1. What do you think is the definition of an acid? Of a base?

Concepts of pH

Why? The level of acidity or basicity affects many important biological and

environmental processes: enzymes function effectively only in narrowly defined ranges of pH; blood pH in part determines respiration rate and must be tightly controlled; acidity in natural waterways affects the health of aquatic ecosystems; soil pH determines whether released toxic metals will be tied up or transported through the environment.

Let’s start with a simple table to see if you know whether some common substances are acidic or basic.

Several samples of liquids were taken to study their acidity. Complete the following table, indicating whether you expect each to be acidic, basic, or neutral.

Sample Acidic, basic, or neutral?

Stomach acid Cola drink

Pure water, 25 degrees C Drain cleaner

Orange juice Soap

Bleach Acid rain Vinegar Blood

The following shows the dissociation reactions of several acids:

Strong acids:

HCl, HBr, HI, HNO₃, HClO₄, H₂SO₄ (these six) HCl  H⁺ + Cl^-

HI  H⁺ + I^- Weak acids:

HCN, HC2H3O2 (acetic acid, also abbreviated HAc), HF, HOCl, HCNO, others

HCN  H⁺ + CN^- HAc  H⁺+ Ac^-

Critical Thinking Questions:

1. What tips you off that the compound you have is an acid? What is the common product of all the acid dissociation reactions?

2. What difference do you note for the dissociation of a strong acid vs. a weak acid?

3. Suppose that you have a 0.1 M solution of both HCl and HCN. Which will have a higher concentration of H⁺ ions? Why?

Critical Thinking Questions:

1. What do you think is the definition of an acid? Of a base?

2. Think back to our titration experiments. What color is phenolphthalein in acid?

In base?

3. Is NaOH soluble or insoluble in aqueous solution? What is the relevant solubility rule? What ions exist when NaOH is placed in an aqueous environment?

4. Which ion will react with H⁺? Write the equation for this reaction. What is the product of the reaction? What is the reaction stoichiometry?

The acidic samples in the list above were titrated to determine the hydrogen ion (H⁺) concentration, which tells us the acidity of the sample. (The molarity is given for basic samples, since different experimental conditions are required.) The pH of each sample is measured with a pH electrode, and the resulting values are tabulated. The following experimental conditions were used for the acidic samples:

1. The sample size of the acidic sample used for the titration is 100 mls.

2. The concentration of the NaOH used to titrate acidic samples to the phenolphthalein endpoint is 0.05 M.

Open Excel and recreate the chart below. Type in the entries in the shaded cells; create formulas to calculate the values in the open cells in columns D, E, and F. The correct answers for these columns are given for row 3 (stomach acid), so you can check your formulas. Remember, the molarity of the titrant is 0.05 M, and the sample size is 100 mls.

1. What formula do you know that allows you to calculate moles if you know molarity and volume?

2. What is the relationship between moles NaOH and moles H⁺? Why?

3. If you know the moles of H⁺ in the sample, what do you need to do to convert this to molarity?

A B C D E F G

2 Sample mls titrant required

Moles NaOH

Moles H+

Molarity

H+ pH

3 Stomach acid _63.24 0.003162 0.003162 0.03162 1.500038

4 Cola drink _{6.32 2.500313}

5 Vinegar _{2.52 2.899629}

6 Orange juice _{0.63 3.501689}

7 Acid rain _{0.063 4.501689}

8

Pure water, 25

degrees C ********************** ************** ********** 0.0000001 7 9 Blood ********************** ************** ********** 5.012E-08 7.3 10 Soap ********************** ************** ********** 3.16E-10 9.500001 11 Bleach ********************** ************** ********** 3.16E-13 12.5 12 Drain cleaner ********************** ************** ********** 1E-13 13

Critical Thinking Questions: (Look at the spreadsheet above to answer the following.) 1. What is the pH of a substance you have identified as neutral?

2. What is the pH range of substances you have identified as acidic?

3. What is the pH range of substances you have identified as basic?

4. As H⁺ gets larger, what happens to pH?

5. Compare the results for stomach acid and cola. What is the ratio of mls titrant for stomach acid to cola? What is the ratio of M H⁺ for stomach acid to cola? What is the difference in pH? You can get these answers easily using Excel; for

example, in any blank cell you could build a formula (=C3/C4) to answer the first question.

