How to Pass This Class

Chemonomicon

How to Pass This Class

1. Don’t skip class

2. Read the syllabus – know how your grade is determined 3. Look up and write down your professor’s office hours

4. Expect this class to require hours of studying every week for the next 16 weeks 5. Pick three

a. School b. Work c. Sleep d. Social life

6. Use a separate 3-ring binder or composition notebook for each class 7. Bring Chemonomicon to class

8. Take notes in class

9. Rewrite and organize class notes when you get home 10. Keeps notes and homework in separate notebooks 11. Read textbooks three times:

a. Skim b. Read c. Skim

12. Outline Chemonomicon as your read it

13. Do Chemonomicon problems as you encounter them 14. Rework examples from class

15. Use test topics list as a study check list 16. Learn to gauge your own learning 17. Learn to read your professors

18. Take responsibility for your own mistakes and learn from them

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Look up your Professors Office Hours

1. Go to https://my.mdc.edu/

2. Log in with your username and student ID 3. Click on the link that says “Faculty Schedule”

4. Write down schedule here

Professor Larry Mailloux Professor

Class CHM1025 Class

Office Location 3291-16 Office Location

Day Office Hours Day Office Hours

Monday Monday

Tuesday Tuesday

Wednesday Wednesday

Thursday Thursday

Friday Friday

Professor Professor

Class Class

Office Location Office Location

Day Office Hours Day Office Hours

Monday Monday

Tuesday Tuesday

Wednesday Wednesday

Thursday Thursday

Friday Friday

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Introduction and the Scientific Method Introduction

Welcome to CHM1025. The purpose of this class is to prepare you to succeed in college level chemistry courses. While some of you may have taken chemistry in high school, college level classes are very different from their high school counterparts. College classes go much faster than high school classes and as a result it is extremely important to follow along with your studying as the course progresses. Falling even a couple days behind can spell disaster! This book has been written to help you follow along with the course by filling it in as the class progresses. Therefore, instead of having all of the practice problems together at the end of each chapter, in this book you will find practice problems spread throughout every section. Do these problems as you encounter them and don’t move on until you have completed and understand each set of problems.

Space (lightly shaded in blue) has been provided in this book for you to write your work for each problem. After you have figured out a problem, carefully rework the problem into the space provided in this book. This way at the end of the semester you will have a well-organized reference to study from for the final exam in your own handwriting.

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Chemistry and the Scientific Method

Chemistry is the branch of science that studies matter and the changes that matter undergoes as it interacts either with another type of matter or energy, such as light. As with any science, chemistry is limited to knowledge about the natural world that can be studied through systematic observation. There are two main types of observations, direct and indirect. Let’s consider the example of a broken leg. Direct evidence would be a broken bone sticking out of the skin, while an example of an indirect observation would be seen using an x-ray machine. Both of these observations indicate the presence of an injury and both allow us to learn about it. While direct observations might be preferable due to their simplicity, for many areas of science we must rely on indirect observations. For example, no one has ever directly seen an electron, but their existence and properties are extremely well understood, since they serve as the basis of all modern electricity and electronic equipment.

Much of what scientists know about the world comes from how matter and energy interact.

Below the interaction of x-rays with different types of tissues allows a doctor to diagnose a broken bone:

An x-ray allows a doctor to diagnose a broken bone Last words, “Hey bro check this out…”

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Just like how a doctor can use how x-rays and matter interact to see inside your body, a chemist can use patterns in how radio waves and matter interact using a technique called NMR

spectroscopy to learn about the structures of molecules. An example of an NMR spectrum is shown below.

Example of an NMR Spectrum

Allows chemists to determine structures of molecules, like the one in the upper left

Just as it takes many years of hard study to become a doctor and learn to properly interpret x- rays, it also takes many years of difficult studying to learn to be a scientist and read things like NMR spectra. It is important that students taking science courses be patient. Students are often very curious about how it is that scientists know the things that they know about the world around them. I will do my best to explain as much as possible in this class, however given that this is only an intro/prep course, the real explanation for many things often makes use of extensive indirect evidence, which requires background knowledge that students simply don’t possess yet. The world around us is much more complex than it often appears, it will take many years of hard work to understand many of these concepts fully.

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The reason that scientists have been so successful at figuring out how the world works over the past few hundred years is because they perform their work using a systematic process known as the scientific method. The scientific method is a series of steps that scientists go though as they learn about the world around them. The figure below, shows the scientific method in the form of a flow chart. Be aware that while figures like this make the scientific method appear simple, in real life scientific research can take years, or even decades, to complete the process. Also, while all branches of science follow the scientific method, the exact process varies from field to field; a psychologist for example performs the steps of the scientific method differently that a chemist would. The diagram below is just a summary of how an idealized scientific experiment is performed. Real research is more complicated.

A flow chart of the scientific method.

Science starts with observations and questions. For example, one morning on September 28, 1928 a biologist working at St. Mary’s Hospital named Dr. Alexander Fleming returned from a vacation to notice that petri dishes in his lab had become contaminated by a kind of mold known as penicillium notatum. Luckily for humanity Fleming noticed that the mold in the dishes was

Observation

Correct

Ask Questions

Background research

Construct Hypothesis

Test w/ Experiments

Analyze Results

Incorrect

Publish Result

Try Again

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preventing the grown of the staphylococci bacteria he had been studying and he hypothesized that something in the mold was killing the bacteria. A hypothesis is a tentative explanation for an observation, in this case Fleming made the educated guess that some chemical released by the mold was somehow preventing the growth of the bacteria in the dish. To test his hypothesis Fleming spent the next few weeks growing more Penicillium mold to allow him to confirm his finding that appeared to show that something in the mold was somehow killing the surrounding bacteria.

Fleming repeating his experiment once however, was far from sufficient evidence to elevate the notion that invisible chemicals released by the mold were actually killing the bacteria to the level of what a scientist would call a theory. In science, a theory is a well-established explanation of observed phenomena. This means confirmation by multiple independent scientists is necessary before something is referred to as the theory. This is unlike the everyday usage of the word

“theory” which is more like a hunch or a guess. Nowadays we take for granted that antibiotics molecules like penicillin kill bacteria, however it took the combined work of many scientists over a decade to identify, isolate, and purify the active ingredient responsible for the bacterial killing abilities of the penicillium notatum mold. In the case of Flemings work, it would not be until 1942 (14 years later) that the active ingredient from the mold (penicillin) would be used to treat an infection in an actual living person.

