How does a log scale work?

A dataset and activity to introduce a log scale in a classic chemistry context.

Introduction

The pH scale is a log scale. What exactly does that mean? 

The standard pH scale, which ranges from 1-14, measures the concentration of H⁺ ions in an aqueous solution. Solutions with a relatively high concentration of H⁺ ions - those with a pH of 1-6 - are considered acidic. Solutions with a relatively low concentration of H⁺ ions - those with a pH of 8-14 - are considered basic.

Because  pH is measured on a log scale, the pH value of a solution is  equal to the negative logarithm of the hydrogen ion concentration. We can use the equation pH = -log[H⁺] where [H⁺] represents the hydrogen ion concentration in solution. Taking the negative log of the H⁺ concentration ensures we end up with a positive pH value. This means that if a solution contains 1x10^-4 M hydrogen ions, its pH = -log(1x10^-4) = 4. Based on the diagram above, the solution might be tomato juice.

Given that pH is measured on a log scale,  how should you interpret differences in pH when comparing different solutions? How much more acidic is a solution of pH 5 compared with a solution of pH 6? How much stronger of a base is bleach as compared to hand soap? How much more acidic are gastric juices as compared to lemon juice?

Work with this dataset containing the pH and hydrogen ion concentrations of common substances to see exactly how differences in pH relate to the differences in hydrogen ion concentrations.

The Dataset

This dataset lists the concentration of hydrogen ions and the pH values of fourteen common substances.

The Activity

1. Open the dataset and look at the Table tab. What does a change from a pH of 8 to a pH of 9 mean in terms of the concentration of H⁺ ions in each solution? How many times more H⁺ ions are in sea water as compared to baking soda.

2. What does a change from a pH of 4 to a pH of 2 mean in terms of the concentration of H+ ions in each solution? Use the values in the dataset to help you calculate this.

3. Make a line graph showing the value of [H+] for each whole number on the pH scale. Paste your graph into this assignment sheet.

Click on the Graph tab at the top of the screen to switch to graph view. Be sure that the Scatter/Box/Bar or Categorical Bubble icon is selected; this will ensure you make a scatter plot. Click the Show buttons beneath the variable names to show them on the graph. Be sure each pH is on X and [H+] is on Y. If not, you can correct that on the panel to the right side of your graph. Finally, check the Connect Dots box.

4. What do you notice about the graph you made, particularly with regard to the points that represent pH 2 through pH 14?

5. Next, click on the box labelled lin that corresponds with the Y-axis of the graph. This will change the scale of the Y-axis from a linear scale to a logarithmic scale.

6. What do you notice about the shape and slope of this new graph?

7. What advantage does the logarithmic scale have over the linear scale that you used for your first graph?

8. Use the pH formula from the introduction (pH = -log[H⁺]) to find the pH value of a substance whose [H⁺] = 3.25 x 10^-4.

9. Based on your calculations in question 9, predict the pH value for a substance whose [H+] = 3.25 x 10^-6. Explain how you determined this predicted value.

10. Use the pH formula to find the pH value of a substance whose [H+] = 3.25 x 10^-6.

11. A solution with a pH of 1 and a pH of 3 are relatively close to each other on the pH scale. Does this mean that they have close to the same concentration of H⁺ ions? Explain.

12. How many times greater is the concentration of H⁺ ions in a solution with a pH of 3 as compared to a solution with a pH of 7?