Readings for Chemistry 11 First Assignment

Evolution of ScienceLet's look at how the discipline of Chemistry was arrived at and where it is today.

By the time of Aristotle (384-322 B.C.), the human quest for the knowledge of nature was being molded into a disciplined investigation. The study of nature, called natural philosophy, sought knowledge through observation and logic, and the proposal of models and theories. By the age of Newton (1642-1727 A.D.) natural philosophy had evolved into science. In addition to investigation by means of observation, logic and models, science includes experimentation and measurement. During the next two centuries several distinct disciplines developed. Among these scientific sub-disciplines were: Biology (the study of living organisms), Physics (the study of interactions between energy and matter) and Chemistry (the study of properties of matter and the changes which matter undergoes).

Disciplines of ChemistryTraditionally, there are five subdivisions used to classify the various fields of study within Chemistry. These are:

1. Analytical Chemistry - The study of the separation and identification of species of matter, usually quantitatively.

2. Biochemistry - The study of the chemical properties and processes of living matter.

3. Organic Chemistry - The study of carbon and its compounds, many of which are derived from animal and vegetable matter. Other organic compounds are synthesized.

4. Inorganic Chemistry - The study of chemicals that are other than organic.

5. Physical Chemistry - The study of the structure, transformation and physical properties of matter.

 

Analytical Chemistry

A branch of chemistry that deals with the development and use of techniques for chemical measurement. These techniques are used in analyzing the chemical composition of substances. Chemical analysis may be qualitative or quantitative. Qualitative analysis involves attempting to identify what materials are present in a sample. Answering the question: "What is it?". Quantitative analysis involves determining how much of a material is present in a sample. Answering the question: "How much is present?".Modern analytical methods make it possible to identify hundreds of components in a single sample and to detect specific substances present in less than one part per million.  

Biochemistry

Biochemistry is the study of the chemical processes and reactions that take place within living organisms. It can be considered a subdivision of both chemistry and biology, although the skills and techniques used within it place great emphasis on traditional chemistry.

For a very long time, it was thought that living and non-living matter were fundamentally different. It was thought that only living beings could create special biological molecules, from other biological molecules obtained through food. These molecules were thought to be imbued with a “vital force” that made life possible. In 1828, the German chemist Freidrich Wöhler put an end to this by accidentally synthesizing the organic chemical urea — a major component of urine — from inorganic precursors. The field of biochemistry was born.

Since 1828, studies in the field of biochemistry have brought us knowledge of the way plants extract energy from the Sun (photosynthesis), animals convert glucose into the energy currency of the body, ATP (glycolysis), why our muscles burn when we vigorously exercise (the production of lactic acid), how proteins are synthesized in the cell (protein expression), and much more. As living things (plants, animals, humans) tend to be the most useful and important arrangements of matter on Earth from our perspective, knowledge of their inner workings through biochemistry has useful applications in many areas, such as medicine, agriculture, molecular biology, etc.

Some molecules studied by biochemists include carbohydrates, proteins, lipids, and nucleic acids. Most of these are organic polymers, meaning they primarily consist simple molecular patterns (monomers) repeated numerous times in a chain, sometimes thousands. The primary elements found in organic compounds are oxygen, hydrogen, nitrogen, calcium, and phosphorous, with trace amounts of chlorine, sulfur, potassium, sodium, magnesium, iron, and a few others.

Many of the molecules in the body serve in structural roles. These include the carbohydrates and proteins. Proteins are manufactured directly based on genetic instructions, and are among the most complex organic molecules. Nucleic acids are the building blocks of our genetic instructions (DNA and RNA) found in all forms of

life, from humans to viruses. The distinct pattern of nucleic acids found in the nuclei of a species’ cells is called its genome. Then, there are the lipids, the catch-all term for many non-water-soluble biomolecules. The fat in our body is made of lipids.

 

Organic Chemistry 

Organic chemistry is a branch of chemistry that involves the study of organic carbon compounds. It encompasses the structures, composition, and synthesis of carbon-containing compounds. In understanding organic chemistry, it is important to note that all organic molecules consist not only of carbon, but also contain hydrogen. While it is true that organic compounds can contain other elements, the bond between carbon and hydrogen is what makes a compound organic.

