Question

A grape has a pH of \(3.5\), and baking soda has a pH of \(8.0\). The hydrogen ion concentration of the grape is how many times that of the baking soda?

Short answer

The hydrogen ion concentration of the grape is \(10^{4.5}\) times greater than that of the baking soda.

Step-by-step solution

Understand the pH scale

The pH scale is a logarithmic, not linear, scale. A unit decrease in pH means the hydrogen ion concentration increases ten-fold. So a solution of pH 3 will have ten times more hydrogen ions than a solution of pH 4.

Calculate difference in pH

The pH of the grape is \(3.5\) and the pH of the baking soda is \(8.0\). Thus, the difference or delta pH (\( \Delta pH \)) is \(8.0 - 3.5 = 4.5\)

Translate pH difference to hydrogen ion concentration

Since pH is logarithmic, a one unit differences means a ten-fold difference. Thus, pH difference of 4.5 implies that the baking soda has an hydrogen ion concentration that is \(10^{4.5}\) times less than the grape.

Key concepts

Logarithmic pH Scale

Understanding the pH scale is crucial for students grappling with the concepts of acidity and alkalinity in solutions. To demystify the pH scale, let's consider it as a ladder, where each rung represents a tenfold change in acidity or basicity. Unlike a linear scale where each step is equal, each step up or down this ladder, or pH scale, means the substance is 10 times more alkaline or acidic. For instance, a lemon with a pH of 2 is not just 'one unit' more acidic than vinegar with a pH of 3, but in fact, it's 10 times more acidic.

To put it into a formula, the relationship between pH and hydrogen ion (H+) concentration is given by the negative logarithm: \[ pH = -\log[H^+] \] This means that as the pH value decreases, the concentration of hydrogen ions increases exponentially, not proportionally. This concept is fundamental when comparing the acidity or alkalinity of different substances and understanding their chemical behavior.

Hydrogen Ion Concentration Calculation

Moving on to how we actually calculate the hydrogen ion concentration. Since the relationship between pH and hydrogen ion concentration is logarithmic, we use the pH definition to find the concentration. Given the pH value, we can use the inverse of the logarithmic function, which is the exponentiation of the base 10. Here is the formula that governs this calculation: \[ [H^+] = 10^{-pH} \] So, if we have a solution with a pH of 7, the concentration of hydrogen ions is \[ [H^+] = 10^{-7} \] mol/L. It's important for students to grasp that with each integer change in pH, the hydrogen ion concentration changes tenfold. This concept is what perplexes many students, but remembering that a pH change is exponential can aid in understanding how to calculate the changes in hydrogen ion concentrations.

pH Difference

To clarify pH difference, consider it as the 'distance' between two pH values on the logarithmic scale we previously discussed. Our exercise comparing the acidity of a grape to baking soda showcases this concept. The grape, with a pH of 3.5, is significantly more acidic than the baking soda with a pH of 8.0. The pH difference is 4.5 units, which doesn't mean 4.5 times difference in hydrogen ion concentration but an exponential difference due to the logarithmic nature of pH.

When we translate this into the hydrogen ion concentration, we use the power of ten, as shown in the formula: \[ 10^{\Delta pH} \] For our example, the pH difference of 4.5 reflects a hydrogen ion concentration difference of \[ 10^{4.5} \], which is 31,623 times. So, the grape's hydrogen ion concentration is 31,623 times that of the baking soda. This stunning number demonstrates the power of the logarithmic pH scale and why even small numerical changes in pH can represent large changes in chemical properties.