6. Compare the results for stomach acid and acid rain. What is the ratio of mls titrant for stomach acid to acid rain? What is the ratio of M H⁺ for stomach acid to acid rain? What is the difference in pH?

7. Compare the results for stomach acid and acid rain. What is the ratio of M H⁺ for stomach acid to soap? What is the difference in pH?

8. Create a graph (XY scatter plot) of M H⁺ (x axis) vs. pH (y axis). Describe the shape of the graph. Where are most of the points relative to the y axis? What type of mathematical function is useful with data such as this?

9. In fact, the pH scale is defined as pH = -log[H⁺]. You can overwrite the value in cell G3 with the formula = -log(F3) and then copy this formula down the column.

Try this to obtain the calculated values. Typical pH values range from 1 for very acidic samples to 14 for very basic samples.

10. Answer the following questions to help you think about the usefulness of logs in developing a concept of the pH scale.

a. What is the range of H⁺ values in the samples above – low to high?

b. What is the range of pH values in the samples above – low to high?

c. Why, then, is a log scale useful for developing a concept of pH?

d. Why do you think it is defined as –log(H⁺) and not +log(H⁺)?

Complete the following chart:

[H⁺], M_____ pH_________

1.0 0.1 0.01 0.001 0.0001

0.5 0.05 0.005

0.2 0.02 0.002

1. How much more acidic is 1.0 M than 0.1 M? What is the difference in pH for these two concentrations?

2. How much more acidic is 1.0 M than 0.01 M? What is the difference in pH for these two concentrations?

3. How much more acidic is 1.0 M than 0.001 M? What is the difference in pH for these two concentrations?

4. How much more acidic is 1.0 M than 0.0001 M? What is the difference in pH for these two concentrations?

5. How much more acidic is 0.1 M than 0.01 M? What is the difference in pH for these two concentrations?

6. How much more acidic is 0.1 M than 0.001 M? What is the difference in pH for these two concentrations?

7. How much more acidic is 0.5 M than 0.05 M? What is the difference in pH for these two concentrations?

8. How much more acidic is 0.5 M than 0.005 M? What is the difference in pH for these two concentrations?

9. How much more acidic is 0.2 M than 0.02 M? What is the difference in pH for these two concentrations?

10. How much more acidic is 0.2 M than 0.002 M? What is the difference in pH for these two concentrations?

11. A certain acid, A, has a pH of 4.5. It is 10 times as acidic as acid B. What is the pH of acid B?

12. Acid C is 100 times stronger as an acid than acid A. What is its pH?

II: Introduction to Logarithms Activity I

Please do not use any calculators for this activity.

Topic: Exponential and Logarithmic functions with base ten

Learning goals: Upon the completion of this activity, the students will

• Evaluate expressions involving the powers of ten

• Solve exponential equations

• See the inverse relation between 𝑦 = 10^𝑥 and x= log (y)

• For the exponential function 𝑦 = 10^𝑥, increasing x by d units, increases y by a factor of 10^𝑑

• For the logarithmic function x=log (y) increasing y by a factor of 10^𝑘, adds k units to the logarithm

Background: In this activity, we will explore the properties of 𝑦 = 10^𝑥.

To do the operations with powers of ten we need two main equations (Note: the third one can be derived by using the first two of them.)

1. 10^𝑚10^𝑛 = 10^𝑚+𝑛 2. ₁₀¹_𝑚= 10^−𝑚

3. ¹⁰₁₀^𝑚_𝑛 = 10^𝑚−𝑛 Examples:

1. Compare 10⁸ to 10⁵.

By using the property 1, we observe that 10⁸ = 10⁵10³, therefore the number 10⁸ is 10³ times greater than 10⁵.

We can also make this observation by looking at the ratio of one term to the other.

Since ¹⁰₁₀⁸₅ = 10⁸10⁻⁵= 10⁸⁻⁵ = 10³, we can say that 10⁸ = 10³∙ 10⁵. 2. Compare 10⁻⁸ to 10⁻⁵.