Originally, organic chemistry was defined as the study of compounds created by living organisms. However, its definition has been enlarged to include artificially synthesized substances as well. Before 1828, all organic compounds were obtained from living organisms. Scientists didn’t believe it was possible to synthesize organic compounds from inorganic compounds. Many attempted to do so and failed. However, in 1828, urea was synthesized from inorganic substances, paving the way for a new definition of organic chemistry.

There are more than six million known organic compounds. In addition to being plentiful, organic compounds are also unique. This is because carbon atoms have the ability to form strong bonds with many different elements. Carbon atoms are also able to bond covalently to other carbon atoms, while simultaneously forming strong bonds with other nonmetal atoms. When carbon atoms bond together, they can form chains consisting of thousands of atoms. They can also form rings, spheres, and tubes.

Many individuals consider organic chemistry to be very complicated and unrelated to daily life. Though the study of organic chemistry may be complex, it is very important to everyday life. In fact, organic compounds are a part of everything, from the foods we eat to the products we use. They are important in the creation of

clothing, plastics, fibers, medications, insecticides, petroleum-derived chemicals, and a long list of products used to support life and to make it more convenient.

 The study of organic chemistry is important, not only to those who are interested in science-related careers, but to every individual alive today and to those who will be born in the future. Organic chemistry is key in developing new products and improving those on which we’ve become dependent. Each year, organic chemists make discoveries that are helpful in improving medicines, aiding agricultural growth, understanding the human body, and performing countless tasks important to the average person.

 

Inorganic Chemistry

Inorganic is a branch of chemistry that deals with the properties and behavior of inorganic compounds. Inorganic compounds are generally those that are not biological, and characterized by not containing any hydrogen and carbon bonds. It is almost easier to discuss inorganic chemistry in terms of what it is not: organic chemistry. Organic chemistry is the study of any chemical reaction that involves carbon, which is the element that all life is based on. It is often said that inorganic chemistry is any type of chemistry that is not organic chemistry.

 

Physical Chemistry

Literally millions of chemical compounds are in existence. The ways in which these compounds and their constituents react and interact with one another is governed by certain physical principles that explain their behavior. Physical chemistry is therefore the foundation upon which all other fields of chemistry rests, and this science is also relevant to virtually all other scientific fields. Physical chemistry is encompassed by four subject areas, including thermodynamics, quantum chemistry, chemical kinetics, and statistical thermodynamics.

Thermodynamics is the study of the conversion of energy into heat and work. In this context, work is defined as the energy transferred by a force; for example, kicking a ball is a form of work in which the person who kicks the ball transfers force from their foot to the ball, causing the ball to move. Thermodynamics also studies ways in which the conversion process can be altered by changing variables such as pressure and temperature within a system.

Quantum chemistry is a theoretical science which describes how molecules bond to one another by applying principles of quantum field theory and quantum mechanics. These principles describe how atoms and subatomic particles behave in various systems, and in turn govern how molecules behave. Theoretically all chemical systems can be described using quantum chemistry, but in practice only very simple systems can be accurately investigated.

Chemical kinetics studies the rates of chemical processes. The rate of a given chemical process is simply the speed at which a chemical reaction occurs. For example, contrast the rate of iron oxidation, which is a very slow process, with the rate of fuel combustion, which is a split-second process. Chemical kinetics also studies how changing variables such as pressure and temperature change the rate at which reactions occur.

These three aspects of physical chemistry are linked by a fourth, called statistical thermodynamics. This field is concerned with energy distribution in chemical systems, and also links the microscopic and macroscopic worlds. The main goal of statistical thermodynamics is to interpret macroscopic properties of various types of

matter in relation to the interactions between their constituent microscopic molecules and particles.

Through the study of these four concepts, modern physical chemistry seeks to understand complex chemical problems in the context of biological, environmental, and materials sciences. Even though these are widely disparate fields, the principles of physical chemistry are relevant to all, including biological as well as physical and chemical sciences. This is in fact a highly multidisciplinary science, precisely because the chemical principles it studies are relevant to all biological and chemical systems.