Another requirement of scientific hypothesis and theories is that they must also be falsifiable, meaning that they make predictions that can be either be proven right or wrong. For example, Flemings could have spent several weeks growing additional mold only to find that his

hypothesis was wrong and that the mold growth was only a coincidence and that something else was killing the bacteria. Luckily that was not the case as penicillin has saved hundreds of millions of lives over the last 80 years since its discovery. For an example of a scientific theory consider evolution. Evolution is an example of a scientific theory because it is supported by tons of scientific evidence from many fields of science including biology, chemistry, geology,

medicine, and more; that is it is well established. Furthermore, evolution can be used to make predictions (about antibiotic resistant bacteria for example) that can then be observed to be either true or untrue allowing the theory to be modified and refined. In the case of bacteria growing

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resistant to antibiotics Fleming himself warned as early as 1945 in a New York Times interview that misuse of penicillin could result in the development of resistant strains. Unfortunately, society did not heed his warning as over use of antibiotics has caused bacteria to become resistant to many different antibiotics, not just penicillin.

After theory there is one more vocabulary word in the scientific process that often causes confusion, and that is the term law. Students often incorrectly believe that a law is some kind of super theory, or that a theory can be promoted to a law. This is incorrect; theories are already well established, there is no promoting to be done. In science, a law is a statement that

summarizes reoccurring facts or observations in nature. Unlike a hypothesis or theory, laws do not try to explain why something happens, but simply state what happens. Scientific laws can be either equations like the ideal gas law (PV=nRT) which you may have learned in high school chemistry, or statements of fact, like the law of conservation of energy which states, “energy is neither created or destroyed.” Like a speed limit sign, no attempt is made to explain why, the rule is simply stated.

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Measurement, Significant Figures, Conversions, and Density

In this class we will make use of several basic mathematics concepts. The purpose of this section is to provide a short review of those concepts. If after completing this section you are still having trouble, ask your instructor for help as quickly as possible so you don’t fall behind. Remember basic mathematics is prerequisite for this course.

Scientific Notation

Oftentimes scientists work with very large numbers, like 300,000,000 m/s (speed of light) or 0.00000000000000000000000000167 kg (the mass of a proton). Because all of the zeros make these numbers cumbersome to write, scientists use a special notation known as scientific notation. In this notation the non-zero numbers are written and the zeros are condensed as the number ten raised to a power equal to the number of places the decimal point needs to be moved to put the decimal between the first and second non-zero number. For small numbers less than one the ten is raised to a negative power, and for large numbers the ten is raised to a positive number. To clarify here are a few examples. Look carefully at how the sign of the exponent and the direction the decimal moves are related.

123,000 1.23 x 10

0.000123  1.23 x 10

When dealing with numbers in scientific notation, scientists refer to the number part as the coefficient, and the times ten part as the exponent as shown below:

6.02 x 10

Coefficient Exponent Decimal

moved five places to the left

- Large number -positive exponent Decimal

between 1^st and 2^nd number

-Small number -negative exponent Decimal

moved four places to the right

Decimal between 1^st and 2^nd number

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We will be working with scientific notation a lot in this class. Fill in the table below and check your answers off the key before proceeding further. For these problems do NOT worry about significant figures (sig figs). If you don’t know what sig figs are don’t worry, you will learn them in a little while.

Scientific Notation Practice 1 – Complete the grey parts of the following table by converting the numbers given into either regular or scientific notation as appropriate.

Regular Notation Scientific Notation

57,800,000

5.44 x 10

3 x 10

14,600,000,000

1300

1.89 x 10

2.22 x 10

0.000005703

0.000000201

Scientists frequently perform calculations using scientific notation. Of course in modern times scientists use calculators to do their math for them, however it is still important to be able to perform basic mathematics with numbers in scientific notation without using a calculator, so that if you make a mistake you will be able to catch it. Also, it is often faster to do things in your head instead of using a calculator. In this class we will limit ourselves to addition/subtraction and

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multiplication/division with scientific notation. For right now we will not worry about significant figures (if you don’t know what these are don’t worry you will learn soon) and instead concern ourselves with just the basic mathematics Significant figures will be covered in a later section.

Math with Scientific Notation – Exponent Manipulation

When working with numbers in scientific notation it is sometimes necessary to adjust the coefficient and exponent. One reason that this may be the case is to put numbers back into scientific notation at the end of a calculation. For example, consider the following number which is NOT written in proper scientific notation:

12.04 x 10

This is not proper notation because the decimal is in the wrong place. In proper scientific

notation the decimal is always put between the first and second numbers (in this case the one and the two), whereas in the number 12.04 x 10

the decimal is between the two and the zero.

Therefore, to fix this we must move the decimal one place to the left, turning the coefficient from 12.04 to 1.204. This has the effect of making the coefficient ten times smaller. To compensate for this, we must make the exponent ten times bigger; in this case increase it from 10

to 10

.

12.04 x 10

 1.204 x 10

Alternatively, if the coefficient is less than one, we will have to make the exponent smaller to compensate. For example, in the number 0.005364 we will need to move the decimal three places to the right. Therefore, we will have to make the exponent smaller by three:

0.005364 x 10

= 5.364 x 10

(negative six is smaller than negative three)

Lastly there are times when it is necessary to adjust the exponent, not to put it into proper scientific notation, but simply to change it in some desired way. The process is exactly the same

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as described above. This is generally done as an intermediate step during some calculations, it is not how final answers to calculations should be reported. For example, the number 1.2345 x 10

can also be expressed…

1.2345 x 10

= 12.345 x 10

= 123.45 x 10

= 1234.5 x 10

= 12345 x 10

You simply adjust the exponent and coefficient as necessary for the problem at hand. Do the two sets of practice problems on the next page before proceeding further.

Scientific Notation Practice 2 –Fill in the grey parts of the table below by putting the following numbers into correct scientific notation.

Original Number Proper Scientific Notation

0.12585 x 10

0.0012585 x 10

0.0045815 x 10

12.04 x 10

662.6 x 10

478.2 x 10

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Scientific Notation Practice 3 –Fill in the grey boxes in the following table by adjusting the exponents as indicated. The first one has been done for you as an example.