Since ¹⁰₁₀⁻⁸₋₅ = 10⁻⁸10⁻⁽⁻⁵⁾ = 10⁽⁻⁸⁾10⁵ = 10⁻³, therefore 10⁻⁸= 10⁻³∙ 10⁻⁵.

Activity:

1. Let A=10⁻¹² and B= 10¹⁶. i. Compare B to A.

ii. Compare A to B.

2. Find the value of x, if 10^𝑥= 10,000.

3. Find the value of x, if 10^7.2 = 10^𝑥10⁵.

4. Find the value of x, if 10^𝑥= 0.01.

5. Fill in the following table and create a scatter plot.

x 𝑦 = 10^𝑥

1 2 3 4 5 6 7 8 9 10

6. Fill in the following table and create a scatter plot.

7. Graph 𝑦 = 10^𝑥, where x is between -10 and 10.

x 𝑦 = 10^𝑥

0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

8. Fill in the following table and graph y= log(10^𝑥)

x 𝒚 = 𝟏𝟎^𝒙 log(y) 𝟏𝟎 ^{𝒍𝒐𝒈(𝒚)}

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

9. Evaluate the following equations:

a) 10^{𝑙𝑜𝑔(222)} = b) 𝑙𝑜𝑔10³⁴³ =

10. What is the value of Y, if Y = log (10^𝐴).

III: Introduction to Logarithms Activity 2:

Topic: Exponential and Logarithmic functions with base ten

Learning goals: Upon the completion of this activity, the students will

• Solve logarithmic equations

• See the inverse relation between 𝑦 = 10^𝑥 and x= log (y)

• Explore the properties of logarithmic functions

Activity:

1. Solve each equation for x by using the information given in the following table.

a. 10^𝑋 = 8 b. 10^−𝑥 = 8 c. 10^−𝑥 = 80 d. 10^−𝑋 = 800 e. 10^−𝑥 = 8000 f. 10^−𝑥 = 80000

2. If Y=10^𝑋 means X=log (Y), solve Y=10^−𝑋 for x.

y Log(y) 10^{𝐿𝑜𝑔(𝑦})

8 0.90309 8

80 1.90309 80

800 2.90309 800

8000 3.90309 8000 80000 4.90309 80000

3. Solve the following equations:

a. 3= -log(y)

b. -2=-log(y)

c. 10=log(y)

4. If 𝑦 = 10^𝑥, complete the following sentences by using expressions among the following expressions:

{a factor of 10, one unit, K units, a factor of 10^𝐾, 1, 10, K units, D units, 10^𝐷, K+1, D+1}

If we multiply y by , X is increased by 1.

If we add to X, it causes Y to be multiplied by 10.

If we multiply y by , X is increased by K units.

If we add to X, it causes Y to be multiplied by D units.

5. Use the information provided in the following table, to find out a. What is the relation between log(AB), log(A) and log(B)?

b. What is the relation between log(A/B) , log(A) and log(B)?

A B log A log B log(AB) log (A/B)

1 10 0 1 1 -1

2 10 0.30103 1 1.30103 -0.69897

3 10 0.477121 1 1.477121 -0.52288

4 10 0.60206 1 1.60206 -0.39794

5 10 0.69897 1 1.69897 -0.30103

6 10 0.778151 1 1.778151 -0.22185

7 10 0.845098 1 1.845098 -0.1549

8 10 0.90309 1 1.90309 -0.09691

9 10 0.954243 1 1.954243 -0.04576

10 10 1 1 2 0

Topic: Exponential and Logarithmic Functions with base 10

Learning Goals: Upon the completion of this activity, the students will

• Evaluate expressions involving powers of ten

• Compare expressions involving powers of ten

• Solve exponential equations

Background:

The Richter scale was developed in 1935 by Charles Richter to compare the sizes of earthquakes. If M represents the Richter magnitude of an earthquake and P represents the relative strength of an earthquake we might express this relation as

𝑃 = 10^𝑀 or

M = log (P) In this activity, we will

• calculate the relative strength of earthquakes with different magnitudes

• compare the magnitudes of earthquakes with different strengths

To do the operations with powers of ten we need two main equations (the third one can be derived by using the first two of them.)

