understanding the logarithmic nature of the pH scale, you can see why small differences in pH represent significant differences in acidity or basicity. Thus, a slight change on this scale corresponds to a substantial change in concentration of hydrogen ions in a solution.

Calculating Hydrogen Ion Concentration

Calculating the concentration of hydrogen ions (\[ H^+ \]) is essential to understand a solution's acidity. The formula to calculate these ions from the pH value is \( H^+ = 10^{-\text{pH}} \). This equation is derived from the properties of logarithms and helps in transforming the logarithmic pH value into a more understandable \( H^+ \) concentration.

For example, if lemon juice has a pH of 2.3, the hydrogen ion concentration is \( H^+ = 10^{-2.3} \), resulting in approximately \( 5.01 \times 10^{-3} \) moles per liter. Similarly, for vinegar with a pH of 3.3, the calculation \( H^+ = 10^{-3.3} \) leads to approximately \( 5.01 \times 10^{-4} \) moles per liter. Knowing these values allows us to compare the relative acidity of different substances.

Understanding the Factor of Acidity

The factor of acidity is an indicator of how much more acidic one solution is compared to another. To determine this factor, you divide the hydrogen ion concentration of the more acidic solution by that of the less acidic one. Since the pH scale is logarithmic, this division gives you a number showing how many times more acidic the first solution is.

In the case of lemon juice and vinegar, lemon juice has a hydrogen ion concentration of \( 5.01 \times 10^{-3} \), while vinegar has \( 5.01 \times 10^{-4} \). Dividing these \( H^+ \) values results in 10, meaning lemon juice is 10 times more acidic than vinegar. This factor is vital in various applications, such as food chemistry and medicine, where understanding the effects of acidity is crucial.