Original Number Adjustment Adjusted Number

1.5782 x 10

Change exponent to 10

0.015782 x 10

1.023 x 10

Change exponent to 10

9.9872 x 10

Change exponent to 10

8.3251 x 10

Change exponent to 10

2.248 x 10

Change exponent to 10

Math with Scientific Notation – Multiplication and Division

Multiplication and division with scientific notation is easy. To multiply you simply multiply the coefficients and add the exponents. For division you divide the exponents and subtract the exponents. For example:

2.45 x 10

x 3.11 x 10

= 2.45 x 3.11 x 10

= 7.62 x 10

8.75 x 10

/ 2.25 x 10

= 8.75/2.25 x 10

= 3.89 x 10

When dealing with negative exponents be careful as subtracting a negative number is the same as adding a positive one. For example:

9.51 x 10

/ 3.77 x 10

= 2.52 x 10

= 2.52 x 10

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Sometimes after multiplying or dividing the coefficient will either be larger than ten or less than one. When this happens it is necessary to put the number back into proper scientific notation by adjusting the exponent. For example, in the problem below the coefficient ends up being 13.4, which or course is greater than ten:

2.22 x 10

x 6.02 x 10

= 2.22 x 6.02 x 10

= 13.4 x 10

Which can be put into proper scientific notation by adjusting the exponent and coefficient giving:

13.4 x 10

 1.34 x 10

Before continuing onto addition and subtraction with scientific notation, do the practice problems below.

Scientific Notation Practice - 4 -Fill in the grey half of the following table by multiplying or dividing the numbers as indicated and give your answer in proper scientific notation. Use the grey space below, and on the next page if necessary, to show your work.

Numbers Answer in Proper Scientific Notation

4.254 x 10

x 2.355 x 10

9.987 x 10

x 2.244 x 10

7.789 x 10

/ 1.689 x 10

8.845 x 10

/ 9.877 x 10

0.04582 x 10

x 4.854 x 10

0.2385 x 10

/ 0.6895 x 10

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Scientific Notation Practice 1 – work cont.

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Math with Scientific Notation – Addition and Subtraction

When adding and subtracting numbers in scientific notation the exponents must be made the same. This has the effect of lining up the decimal mathematically. After the exponents have been made the same the coefficients are simply added, the exponents are left unchanged. Lastly, adjust the decimal to put the number back in scientific notation when necessary.

For example, consider adding the numbers 1234.5 and 678.9. When performed with regular notation we get:

1234.5 +678.9 1913.4

However, when the numbers are in scientific notation lining up the decimal is not as easy.

Consider the same numbers as before written in scientific notation:

1.2345 x 10

+6.789 x 10

Here the decimals may be physically lined up, but they are NOT mathematically lined up because the exponents are different. To mathematically line up the decimals we must make the exponents the same. Then we can add the coefficients to get the answer. For example, if we make both of the exponents 10

our coefficients add to 1.9134 which combined with our exponent of 10

gives our answer in scientific notation.

1.2345 x 10

 1.2345 x 10

+6.789 x 10

 +0.6789 x 10

1.9134 x 10

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Alternatively, we could have made both the exponents 10

. Doing this would have given us 19.134 as our coefficient and 19.134 x10

as our answer, which we must then put into correct scientific notation.

1.2345 x 10

 12.345 x 10

+6.789 x 10

 +6.789 x 10

19.134 x 10

 1.9134 x 10

Before proceeding further perform the practice problems that follow in the space provided. Then check your answer off the key before proceeding.

Scientific Notation Practice - 5 - Adjust exponents to add or subtract the following numbers. For these problems do not worry about significant figures. Use the grey area to show your work.

8.1456 x 10

- 2.55 x 10

7.4562 x 10

+ 8.845521 x 10

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1.2587 x 10

+ 4.452 x 10

5.1250 x 10

+ 4.22 x 10

1.6789 x 10

- 1.12 x 10

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Measurement Quality and Significant Figures

In this section we will learn about significant digits. Significant digits, sometimes referred to as significant figures, or sig figs for short, are a way to telling the reader about the quality of a measurement. The quality of a measurement is just as important as the value of the measurement itself; and we use sig figs to communicate that information to the reader.

When reading this section, try not to think of your everyday usage of the word significant, which relates to how meaningful or important something is. Here the word significant is best thought of meaning “actually measured.” Consider the number one hundred and fifty (150) for example.

What this number means depends upon context, consider the following situations.

Situation one, I see a person and guestimate that they weigh 150 pounds.

Situation two, I watch a person at the gym get onto a scale which reads 150. pounds.

Clearly these two situation are not the same, in the first situation the one hundred and fifty is only a rough measure of the person’s weight whereas in the second situation a scale measured the person’s weight and found it to be (to the nearest pound) one hundred and fifty pounds.

Significant figures give scientists a way to let the reader know the quality of a measurement.

If you look at situation one above you will notice that I wrote that the person weighed 150 pounds (no decimal), but for situation two I wrote 150. with a decimal after the zero. This was not a typo, the decimal is to let the reader know that in situation two the zero in the one hundred and fifty was actually measured (that is the scale read 150.) whereas with situation one the zero was not measured, as I only estimated the person’s weight to the nearest ten pounds, therefore no decimal was used. This is the essence of sig figs, they let the reader know how carefully a

measurement was obtained, we will now consider the details.

When learning about sig figs students often have difficulty because they are not used to numbers representing actual measurements in the real world. Up until this point in your studies, math was

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done in math class and in math class the numbers you are given are just that, numbers. You would take the number that you were given and then after doing some algebra you would put the numbers into your calculator and whatever the calculator spits out was the “correct” answer.

Furthermore, the more decimals you wrote down from your calculator screen the more

“accurate” your answer was. You then would write down however many decimals your teacher asked you to write down. For the following example we will divide our number one hundred and fifty by 6.92 (an arbitrary number). Let’s assume some hypothetical math teacher wants all problems to be rounded to two decimal places, then in this case solving the following problem for x gives:

150 = 6.92(x) x = 150/6.92

x = 21.67630058  21.68 (rounded to two decimals)

Which because your teacher asked for two decimal places, you would report your answer as 21.68. This is all well and good in a math class where all the numbers are just numbers.

However, the main reason our math teacher asked all students to report their answers to two decimal places was so that everyone’s papers would have two decimals to make grading easier.