 

Interdisciplinary Subdivisions

There are also subdivisions of Chemistry that combine Chemistry with other disciplines such as biology, geology, astronomy and physics. These are interdisciplinary subdivisions and they include:

Descriptive Chemistry - The study of the qualities of matter. Environmental Chemistry - The study of the effects of chemicals on the

ecosphere. Food Chemistry - The study of the chemical composition and effects of

foodstuffs. Nuclear Chemistry - The study of the chemistry of atomic nuclei. Drug Chemistry - The study of the chemical composition and changes of

drugs. Petroleum Chemistry - The study of the chemicals and changes in

petroleum and petroleum products. Radiochemistry - The study of the effects of nuclear radiation on chemical

composition and changes. Astrochemistry - The study of the origin and interactions of chemicals in the

universe, especially interstellar matter

The Scientific Method

The scientific method starts with having a purpose for an experiment - a reason why the experiment is done. The scientific approach is systematic:

The problem is precisely defined. Predictions are made. The experiment is designed. Results are collected. Results are interpreted. Theories or models are proposed.

 

Problem or Question

Ask a Question: The scientific method starts when you ask a question about something that you observe: How, What, When, Who, Which, Why, or Where?

And, in order for the scientific method to answer the question it must be about something that you can measure, preferably with a number.

 

Observations or Research

Do Background Research or make observations: Rather than starting from scratch in putting together a plan for answering your question, you want to be a savvy scientist using library and Internet research to help you find the best way to do things and insure that you don't repeat mistakes from the past.

Sometimes scientists begin with observations of a particular situation and this will lead to the question.

 

Form a Hypothesis

Construct a Hypothesis: A hypothesis is an educated guess about how things work:

"If _____[I do this] _____, then _____[this]_____ will happen."

You must state your hypothesis in a way that you can easily measure, and of course, your hypothesis should be constructed in a way to help you answer your original question. In a way it is prediction of the outcome of your results for your experiment. Here is an example.

According to the concept of acid-base neutralization, a base should remove the soft drink stain because most foods are acidic and bases are known to neutralize acids.

 

Conduct an Experiment

Test Your Hypothesis by Doing an Experiment: Your experiment tests whether your hypothesis is true or false. It is important for your experiment to be a fair test. You conduct a fair test by making sure that you change only one factor at a time while keeping all other conditions the same.

You should also repeat your experiments several times to make sure that the first results weren't just an accident.

 

Analyze Results and Draw Conclusions

Once your experiment is complete, you collect your measurements and analyze them to see if your hypothesis is true or false.

Scientists often find that their hypothesis was false, and in such cases they will construct a new hypothesis starting the entire process of the scientific method over again. Even if they find that their hypothesis was true, they may want to test it again in a new way.

 

Communicate the Results

To complete your lab you will communicate your results to others in a final report. Professional scientists do almost exactly the same thing by publishing their final report in a scientific journal.

 

 

Models and Theories

Establishing relationships among observations is an important part of the scientific method. It enables observations to be organized efficiently and it leads to explanations of why things are as they are. Such scientific explanations are called theories or models.

A scientific model is not like a model train. It is more like a comparison or an analogy. Analogies such as "The car took off like a jack-rabbit," or "She is as thin as a rail" help to describe reality. Such comparisons are not exact. Cars do not have two long ears and a short tail like a jack-rabbit, but the analogy does help us to understand the situation. Scientific models are like analogies. They compare something whose behaviour is not understood, like atoms, to something that is understood, like marbles. Models do not portray reality exactly, but they sharpen perception of it.

A well-established scientific model is often called a theoretical model or just a theory. Some examples are the atomic theory, the theory of evolution, the theory of continental drift, and the "big bang" theory of how the universe evolved. A theory is an approximation that works. By "work" we mean that is is able to combine and explain most related data without too many alterations or gaps.

Predictions and ModificationsOnce a scientist has what she believes to be a good model she makes a prediction on the basis of this model, then she devises another experiment to test her prediction. If, for example, a model for atoms is established which compares them to marbles, two things might be predicted: (1) the faster an atom (marble) travels the more pressure it will exert when it hits the walls of its container, and (2) moving atoms (like marbles) will gradually slow down and stop. Experiments would then be performed to test these predictions, and it would be found that the first one is true and the second one is not. The model would then have to be modified to take into account the second observation. Theoretical models are never rigid but are modified gradually in the light of new knowledge obtained from the experiments designed to test them. Very rarely, a model may be abandoned completely because it fails to explain some crucial observation. More often it is changed to accommodate the new facts. Scientists tend to cling to old theories and do not blithely throw them away when discoveries are made that conflict with them. The scientific method then, involves:

Scientific observation, characterized by: use of all appropriate senses; absence of premature conclusions, use of quantitative statements, attention to conditions of observation, attention to stages of a process (before, during, after), and attention to "missing" traits.

