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# Calculating pH, A Look at Logarithms (30 Favorites)

LESSON PLAN in Concentration, Acid & Base Theories. Last updated March 25, 2020.

### Summary

In this lesson, students will be introduced to a base-10 logarithmic scale and use it to calculate pH from hydrogen ion concentration. Often students are able to calculate pH by pushing the correct buttons on their calculators, but they don’t understand what the values mean. This lesson attempts to bridge that gap using a guided inquiry model.

### Grade Level

High School

### NGSS Alignment

This lesson will help prepare your students to meet the performance expectations in the following standards:

**Scientific and Engineering Practices**:- Using Mathematics and Computational Thinking

### Objectives

By the end of this lesson, students should be able to

- State to purpose of using logarithms to express pH.
- Estimate pH value from hydrogen ion concentrations using knowledge of logarithms.
- Using a calculator, compute pH from hydrogen ion concentrations and vice versa.
- Classify a substance as acidic, basic, or neutral based on pH or hydrogen ion concentration.

### Chemistry Topics

This lesson supports students’ understanding of

- acids and bases
- pH and pOH
- acid/base theories
- logarithms
- concentrations

### Time

**Teacher Preparation**: 5 minutes

**Lesson**: 45 minutes

### Materials

- Copies of student handout (1 per student)
- Copies pre-quiz and post-quiz number values (1 per group; optional)
- Scientific or graphic calculators

### Safety

- No specific safety precautions need to be observed for this activity.

### Teacher Notes

- Students should be introduced to various theories of acids and bases, such as Arrhenius, Bronsted-Lowry, and/or Lewis, as well as the autoionization of water before completing this lesson. They should also be able to understand that H
^{+}and H_{3}O^{+}can be used interchangeably. - Though I prefer to use this lesson as a group activity, it may also be used as an individual or whole-class activity.
- In my experience, student math levels vary within a class. I suggest surveying the class by asking who has used logs in math class. I then group those kids together and group together the ones who have never heard of logs. The ones who have already been introduced to logs get through the activity quickly, but get stuck on the significant figure portion of it. I spend most of my time facilitating the groups for which logs are a new topic. The practice problems offer an opportunity for students to figure out which equations to use in order to solve for the unknown. In some cases, there are various pathways to the correct answer.
- Suggested
values for the
**pre-quiz**(in order, so mix them up when written on the board/sharing with students): 7.89 x 10^{-9}, 9.5 x 10^{-8}, 5.8 x 10^{-5},^{ }6.45 x 10^{-5}, 0.00089, 7 x 10^{-3}, 0.0995, 8.4 x 10^{-1}. Instead of writing them on a part of the board that can be covered, you could also write them on sheets of paper that get distributed to each group. The teacher should time each group when they raise their hand (and compare this with the time it takes for the post-quiz). Values for**post-quiz**(in order, so mix them up when written on the board/sharing with students): 0.08, 1.002, 2.2, 3.05, 4.190, 4.24, 7.02, 8.10. The teacher should also time the groups with putting these values in order. The pre and post quizzes should be done at the same time for the entire class. - Be sure to look at each student’s calculator and explain how to enter log/antilog on it. (This is entered on most graphing and scientific calculators similarly; older versions sometimes differ.)
- For lower level classes, leave off the significant figure portion. Have students report all pH values to 2 decimal places.
- A suggested extension activity: Write the following problems on the board and have students predict answers
__without__the use of a calculator. Log 0.0001, -log 1 x 10^{-13}, log 0.05, log 6000, log 4000. - Answer Key document is available for this activity.

### For the Student

### Lesson

### Pre-Quiz

Your teacher will reveal a set of numbers on the board. Your goal is to write the numbers down in numerical order (smallest to largest) as fast as you can. Raise your hand when finished.

Write the 8 numbers in order below.

Time it took to put them in order: _______

### Background

The goal of this activity is to introduce the base 10 logarithmic scale, show that it is a more efficient method of expressing very small or large numbers, and to practice calculating pH values. pH values are used to show how acidic or basic a substance is.

Necessary equation: Recall the autoionization constant for water, 1.0 x 10^{-14}, and the corresponding equation, K_{w} = [H^{+}][OH^{-}].

### Activity

Note how easy the above problems are when put into scientific notation.

Explain why the following problem is more complicated than the previous ones.

Log 650 =

The number 650 is between 100 and 1000, so the answer must be between ___ and ____.

__Using your Calculator__

Log 650 =

Directions on how to enter this into my calculator:

__Calculating pH__

pH = - log [H |

Find pH when [H^{+}] = 2.6 x 10^{-6} M

Find pH when [H^{+}] = 1.8 x 10^{-6} M

Find pH when [H^{+}]
= 3.9 x 10
^{-4} M

Notice that it is easier to compare pH values since concentration values are so small.

The _________ the [H^{+}], the _________ the pH.

Note that pH values do not have units.

__Acidic or Basic?__

When [H^{+}] > [OH^{-}], the solution is acidic

When [H^{+}] < [OH^{-}], the solution is basic

When [H^{+}] = [OH^{-}], the solution is neutral

Find [H^{+}] in a neutral solution using the equation for K_{w}:

What is the pH of this neutral solution?

Therefore, when pH <______, the solution is __________

When pH > __________, the solution is ____________

When pH = __________, the solution is ____________

__A note on Sig Figs__

For pH: Since the number to the left of the decimal point is simply an exponent, it does not count as a significant figure. Therefore, __only numbers to the right of the decimal point count__.

For [H^{+}]: Regular significant figure rules apply because it is a measured value.

6.8 → ____ sf

12.672 → _____ sf

[H^{+}] = 1.00 x 10^{-2}, find pH________ → pH has ____ sf, even though there are ___ digits

__Antilog__

10 |

Use antilog when you have the pH and are trying to find [H^{+}].

If you ever get confused, remember that log 1 = 0, so antilog of 0 is 1, or 10° = 1

Find [H^{+}] of a solution if pH = 4.97.

Directions on how to enter this into my calculator:

__Post-Quiz__

Your teacher will reveal a set of numbers on the board.Your goal is to write the numbers down *in numerical order* (smallest to largest) as fast as you can.Raise your hand when finished.

Write the numbers in order below.

Time it took to put them in order: _______

Compare your time for the pre-quiz with that of the post-quiz.If they differed, why do you think that was the case?

Propose a reason for the negative sign in the calculation of pH.

__Practice Problems__

Use the following relationships to solve the problems below.

K pH = - log [H pOH = - log [OH pH + pOH = 14 |

- pH = 1.34, find pOH
- [H
_{3}O^{+}] = 8.20 x 10^{-9}M, find [OH^{-}] - [H
_{3}O^{+}] = 8.20 x 10^{-9}M, find pH - [OH
^{-}] = 6.4 x 10^{-12}M, find pOH - [H
_{3}O^{+}] = 3.65 x 10^{-2}M, find pOH - The pH of black coffee is 5.66, find [H
_{3}O^{+}] - Some battery acid has pOH = 13.06, find [H
_{3}O^{+}] - Circle the acidic solutions in the above problems.
- Coca-Cola has a pH value of 2.53, while Barq’s Root Beer has a pH of 4.61.
- Find [H
_{3}O^{+}] of each soft drink. - Which drink is more acidic?
- How many times more acidic is the drink you chose in Part b?

- Find [H
- Briefly research the Richter scale, used to describe the magnitude of earthquakes.Explain its significance to today’s topic.