It would have been just as mathematically valid to ask for one or three decimals because the numbers are just numbers, they don’t actually represent anything real. In chemistry this is not the case. The numbers you will encounter in this book are meant to represent actual measurements, obtained by actual people, using actual instruments. This means that in our class the correct answer to the problem above would depend upon how carefully the measurements were obtained, and this is where sig figs come in. For example:

In the case of situation one, the one hundred and fifty was only a rough estimate (150, two sig figs) thus for the problem above we would get:

150 = 6.92(x) x = 150/6.92

x = 21.67630058  22 (rounded to two sig figs)

Whereas in situation two the one hundred and fifty was carefully measured to the nearest pound (150. three sig figs) we would get

150. = 6.92(x) x = 150/6.92

x = 21.67630058  22.7 (rounded to three sig figs)

Notice how similar the two examples above are, the only difference is 150 vs 150. and yet this tiny difference in notation results in a different rounding of the final answer. Details matter. The number 150. is ten times as accurately measured as the number 150 and therefore the answer is reported to one more decimal place because it is based upon higher quality measurements. In the section that follows we will learn how to tell how many sig figs a number has based upon how it is written. Then we will learn how to apply the concept of sig figs to addition, subtraction, multiplication, and division.

Identification of Significant Figures

To determine the number of significant figures (sig figs) in a number chemists apply a set of rules. When working with numbers in scientific notation, it is only the coefficient that impacts sig figs, the exponent does not impact sig figs. Below are the rules for assigning sig figs along with several examples.

1. All non-zero numbers are significant.

a. 1234.5 = 5 sig figs b. 645.332 = 6 sig figs c. 6.626 x 10

= 4 sig figs 2. Trapped zeros are significant

a. 305 = 3 sig figs b. 7004.2 = 5 sig figs c. 6.02 x 10

= 3 sig figs

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3. Zeros to the right of a decimal, and after numbers, are significant a. 123.4 = 4 sig figs

b. 123.40 = 5 sig figs c. 123.400 = 6 sig figs d. 123.4000 = 7 sig figs e. 6.60080 = 6 sig figs f. 5.4 x 10

= 2 sig figs g. 5.40 x 10

= 3 sig figs h. 5.400 x 10

= 4 sig figs

4. Zeros to the right of the decimal and before numbers are not significant a. 0.0123 = 3 sig figs

b. 0.00123 = 3 sig figs c. 0.000123 = 3 sig figs d. 0.0001230 = 4 sig figs e. 0.01230 = 4 sig figs

f. 0.012030 = 5 sig figs (zero between the 2 and 3 is “trapped” and thus significant) 5. Zeros to the left of the decimal and after numbers are confusing. To tell the reader that a

zero is significant put a decimal after the zero a. 150 = 2 sig figs

b. 150. = 3 sig figs c. 150.0 = 4 sig figs d. 150.00 = 5 sig figs e. 20 = 1 sig fig f. 20. = 2 sig figs g. 20.0 = 3 sig figs h. 20.00 = 4 sig figs i. 3500 = 2 sig figs j. 3500. = 4 sig figs k. 3500.0 = 5 sig figs

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6. Small counted quantities are exact – they do not impact sig figs a. A chemistry class has 24 students

b. A hand has five fingers

7. Very large counted quantities are estimates, they do impact sig figs. For example:

a. Populations of animals in field studies

i. American Crocodile population = 15,000 = 2 sig figs ii. Florida Panther Population = 230 = 2 sig figs

b. Large human population measurements

i. There were 3500 cyclists at critical mass = 2 sig figs ii. The population of India is 1.324 billion people = 4 sig figs

Occasionally there are times when the only way to write a number with a given number of sig figs is to use scientific notation. For example, look at rule five examples e-g. To express the number three thousand five hundred to three sig figs, I would have to write 3.50 x 10

.

Sig Figs Practice 1 – Complete the grey parts of the following table by determining how many sig figs each of the following numbers has.

Number Significant Figures

62.4

0.00536010 1200 1404.0 110.0 1200.

6.6710 x 10

5.60 x 10

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The notation used to communicate a number has nothing to do with how carefully the measurement was obtained. As a result, when converting a number into our out of scientific notation, the number of sig figs should not change.

Sig Figs Practice 2 – Fill in the grey boxes in the following table by determining the number of significant digits in each number and converting the numbers into our out of scientific notation as appropriate. Be sure that the number of sig figs does not change when you switch notations.

The first row has been done for you as an example

Number Sig Figs Regular Notation Scientific Notation

2 71000 7.1 x 10

71000.

1.88 x 10

1.880 x 10

1.8800 x 10

1880.00

0.009500 0.00950 0.0095

Oftentimes it will be necessary to round a number to a specified number of sig figs. This can be stated in a number of ways. For example, to ask someone to round the number 137.8 to two sig figs is the same as asking that person to round 137.8 to the tens place. In both cases the correct answer is 140 with no decimal after the zero.

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Sig Figs Practice 3 – fill in the grey boxes in the following table by rounding the given number as asked. The first couple rows have been done for you as an example

Given Number Rounding Instruction Rounded Number

6588.229 To the hundredth place 6588.23

6588.229 To two sig figs 6600

45.02678 To the hundredth place 0.0045602 to four sig figs

153.45 to two sig figs 6.626 x 10

to three sig figs 45.02678 to five sig figs 5837.2 to one sig fig

5837.2 to the hundreds place

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Math with Significant Figures – Multiplication and Division

A chain is only as strong as its weakest link. Likewise, the result of a calculation is only as accurate as the least accurate measurement. In order to ensure that they do not report answers to a level of precision that their measurements can’t back up, scientists must keep track of

significant figures during calculations. Every mathematical operation has its own rules for

keeping track of significant figures. In this class we will concern ourselves with two, the rules for addition/subtraction and the rules for multiplication/division. The rules for multiplication and division are easier, so we will start with those rules first.

The results of calculations performed using significant figures are often very confusing to students. For example, in our course the answer to the calculation 2 x 12 = 20, not 24! It appears that all the rules of math you have learned just went right out the window. Don’t worry, they have not. Sig figs do not change the rules of mathematics, rather they tell scientists to what value they can round their answers such that there can be no doubt regarding the precision with which the answer is stated. To help you understand why 2 x 12 = 20 and not 24, read the section below carefully.

When multiplying or dividing two numbers, the answer may not have more significant figures that the measurement with the least number of sig figs. To understand why this is the case, consider the rectangle shown below (I have used inches because students are familiar with inches however the box below is not to scale) that has been measured to be two in wide and 12 in long.