Seeking relationships in observed data or processes.

Proposing models or theories which would explain the observed behaviour and relationships.

Making predictions based on these models and testing these predictions experimentally to see if they agree with results.

Modifying the models or theories on the basis of observed experimental results.

Accelerated progress in the understanding of nature since the 17th century can be largely attributed to the introduction of the scientific method.

SELF CHECK: The Scientific Method Answer the following questions and check your answers below:

1. List the steps of the scientific method in order,2. When actually doing their research, do scientists always follow the scientific method

in a linear (straight-ahead) order?3. In what two sections of the scientific method do we find an answer to the question

asked in the Problem section?

Answers:

1. Problem/Question, Observation/Research, Hypothesis, Experiment, Collect and Analyze results, Conclusion, Communicate the results.

2. No, not necessarily. Scientists may circle back to a previous procedure if they encounter difficulties or anomalies.

3. Collect and analyze results and the Conclusion.

Scientific NotationWatch the following Video: https://youtu.be/JAIHikod-Q0

Large and Small Numbers 

As scientists, we deal with some extremely big and extremely small numbers. For example, we'll be dealing with the mass of the Earth (5 973 700 000 000 000 000 000 000 kg). We'll also be dealing with the mass of an electron (0.000 000 000 000 000 000 000 000 000 000 910 938 kg). 

As you can see, as scientists, we could get really tired of writing zeros.

Fortunately, physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest. We call it scientific notation. 

mass of Earth = 5.973 × 1024 kg

mass of electron = 9.10938 × 10-31 kg

Much better! 

If the number you’re working with is greater than ten, you’ll have a positive exponent in scientific notation; if it’s less than one, you’ll have a negative exponent. As you can see, handling super large or super small numbers with scientific notation is easier than writing them all out, which is why calculators come with this kind of functionality already built in.

Very Small Numbers

In scientific notation all numbers are written in this form:

a × 10b

We'd say ¨a times ten to the power of b.¨

a is any real number between 1 and 10 b is adjusted to represent the magnitude of the number if b is positive, we're talking about a number greater than 10 if b is negative, we're talking about a number less than 1

Generally, a physicist would use scientific notation for numbers that are either:

greater than 10 000orless than 0.001

Warning

Although, there is plenty of flexibility with this rule, you'll find that if you answer a question in this course as: 20 200 000 or 0.0000324, you'll probably by docked marks for not being smarter about your numbers.

Precision, Uncertainty, and AccuracyWatch this video clip: https://youtu.be/-Ue-o_txQAw

Measurements & Uncertainty

There is some uncertainty in any measurement process. This uncertainty is contributed to by the limitations of both the instrument used and the skill of the user. Assuming a measurement is made with skill and care, the uncertainty will be primarily due to the limitations of the instrument.

Measurement in Centimeters

Consider the two scales and the shaded object below. The upper scale is graduated in centimetres. The lower scale is graduated in centimetres and millimetres.

Using the upper scale, it is easy to see the object has a length greater than 2 cm but less than 3 cm. The length appears closer to 3 cm, so estimates of 2.6 cm, 2.7 cm or 2.8 cm would be reasonable. Estimation is a standard procedure in scientific measurement. It is understood that for a measurement recorded as 2.7 cm, the last digit (7) is an estimate. Let's adopt 2.7 cm as the measured value.

When recording measurements, always record one decimal place past the smallest division on the scale, even if the number is a zero. This process ensures that all the certain digits, plus one uncertain or estimated digit is recorded. In this case, the smallest division is 1 cm, so the measurement must be recorded to 0.1 cm.

Precision and AccuracyPrecision and accuracy have very distinct meanings to a scientist. Precision measures the agreement between results of repeated measurements and is usually controlled by the size of the scale of the measuring device. A balance that can read mass to 0.0001 grams should be more precise than one that reads mass to 0.1 grams.