4. 10^𝑚10^𝑛 = 10^𝑚+𝑛 5. ¹

10^𝑛 = 10^−𝑛

6. y=10^𝑋 means x=log (y), and x=log (y), means y=10^𝑋. 7. 10^log(𝐴) = 𝐴

8. 𝑙𝑜𝑔10^𝐵 = 𝐵

Example 1: An earthquake measured 7.3 in magnitude is 10 times as powerful as an earthquake measured 6.3 in magnitude, since

10^7.3= 10^6.3+1 = 10 ∙ 10^6.3

Example 2: If the strength of an earthquake, P, is 10,000, what is its magnitude M, in the Richter scale?

M =log(10,000) = log ( 10⁴) = 4

Activity 1:

3. The Richter magnitude of 1989 San Francisco earthquake registered 7.1 on the Richter scale. Find the relative strength of this earthquake.

4. One earthquake’s magnitude measures 4.5 on the Richter scale and another one measures 6.5. How do the power of two quakes compare?

5. The magnitude of one earthquake has a Richter scale 4.2. A second earthquake is 100 times as powerful. What is the magnitude of the second earthquake on the Richter scale?

6. If the strength of an earthquake, P, is 1000, what is its magnitude M, in the Richter scale?

7. If the strength of an earthquake, P, is 0.1, what is its magnitude M, in the Richter scale?

Activity 2:

The relation between the magnitude M of an earthquake and the energy E it releases is given by

log (E) = 1.5 M

1. The largest earthquake in the United States measured 9.2 and occurred in Alaska in 1964, which was the second largest earthquake recorded.. The largest earthquake occurred in Chile in 1960 and measured 9.5.

a) Calculate the strength of the

i. 1960 Chile earthquake

ii. 1964 Alaska earthquake

b) How much stronger the 1960 Chile earthquake than the1964 Alaska earthquake?

c) Calculate the energy released by the i. 1960 Chile earthquake

ii. 1964 Alaska earthquake

d) Compare the energy released by the 1960 Chile earthquake to the 1964 Alaska earthquake.

Topic: Logarithmic Functions Activity III:

1. Write the following equations in logarithmic form.

a) 10^𝑀 = 𝑁

b) 10^𝑦 = 5𝑡

c) 10^3𝑡 = 𝑧

d) 10⁰ = 1

e) 10⁻⁵= 𝑥

f) 10^𝑦 = 𝑥⁻¹

2. Write the following equations in exponential form.

a) 𝑦 = log 𝑀

b) 𝑦 = 𝑙𝑜𝑔𝑥²

c) 𝑦 = log 10

d) 𝑦 = 𝑙𝑜𝑔1

e) 𝑦 = 2𝑙𝑜𝑔𝑥

𝒚 = 𝒍𝒐𝒈 𝒙 𝑎𝑛𝑑 𝒙 = 𝟏𝟎^𝒚

Background: The definition of common logarithm means that the equations

are equivalent, that is they mean the same thing. The first one is called the logarithmic and the second one is called the exponential form.

f) 𝑦 = −𝑙𝑜𝑔𝑥

g) −3𝑦 = 𝑙𝑜𝑔𝑥

3. Calculate the following equations without using a calculator a) 𝑙𝑜𝑔√10

b) 𝑙𝑜𝑔(10,000)

c) 𝑙𝑜𝑔₁₀₀₀¹

d) 𝑙𝑜𝑔(0.0001)

e) 𝑙𝑜𝑔 (𝑙𝑜𝑔 10¹⁰)

4. Solve the following equations without using a calculator.

a) 𝑙𝑜𝑔𝑥 = 3

b) 𝑙𝑜𝑔100^2𝑥 = −2

c) 𝑙𝑜𝑔√𝑥 = 3

d) 𝑙𝑜𝑔10^{𝑙𝑜𝑔𝑥} = 5

e) log(log(𝑥 − 2)) = 1

f) 100^2𝑥 = 0.01

g) 0.001 = 1000^5𝑥−2

5. Fill in the following table:

x y = log x 10⁻³

10⁻² 10⁻¹ 1 1010² 10³

CONCLUSION: If 𝑦 = 𝑙𝑜𝑔10^𝑀, then y =