If this was a math class and we wanted to calculate the area of this rectangle we would simply multiply the length and width of the rectangle to get A = L x W = 2 in x 12 in = 24 in

and we would be all done; unfortunately this is not math class. In a math class the goal is for students to learn how to calculate areas using formulas from geometry. The numbers used in a math class,

Area = 20 in² (not 24 - keep reading)

Length = 12 in (one sig fig) Width = 2 in

(one sig fig)

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are simply numbers, where they come from is unimportant. In chemistry (and other branches of science and engineering) however the numbers represent actual measurements of real world objects and our goal is for students to understand how the accuracy/precision of measurements impacts the accuracy/precision of the calculations based on those measurements. Because no measurement is perfect we must consider the quality of the measurements that go into our calculations, or we risk stating answers that our data is not good enough to back up. In this case the length has been measured to two significant figures (12) but the length has only been measured to one sig fig (2), therefore we only have good enough data to express our answer to one sig fig (in this case the nearest tens), therefore 24 rounds to 20 with no decimal point.

The number two (2) is not a very accurate measure of the width of our box. If we wanted to get a better measure of the area of our rectangle we would need to improve the quality of our

measurements. Let’s suppose that the width of our box was measured again using a better ruler giving a width of 2.0 in. Now both our length and width have two sig figs so we can now report our area to two sig figs.

A = L x W = 2.0 in x 12 in = 24 in

(two sig figs)

Now let us suppose that the dimensions of our box were measured again, this time very very carefully to four sig figs, giving a width of 2.049 in and a length of 12.04 in. Because our prior measurements were not as careful we only measured two and twelve, but now that we know the dimensions much more accurately, we can also know the area much more accurately. Using these dimensions, we can calculate our area to four sig figs giving:

A = L x W = 1.965 in x 12.04 in = 23.6586 in

 23.66 in

(4 sig figs)

Length = 12 in (two sig figs) Width = 2.0 in

(two sig figs)

Area = 23.66 in² (four sig figs)

Length = 12.04 in (four sig figs) Width = 1.965 in

(four sig figs)

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Take a look at the three area examples given so far to help understand why sig figs matter. Our first set of measurements (2 in and 12 in) were not very accurate being only to the nearest inch.

As a result, we could only report the area as 20 because our length measurement was only one sig fig. In the second example the width of the box was more carefully measured to the nearest tenth on an inch, this allowed us to determine the area more accurately to two sig figs (24 in

).

Lastly when the box was very carefully measured to the nearest thousandths of an inch we obtained a width of 1.965 in and a length of 12.04 in. Because the number 1.965 rounds to 2 and 12.04 round to 12, when our box was not measured as carefully we simply obtained 2 and 12 as our dimensions leading us to believe the area of the box was simply 24, which because of sig figs we could only report at 20 because the four was uncertain. However, using our very careful measurements showed us that the actual area was of 23.66 in

. The more carefully we obtain a measurement, the more accurate the measurement will be and the closer the area we calculate will be to the “real” area of the box.

Hopefully after reading the proceeding section you have some understanding of why the multiplication and division sig figs rule are the way they are. Now we will consider some example that are more similar to what you will encounter throughout this class.

Some multiplication examples

0.0023 x 786.2 = 1.80826 = 1.8

0.00230 x 786.2 = 1.80826 = 1.81

4.53120 x 10

x 2.24 x 10

= 1.0149888 x10

= 1.01 x 10

2 s.f. 4 s.f. 2 s.f.

3 s.f. 4 s.f. 3 s.f.

6 s.f. 3 s.f. 3 s.f.

38

And some division examples:

72300 / 45.892 = 1575.437 = 1580

7.230 x 10

/ 45.892 = 1575.437 = 1575

8.4526 x 10

/ 2.03 x 10

= 4.16384 x 10

= 4.16 x 10

5 s.f.

3 s.f. 3 s.f.

4 s.f. 5 s.f. 4 s.f.

5 s.f. 3 s.f. 3 s.f.

39

Practice Math with Sig Figs 1 – Perform the following multiplication and division with proper significant figures. Use the grey space to show your work.

64.2 x 0.08206

64.2 / 0.08206

8.314 x 22.4

101.3 x 70.

101.3 x 70

(8.16626 x 10

) x ( 1.21 x 10

)

(8.7456 x 10

) / (1.23 x 10

)

40

X

Math with Significant Figures – Addition and Subtraction

Let us now learn about how to add and subtract using significant figures. The rule for addition and subtraction with sig figs states that the answer may only have as many decimal places as the number with the least number of decimal places. To understand why this is the case consider weighing a cat. Because a cat won’t sit on a scale by command, weighing a cat must be done by difference as shown below:

- =

15.1 pound cat

Let us look at the significant figures in this problem. The weight of the person plus the cat (213.7) has four sig figs and the weight of the person alone also has four sig figs (198.6), however when we subtract we find that we only have three sig figs (15.1) for the weight of our cat. The extra sig figs from the weight of the person subtracted away leaving the weight of our cat to the nearest tenth of a pound with only three sig figs. This is only a silly example of course, but if we wanted to use the weight of the cat in a calculation, we would only have three sig figs to work with.

When performing addition or subtraction with sig figs it is necessary to line up the decimal place. With our cat example that would simply be

213.7 - 198.6

= 15.1

Things become more interesting however when not all of the numbers have the same number of decimal places. Let’s consider our cat example again except this time imagine that the weight of our person alone was measured using a cheaper scale that only reads to the nearest pound. Now because we only know the weight of our person to the nearest pound we can only know the weight of our cat to the nearest pound.

213.7 lbs 198.6 lbs

42

- =

??? lbs cat

To show our work for this situation we once again line up the decimal. Then we place an

imaginary rounding line (dotted red line) after the first decimal place for which we do not have a significant figure. We then round our answer accordingly, in this case we round the weight of our cat to the nearest pound.