Accuracy, measures the agreement between a measurement and the accepted standard value. If a balance consistently gives a value of 3.64 g on repeated measurements, its precision is good. However, if the actual mass is 3.75 g, its accuracy is poor! A finely calibrated instrument will usually be precise and accurate, assuming no error is introduced by the user.

Precision measures the agreement between results of repeated measurements.

Accuracy, measures the agreement between a measurement and the accepted standard value.

Percent Error

Consider the two graduated cylinders diagrammed below. Since most meniscuses are valleys, this usually means reading to the bottom of the meniscus. Some liquids like mercury, however, have inverted meniscus and are read to the tip of the meniscus. The eye should be at the same level as the meniscus when reading.

The smallest division on the first graduated cylinder is 10 mL, so the volume must be recorded to the nearest 1 mL. The volume is estimated at 288 mL. The 2 and the first 8 are certain digits and the last 8 is estimated and is uncertain.

The scale on the second graduated cylinder is much finer. 0.01 mL is the smallest division. The volume is recorded to the nearest 0.001 mL as 0.190 mL. The final zero must be included to indicate which decimal place has been estimated and is uncertain.

To determine the percentage error in an experimental result, use the following formula:

%error = 

The density of water was found to be 0.96 g/mL in an experiment. The actual value is 1.00 g/mL. The experimental error is:

 = 4% error

Complete this quick quiz to test your understanding of measurement and uncertainty.  You will also need to know how to define accuracy and precision.

http://antoine.frostburg.edu/cgi-bin/senese/tutorials/sigfig/index.cgi

Significant Figures

Tutorial: Significant FiguresWatch the following video: https://youtu.be/MCy3UhM_vq8

Before looking at a few examples, let's summarize the rules for significant figures.

1. Non-zero numbers (1,2,3,4,5,6,7,8,9) are significant. (3.14 -> 3 sig figs)2. Zeroes between non-zero numbers are significant. (300.104 -> 6 sig figs)3. Following zeros to the right of non-zero numbers without a decimal are not

significant.  With a decimal, they become significant. (200 -> 1 sig fig, 200. -> 3 sig figs)

4. Following zeroes which are to the right of the decimal point and following non-zero numbers are significant, even at the end. (7.0300 -> 5 sig figs)

5. A zero used to show a decimal point (ie. a zero by itself on the left side of  a decimal) is not significant.  All zeros between a decimal and the first non-zero numbers are not significant.

Example #1:

How many significant figures in 3000 m? How about 3000. m? Example #2:

How many significant figures in 2.312 x 105 m/s? How about 231 200 m/s?

 

Example #3:

How many significant figures in 2.312 x 10-4 kg? How about 0.00023120 kg? Example #4:

Round off 2142.22 to 2 sig figs. Round off 0.0023421 to 3 sig figs.Answers: 1. 1, 42. 4, 43. 4, 54. 2.1 x 103 (or 2100), 2.34 x 10-3

 

Significant Figure Rules in Detail

There are three rules on determining how many significant figures are in a number:

1. Non-zero digits are always significant.2. Any zeros between two significant digits are significant.3. A final zero or trailing zeros in the decimal portion ONLY are significant.

Focus on these rules and learn them well. They will be used extensively throughout the remainder of this course. You would be well advised to do as many problems as needed to nail the concept of significant figures down tight and then do some more, just to be sure.

Please remember that, in science, all numbers are based upon measurements (except for a very few that are defined). Since all measurements are uncertain, we must only use those numbers that are meaningful.  A common ruler cannot measure something to be 22.4072643 cm long. Not all of the digits have meaning (significance) and, therefore, should not be written down. In science, only the numbers that have significance (derived from measurement) are written.

Rule 1: Non-zero digits are always significant.

Hopefully, this rule seems rather obvious. If you measure something and the device you use (ruler, thermometer, triple-beam balance, etc.) returns a number to you, then you have made a measurement decision and that ACT of measuring gives significance to that particular numeral (or digit) in the overall value you obtain.

Hence a number like 26.38 would have four significant figures and 7.94 would have three. The problem comes with numbers like 0.00980 or 28.09. 

Rule 2: Any zeros between two significant digits are significant.