213.7 lbs. person + cat - known to nearest tenth of a pound) - 199.? lbs. person - known to nearest pound

= 14.7 lbs. cat - can only know to nearest pound

Let’s look at a couple more examples with larger numbers

1200 - known to nearest hundred (no decimal, zeros not significant) + 456.3 - known to nearest tenth

= 1656.3 - can only know to nearest hundred

Compare the example above with the example below

1200. - known to nearest ones (decimal point means zeros are significant) + 456.3 - known to nearest tenth

= 1656.3 - can only know to nearest ones

213.7 lbs  (good scale)

199 lbs   (cheap scale)

15 lbs. rounded to nearest pound

1700 - rounded to nearest hundred

1656 - rounded to nearest ones

And one more example

1200.0 - known to nearest tenth (zero after decimal is significant) + 456.3 - known to nearest tenth

= 1656.3 - can only know to nearest tenth

When working with scientific notation we must first adjust the exponents to make them the same. We then add the coefficients together and round them for sigfigs just like the previous three examples. Consider subtracting the following numbers

8.7498 x 10

- 2.1523 x 10

First we must adjust the exponents to make them the same. It does not matter if you make both exponents 10

or 10

. We will consider both ways, let’s do 10

first.

8.7498 x 10

- 0.021523 x 10

Now we line up the decimal places, put in our rounding line, and subtract

8.7498 x 10

-0.021523 x 10

8.728277 x 10

And then round accordingly

8.7283 x 10

1656.3 - no rounding needed in this case

Let’s look at what would have happened had we made both exponents 10

instead

874.98 x 10

- 2.1523 x10

Now we line up the decimal places, put in our rounding line, and subtract.

874.98 x 10

- 2.1523 x 10

872.8277 x 10

We then round our answer accordingly and put the number back into proper scientific notation giving the same answer as when we made both exponents 10

initially.

872.8277 x 10

 872.83 x 10

 8.7283 x 10

Practice Math with Sig Figs 2 – Perform the following addition and subtraction with proper significant figures. Use the grey space to show your work.

62.4 + 8.314

0.08206 + 22.4

45

46

Practice Math with Sig Figs 2 – Cont.

0.08206 - 22.4

150 + 1.611 + 3.11

150. + 1.611 + 3.11

7.3246 x 10

– 4.1 x 10

47

Practice Math with Sig Figs 2 – Cont.

8.8216 x 10

+ 5.51 x 10

6.6724 x 10

+ 1.6 x 10

8.16626 x 10

– 1.21 x 10

Math with Significant Figures – Mixed Operations

Multiplication and division or addition and subtraction rarely occur alone in the complex calculations involved in the sciences, thus it is necessary to know how to perform calculations involving a mixture of operations. To perform these calculations, one follows order of operations and keeps note of the sig figs in the result of each intermediate calculation; however, one does not round for sig figs until the very end of the calculation. Never round partway through a calculation because doing so can result in building up of rounding errors causing them to get larger with each calculation. Let’s consider a few different examples.

780.74 – (5752.7 / 2.21)

Here we have a five sigfigs number from which the quotient of a five sig figs and a three sig fig number is subtracted. We do parentheses first according to order of operations. Using a

calculator gives

780.74 – 2603.031674

Notice in the result obtained above I underlined the first three significant figures in the quotient.

The remaining digits are not significant, but we will keep them until the end to minimize rounding errors. Next we set up our result as a subtraction problem and place our rounding line.

In this case the rounding line goes between the ones and tens place.

780.74 - 2603.031674 - 1822.291674

Then rounding accordingly gives the correct answer below.

- 1820

Now let’s consider 780.74 – (5752.7 / 2.210). The numerical result will be the same, however the final answer will not round to be the same because our denominator now has more sig figs

because the (in this case hypothetical) measurement behind this number was performed more carefully and accurately. Now if we follow sig figs we get:

780.74 – (5752.7 / 2.210)  780.74 – 2603.031674

Then subtraction gives

780.74 - 2603.031674 - 1822.291674

Which then rounds to - 1822

Let’s consider a second example

74922 x [2425 – (2.756 x 140.)]

First we multiply the numbers in parenthesis, noting that 140. has three sig figs due to the

decimal point after the zero. Underlining the significant figures in our product as before we keep the extra digits to minimize rounding errors.

74922 x [2425 – 385.84]

Then we subtract the numbers in brackets according to sig figs

2425 -385.84

2039.16  2039.16 (sig figs underlined, decimals kept to minimize rounding error)

Now we are left with the multiplication of a number with five sig figs (74922) and a number with four sig figs (2039.16)

74922 x 2039.16

Using a calculator and rounding to four sig figs gives our final answer

1.527779455 x 10

 1.528 x 10

Practice Math with Sig Figs 3 – Perform the following mixed calculation problems with proper sig figs. Use the grey space to show your work.

420.6 + (12.64/0.0041)

426.42 – (4725/4.1)

53

54

Practice Math with Sig Figs 3 – Cont.

(3.568 x 10

– 4.231 x 10

) x 3.5682

(0.0044 x 52426.2) + (3345 x 70)

685 x [5845 - (9.62 x 170.)]

Measurement, Conversions, and the Factor Label Method (FLM)

The scientific method starts with observations. Many observations take the form of numerical measurements. For example, using a speedometer we can measure how fast a car is driving. All measurements require the use of units. In the case of a speedometer the units we would

commonly use in the United States are miles per hour; or in any other country in the world, kilometers (km) per hour. Because units are so important in science, several systems of units have been developed over the years. Unfortunately, there is no one single agreed upon set of units, therefore we must learn how to use multiple systems of units and how to convert measurements from one system to another. In the sections that follow we will learn how to convert measurements from one unit to another as well as reviewing the systems of units we will be using in this course.

Scientists and engineers use relationships between units (known as conversion factors) to convert measurements from one unit of measure to another. To do this, scientists use a method known as the Factor Label Method, or FLM for short. Some books may refer to this as dimensional

analysis or the line mole method, these are just more names for the same method. Whatever you want to call it we will be using this method throughout the entire semester. You MUST learn this method if you wish to study any field of science or engineering. If you have “another way” of doing conversions, great… you can use it to check the answer you get from using the FLM!

Let us consider an example. Below is a picture (obtained from Wikipedia) of Cream Puff, the oldest cat ever to live.

57

Amazingly Cream Puff lived to be 38 years old! Let’s find out how many seconds that is. If you happen to know how many seconds are in a year you could simply multiply that number by 38 to get the answer. However, the number of seconds in a year is not a number that is commonly used. Instead we will use conversion factors to build a bridge between our known (38 years) and our unknown (number of seconds). In this case we will use the following relationships:

1 yr = 365 days 1 day = 24 hr 1 hr = 60 min 1 min = 60 sec

By putting these conversions together, we can use relationships we already know to solve this problem. To begin our calculation we set up a fraction with our known (38 years) on top and a one (usually implied) on the bottom.

� 38 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦

1 �

Next we use our conversion factors such that the unit on top is canceled by the same unit on the bottom, the one is understood.

� 38 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦

� � 365 𝑑𝑑𝑦𝑦𝑦𝑦𝑦𝑦

1 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 � = 13870 𝑑𝑑𝑦𝑦𝑦𝑦𝑦𝑦

If we were to stop at this point we would obtain the cats age in days: 38 x 365 = 13870 days (not rounded for sig figs). However, since we are interested in seconds, we continue to use conversion factors, canceling along the way, until the desired unit(s) is/are all that is left.

� 38 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦

� � 365 𝑑𝑑𝑦𝑦𝑦𝑦𝑦𝑦 1 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 � �

24 ℎ𝑜𝑜𝑜𝑜𝑦𝑦𝑦𝑦 1 𝑑𝑑𝑦𝑦𝑦𝑦 � �

60 𝑚𝑚𝑚𝑚𝑚𝑚𝑜𝑜𝑚𝑚𝑦𝑦𝑦𝑦 1 ℎ𝑜𝑜𝑜𝑜𝑦𝑦 � �

60 𝑦𝑦𝑦𝑦𝑠𝑠𝑜𝑜𝑚𝑚𝑑𝑑𝑦𝑦

1 𝑚𝑚𝑚𝑚𝑚𝑚𝑜𝑜𝑚𝑚𝑦𝑦 � = 1,198,368,000 𝑦𝑦𝑦𝑦𝑠𝑠𝑜𝑜𝑚𝑚𝑑𝑑𝑦𝑦 (not rounded for sig figs yet)

58

Telling someone a cat is 38 years old is not very specific. The cat could have just turned 38, or it could be 38 years and 364 days old. Without more information we simply cannot state the age of the cat very accurately in seconds given that we barely know the cat’s age in years. Therefore, we must round for significant figures. In rounding conversions for significant figures it is the number of significant figures in the original measurement (38 years, two sig figs) that determines how many sig figs we can express in our answer. The conversion factors used are exact by definition. That is there are exactly 365 days in one year (not worrying about leap years), and there are exactly 24 hours in one day. Because these conversions are exact they do NOT impact sig figs. Rounding our answer to two sig figs gives:

1,198,368,000 seconds  1,200,000,000 seconds or 1.2 x 10

seconds

In other words, because we only know the age of the cat to the nearest year, we can only state the age of the cat to the nearest hundred million seconds.

While the age of the cat example is simple enough that many students can perform the

calculations without using the factor label method, it is imperative that you learn this technique.

The calculations later in the course will not be so simple and they will not involve units that you

are familiar with such as years, days, hours, minutes, and seconds.

Standard Units (also known as English or Imperial Units)

The standard unit system is the system of units used in the United States. The US is the only country in the world that still uses these units; despite this fact they are still known as “standard”

units. The table below summarizes the commonly encountered standard units used in the United States.

Units of Volume Units of Mass Units of Length

Gallon (gal) Pound (lbs) Mile (mi)

Quart (qt) Ounce (oz) (a.k.a. ounce mass) Foot (ft)

Pint (pt) Inch (in)

Cup Yard (yd)

Ounce (oz) (a.k.a. fluid ounce) Teaspoon (tsp)

Tablespoon (tbsp)

The relationships between these units are as follows

Volume Conversions

Mass Conversions

Length Conversions 1 gal = 4 qts 1 pound = 16 oz 1 mi = 5280 ft

1 qt = 2 pts 1 ft = 12 in

1 pt = 2 cups 1 yd = 3 ft

1 cup = 8 oz 1 tbsp = 3 tsp

The process of converting between these units is exactly the same method we used to calculate Cream Puff’s age. For an example, prof Mailloux rides his bike to work seven miles each way.

How many inches is that?

�

� �

� �

� = 443520 𝑚𝑚𝑚𝑚 = 440,000 𝑚𝑚𝑚𝑚 (𝑦𝑦𝑜𝑜𝑜𝑜𝑚𝑚𝑑𝑑𝑦𝑦𝑑𝑑 𝑚𝑚𝑜𝑜 𝑚𝑚𝑡𝑡𝑜𝑜 𝑦𝑦𝑚𝑚𝑠𝑠 𝑓𝑓𝑚𝑚𝑠𝑠𝑦𝑦)

60

That is a lot of inches, which is reasonable given how small an inch is and how far seven miles is in comparison. It is important that you check that your answer is reasonable. This is easy to do using units you are familiar with like feet and inches; however, doing so with units you are less familiar with will take some getting used to.

A word on notation. To save time when writing out the factor label method on paper some instructors (myself included) use a slightly different notation to write their fractions when doing their calculations on paper. Using this notation, the above calculation would be written as:

= 443520 in = 440,000 in (2 sig figs)

You may use either notation you wish in this course. Because the fraction style notation is easier to type on a computer it is the one that will be used in this book, however when looking at the handwritten parts of the answer key I will use the second notation.

For a second example lets calculate how many ounces are in a 12.0 gallon tank of gas.

.

1536 1540 (3 sig figs) 7.0 mi 5280 ft 12 in

1 mi 1 ft

Conversions Practice 1 -Use the grey space provided below to perform the following standard- standard conversions. Be sure to round your answer to the correct number of sig figs based upon the number of sig figs in the original measurement.

0.557 gallons to _____ ounces

6’4” to ____ yards (treat this as two sig figs)

1990 mi to inches

3.14159 gallons to cups

62

Conversions Practice 1 – cont.

1.2545x10

ounces to quarts

25.0 miles to yards

0.0240 gallons to pints

25.0 miles to yards

0.0240 gallons to pints

64

65

SI Units and the Metric System

Feet and miles may be good enough for everyday usage, but for scientific work a more

systematic and uniform system is necessary. Through international agreement a universal system of measurement known as the International System of Units, or SI units for short, has been created. This system is based upon seven base units that have been designed to work together.

These units are listed below:

Unit Symbol Used to Measure

Meter m Distance

Kilogram kg (not g) Mass Second s (or sec) Time

Kelvin K (not °C) Temperature

Mole mol (or mole) Amount of substance

Ampere A Electric Current (not used in our class) Candela Cd Luminous intensity (not used in our class)

Taken together these units make up what is known as the meter-kilogram-second (MKS) system of units. These units are meant to work together and as a result most equations in chemistry, physics, and engineering require use of these units to obtain the correct answer. These are the fundamental units that underlie all science. You must KNOW these base units.