Suppose you had a number like 406. By the first rule, the 4 and the 6 are significant. However, to make a measurement decision on the 4 (in the hundred's place) and the 6 (in the unit's place), you HAD to have made a decision on the ten's place. The measurement scale for this number would have hundreds and tens marked with an estimation made in the unit's place. Like this:

 

Rule 3: A final zero or trailing zeros in the decimal portion ONLY are significant.

This rule causes the most difficulty with students. Here are two examples of this rule with the zeros this rule affects in boldface:

0.00500

0.03040

Here are two more examples where the significant zeros are in boldface:

2.30 x 10¯5

4.500 x 1012

What Zeros are Not Discussed AboveZero Type #1: Space holding zeros on numbers less than one.

Here are the first two numbers from just above with the digits that are NOT significant in boldface:

0.00500

0.03040

These zeros serve only as space holders. They are there to put the decimal point in its correct location. They DO NOT involve measurement decisions. Upon writing the numbers in scientific notation (5.00 x 10¯3 and 3.040 x 10¯2), the non-significant zeros disappear.

Zero Type #2: the zero to the left of the decimal point on numbers less than one.

When a number like 0.00500 is written, the very first zero (to the left of the decimal point) is put there by convention. Its sole function is to communicate unambiguously that the decimal point is a deciaml point. If the number were written like this, .00500, there is a possibility that the decimal point might be mistaken for a period. Many students omit that zero. They should not.

Zero Type #3: trailing zeros in a whole number.

200 is considered to have only ONE significant figure while 25,000 has two.

This is based on the way each number is written. When whole number are written as above, the zeros, BY DEFINITION, did not require a measurement decision, thus they are not significant.

However, it is entirely possible that 200 really does have two or three significnt figures. If it does, it will be written in a different manner than 200.

Typically, scientific notation is used for this purpose. If 200 has two significant figures, then 2.0 x 102 is used. If it has three, then 2.00 x 102 is used. If it had four, then 200.0 is sufficient. See rule #2 above.

How will you know how many significant figures are in a number like 200? In a problem like below, divorced of all scientific context, you will be told. If you were doing an experiment, the context of the experiment and its measuring devices would

tell you how many significant figures to report to people who read the report of your work.

Zero Type #4: leading zeros in a whole number.

00250 has two significant figures. 005.00 x 10¯4 has three. 

Exact NumbersExact numbers, such as the number of people in a room, have an infinite number of significant figures. Exact numbers are counting up how many of something are present, they are not measurements made with instruments. Another example of this are defined numbers, such as 1 foot = 12 inches. There are exactly 12 inches in one foot. Therefore, if a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision of the expression. Some more examples:

There are 100 years in a century.

2 molecules of hydrogen react with 1 molecule of oxygen to form 2 molecules of water.

There are 500 sheets of paper in one ream.

Interestingly, the speed of light is now a defined quantity. By definition, the value is 299,792,458 meters per second. 

Math With Significant Figures

Addition and Subtraction

In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement. 

For addition and subtraction, look at the decimal portion (i.e., to the right of the decimal point) of the numbers ONLY. Here is what to do:

1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.)

2) Add or subtract in the normal fashion.

3) Round the answer to the LEAST number of places in the decimal portion of any number in the problem.  

Multiplication and Division

In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement. 

The following rule applies for multiplication and division:

The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer.

This means you MUST know how to recognize significant figures in order to use this rule. 

Example #1: 2.5 x 3.42.

The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?

2.5 has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures. 

Example #2: How many significant figures will the answer to 3.10 x 4.520 have?

You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundedth's place is not recognized as significant when, in fact, it is. 3.10 has three significant figures.

Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.

Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are.

Four is also an incorrect answer given by some ChemTeam students. It is too many significant figures. One possible reason for this answer lies in the number 4.520. This number has four significant figures while 3.10 has three. Somehow, the student (YOU!) maybe got the idea that it is the GREATEST number of significant figures in the problem that dictates the answer. It is the LEAST.

Sometimes student will answer this with five. Most likely you responded with this answer because it says 14.012 on your calculator. This answer would have been correct in your math class because mathematics does not have the significant figure concept. 

Example #3: 2.33 x 6.085 x 2.1. How many significant figures in the answer?

Answer - two.

Which number decides this?

Answer - the 2.1.

Why?

It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. 

Example #4: (4.52 x 10¯4) ÷ (3.980 x 10¯6).

How many significant figures in the answer?

Answer - three.

Which number decides this?

Answer - the 4.52 x 10¯4.

Why?

It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role.

WARNING: the rules for add/subtract are different from multiply/divide. A very common student error is to swap the two sets of rules. Another common error is to use just one rule for both types of operations.

Practice ProblemsIdentify the number of significant figures:

1) 3.0800

2) 0.00418

3) 7.09 x 10¯5

4) 91,600

5) 0.003005

6) 3.200 x 109

7) 250

8) 780,000,000

9) 0.0101

10) 0.00800

11) 3.461728 + 14.91 + 0.980001 + 5.2631

12) 23.1 + 4.77 + 125.39 + 3.581

13) 22.101 - 0.9307

14) 0.04216 - 0.0004134

15) 564,321 - 264,321

16) (3.4617 x 107) ÷ (5.61 x 10¯4)

17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]. Watch your order of operations on this problem.

18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]

19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]

Answers to Problems 

1) 3.0800 - five significant figures. All the rules are illustrated by this problem. Rule one - the 3 and the 8. Rule Two - the zero between the 3 and 8. Rule three - the two trailing zeros after the 8.

2) 0.00418 - three significant figures: the 4, the 1, and the 8. This is a typical type of problem where the student errs by giving five significant figures as the answer.

3) 7.09 x 10¯5 - three significant figures. When a number is written in scientific notation, only significant figures are placed into the numerical portion. If this number were taken out of scientific notation, it would be 0.0000709.

4) 91,600 - three significant figures. The last two zeros are not considered to be significant (at least normally). Suppose you had information that showed the zero in the tens place to be significant. How would you show it to be different from the zero in the ones place, which is not significant? The answer is scientific notation. Here is how it would be written: 9.160 x 104. This CLEARLY indicates the presence of four significant figures.

5) 0.003005- four significant figures. No matter how many zeros there are between two significant figures, all the zeros are to be considered significant. A number like 70.000001 would have 8 significant figures.

6) 3.200 x 109 - four significant figures. Notice the use of scientific notation to indicate that there are two zeros which should be significant. If this number were to be written without scientific notation (3,200,000,000) the significance of those two zeros would be lost and you would - wrongly - say that there were only two significant figures.

7) 2

8) 2

9) 3

10) 3

11) 3.461728 + 14.91 + 0.980001 + 5.2631

12) 23.1 + 4.77 + 125.39 + 3.581

In each of these two problems, examine the decimal portion only. Find the number with the LEAST number of digits in the decimal portion. In problem 1 it is the 14.91 and in problem 2 it is 23.1.

That means problem 1 will have its answer rounded to the 0.01 place and problem 2 will have its answer rounded to the 0.1 place. The correct answers are 24.61 and 156.8.

13) 22.101 - 0.9307

The answer is 21.170. The first value in the problem, with three significant places to the right of the decimal point, dictates how many significant places to the right of the decimal point in the answer.

14) 0.04216 - 0.0004134

The answer is 0.04175.

15) 564,321 - 264,321

This problem is somewhat artifical. The correct answer is 300,000, BUT all of the significant figures are retained. The most correct way to write the answer would be 3.00000 x 105.

16) (3.4617 x 107) ÷ (5.61 x 10¯4)

The calculator shows 6.1706 x 1010 which then rounds to 6.17 x 1010 - three significant figures. The value which dictates this is in boldface.

17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]

The calculator shows 1.3398 x 101 which then rounds to 13.40 - four significant figures. In this problem pay attention to order of operations, since division is not commutative. There are two ways to do this problem using the calculator: 1) multiply the last two numbers, put the result in memory, multiply the first two, then divide that by what is in memory or 2) multiply the first two numbers then do two divisions.

18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]

The calculator shows 4.625 x 10¯22, which then rounds to 4.62 x 10¯22 - three significant figures. Notice the use of the rounding with five rule.

19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]

The calculator shows 2280.3972, which rounds off to 2280, three significant figures. In scientific notation, this answer would be 2.28 x 103.

Note this last use of scientific notation to indicate significant figures where otherwise you might not realize they were significant. For example, 2300 looks like it has only two significant figures, but you know (from the problem) it really has three. How do you show this. One way is to use scientific notation like this: 2.30 x 103. Now the 2.30 portion clearly has three significant figures.