A meter is about three feet long. Therefore, to measure very small or very large things with meters is not always practical. To adjust for the size appropriate for a given measurement, metric units are modified by prefixes that are based upon powers of ten. There are many of these

prefixes, the most common ones that we will make use of in this class are given in the table below. You will not be given this information on exams, so put it in your brain.

66

Prefix Symbol Meaning mega M 1,000,000 (or 10

) x base unit kilo k 1,000 (or 10

) x base unit deci d 1/10

(or 10

) x base unit centi c 1/100

(or 10

) x base unit milli m 1/1,000

(or 10

) x base unit micro μ or mc 1/1,000,000

(or 10

) x base unit nano n 1/1,000,000,000

(or 10

) x base unit

Converting between measurements in the metric system is easy once you get used to it. You always convert from the first prefix back to the base unit, and then convert to the second prefix:

1

prefix  base unit  2

prefix

For an example let us convert 183 cm to mm:

.

1832

By relating everything back to the base unit we can minimize the amount of material that we need to memorize. If you know that there are 100 cm in a meter and that 1000 mm equals one meter, it is not necessary to know that there are 10 mm in a cm, the FLM takes care of it for you.

Let’s do another example. Professor Mailloux is 193 cm tall. How many kilometers is this?

193 1

100

1

1000 0.00193

Conversions Practice 2 – Perform the conversions below with correct sig figs.

62.4 km to nm

6.02 x 10

mm to dm

8.314 μL to cL

6.626 x 10

ns to cs

0.08206 cg to kg

68

69

Conversions Practice 2 – cont.

0.005066 Mg to kg

454520. mcL to nL  

143.501 ng to mg

0.0001430 mcs to ns  

1.40 x 10

ds to ks  

71

Metric to Standard Conversions

It would be nice if the whole world used only one system of units, unfortunately there is simply too much institutional inertia to allow the United States to easily switch to the metric system. As a result you will need to learn to convert between metric and standard units. To do this you will need to memorize three conversions factors to convert between metric and standard; one for mass, one for volume, and one for distance. Below are the conversions that we will make use of in this class:

Mass Volume Distance

1 pound = 453.59 g 0.946 L = 1 quart 2.54 cm = 1 inch

Like other conversion factors we have learned about thus far, these conversion factors do not impact sig figs either. This is because these conversions are either exact by international agreement (2.54 cm is exactly one inch) or while not exact the conversions could always be looked up in real life to as many digits as needed. For example, a more exact conversion between liters and quarts is 1 quart = 0.946353 L. For our purposes 0.946 is accurate enough, however if a scientist were doing very careful work they would simply look up the conversion to as many decimals as are necessary for the work they are doing. Therefore, in this class we will only look at our original measurement for sig figs, and not at the conversion factors. Below are examples of metric-standard conversions using each of our three conversion factors listed above.

Convert 3.1 miles to km (this is the length of a “5 K race”)

3.10 5280

1

12 1

2.54 1

1 100

1

1000 4.99

Note that the last three steps are simply a metric-metric conversion like the ones you just finished in the prior section.

Consider a couple more examples:

Convert 150 pounds to kg

150 453.59

1

1

1000 68

Convert 18.5 gallons to mL

18.5 4

1

0.946 1

1000

1 70004 7.00 10

Notice that in the last example it was necessary to use scientific notation in order to express our answer to three significant figures.

Conversions Practice 3- Perform the following conversions using the correct number of significant figures. Use the grey space to show your work.

25.4 quarts to mL

5.0 tbsp to mL

85 kg to lbs

74

Conversions Practice 3 – Cont.

1.25 cg to ounces

15 miles to km

3.90 Mm to yards

Fractional Conversions

Sometimes it is necessary to convert two units in a single conversion calculation, for example we

may wish to convert miles per gallon to km per liter. Fractional conversions are no more difficult

than regular conversions. For an example Prof Mailloux’s car gets 32.0 miles per gallon, what

would this be in km per liter? To do this conversion we will need to convert miles to km and

gallons to liters. You may convert the miles first or the gallons first, the choice is up to you. The

calculations below demonstrate how to perform this conversion both ways, to help you see the

two different parts of the conversion, I have colored the text for you:

76

Miles first

� 32.0 𝑚𝑚𝑚𝑚

1 𝑠𝑠𝑦𝑦𝑙𝑙 � � 5280 𝑓𝑓𝑚𝑚

1 𝑚𝑚𝑚𝑚 � � 12 𝑚𝑚𝑚𝑚

1 𝑓𝑓𝑚𝑚 � � 2.54 𝑠𝑠𝑚𝑚

1 𝑚𝑚𝑚𝑚 � � 1 𝑚𝑚

100 𝑠𝑠𝑚𝑚 � � 1 𝑘𝑘𝑚𝑚

1000 𝑚𝑚 � � 1 𝑠𝑠𝑦𝑦𝑙𝑙

4 𝑞𝑞𝑚𝑚𝑦𝑦 � � 1 𝑞𝑞𝑚𝑚

0.946 𝐿𝐿 � = 13.6 𝑘𝑘𝑚𝑚/𝐿𝐿

Gallons first

� 32.0 𝑚𝑚𝑚𝑚

1 𝑠𝑠𝑦𝑦𝑙𝑙 � � 1 𝑠𝑠𝑦𝑦𝑙𝑙

4 𝑞𝑞𝑚𝑚𝑦𝑦 � � 1 𝑞𝑞𝑚𝑚

0.946 𝐿𝐿 � � 5280 𝑓𝑓𝑚𝑚

1 𝑚𝑚𝑚𝑚 � � 12 𝑚𝑚𝑚𝑚

1 𝑓𝑓𝑚𝑚 � � 2.54 𝑠𝑠𝑚𝑚

1 𝑚𝑚𝑚𝑚 � � 1 𝑚𝑚

100 𝑠𝑠𝑚𝑚 � � 1 𝑘𝑘𝑚𝑚

1000 𝑚𝑚 � = 13.6 𝑘𝑘𝑚𝑚/𝐿𝐿

Conversions Practice 4 - Perform the following conversions using the correct number of significant figures. Use the grey space to show your work.

45.1 mi/gal to km/L

0.03200 km/L to in/gallon

55.60 sec/cup to days/L

77

78