CHAPTER 2 –MEASUREMENT NOTES



CHAPTER 2 –MEASUREMENT NOTES

I. How to use your calculator – Individual instruction in class

HINT: When performing scientific notation, always use the EE or EXP keys! Find one of these on your calculator.

II. Scientific Notation

A. A. Used to represent either very large or very small numbers in science.

B. B. Uses a number > 1 < 10 multiplied by powers of 10 (exponents)

C. C. Numbers larger than 1 are represented with positive exponents

D. D. Numbers less than one are represented with negative exponents.

E. E. Form is as follows:

a. 1) only 1 number to the left of any decimal

b. 2) use “X” as the multiply symbol. Do NOT use a DOT!

F. F. Examples:

a. 1.) 5.6 x 104

b. 2.) 9.8 x 10-5

G. G. Changing “long form” numbers to scientific notation:

a. 1.) Move the decimal until there is just one number to the left of the decimal.

b. 2.) Count the number of places you moved. That is the exponent to use for the power of 10.

c. 3.) If the “long form” number was less than 1, make the exponent negative.

d. 4.) If the long form” number was 1 or greater, make the exponent positive.

e. 5.) The exponent of “0” is equal to multiplying by 1.

H. H. Examples:

a. 1.) 1,588,293 = 1.588293 x 106

b. 2.) 0.4976 = 4.976 x 10-1p

c. 3.) 19 = 1.9 x 10 0

I. I. Changing scientific notation numbers to “long form:”

a. 1.) Move the decimal the number of places specified by the exponent.

b. 2.) Move the decimal to the right if the exponent is positive (number is 1 or greater).

c. 3.) Move the decimal to the left if the exponent is negative (number is less than 1.)

J. Examples:

a. 1. 8.4 x 10 7 = 84,000 000)

b. 2. 3.923 x 10 –4 = 0. 0003923

K. REMEMBER!! When entering problems in your calculator in scientific notation, NEVER actually type in “10 x…..” This will always cause error. Your EXP or EE key does this for you!

III. Metric (SI) System

A. In science, we always use the metric system, or as it is called, the SI system, for measurements.

1. SI is short for “Le Systeme Internationale d’Unites”

2. Used universally except in the United States and some undeveloped nations!

B. How it works

1. Uses base units with values of “1” of something (meter, second, gram, Liter, mole) and then uses powers of 10 to indicate more or less than 1. There is a base unit chart on page 34. You are responsible for learning the following base units: length, mass, time, temperature, and amount of substance.

2. Each power of ten has a name associated with it, which is used as a prefix to denote more or less than 1 of that particular item.

3. There is a chart on page 35 that contains the metric prefixes and their values. You are responsible for learning this chart! This means memorizing the prefixes and their values.

C. Examples

1. Lets say you have 1 gram of copper. Then you acquire 9 more grams, for a total of 10 grams. That is one power of ten. So that could be written as either 10 g OR 1 dekagram. Deka is the prefix that means “10 x” so, that is 1 dekagram (dag).

2. Now we decrease the amount of copper we have. Let’s say we have only one-tenth of a gram. We have decreased the amount by 1 power of ten. So we could write it 0.1 gram, OR we could say “1 decigram.” Deci (d) is the prefix that means 1/10 or 0.1 of an item. So it is correctly written as 1 dg.

3. If we have one million (1,000,000) seconds, we can say we have 1Megasecond.

4. If we have only one millionth of a second (1/1,000,000), we have a microsecond.

D. Converting in the metric system

1. To convert within the metric system, you have to keep up with the number of decimal places you must move. You can do this in your head, visualizing the metric staircase, or you can use your calculator (preferred).

2. For instance, if you have 315 grams of copper, and want to convert that to milligrams (mg), you would have to know that “milli” means 1/1000 of an item. That is 3 powers of ten. So I have to move 3 decimal places to the right. I could have just counted those in my head, or I could have multiplied 315 by 1 x 103 . Notice I did not multiply by 1 x 10 -3 . We would use the absolute value of the power of ten. We could correctly write this as “315,000 mg.”

3. Why did we multiply and not divide in the example in #2? It is because a gram is much larger than a milligram. It takes lots of little things to make a big thing! So there are many more tiny milligrams in the number, and you multiply to make the number larger.

4. Let’s take an opposite example. Using the same 315 grams of copper, let’s figure out how many megagrams (Mg) that would be. A “mega” something is one MILLION times larger than 1 of something. That is 1 x 10 6 ! So now, the opposite reasoning is true. I only need a small portion of that “big something” to represent the smaller quantity. So this time, I will either move my decimal 6 places to the LEFT or DIVIDE by 1 x 10 6 to get my smaller value. Dividing 315 by 1 x 10 6 gives me 0.000315 megagrams (0.000315 Mg).

5. What if you want to convert a quantity NOT from the “base” position? You still need to know how many powers of ten you need to move the decimal. There are a couple of ways to do this. You could look at a metric staircase and actually count the “steps” between the two quantities. Then you could move the decimal that many places. You would move the decimal to the right if you were going from a larger to a smaller prefix, and you would move it to the left if you were moving from a smaller to a larger prefix.

However, it is still preferred that you use your calculator to do this. Here are two rules for determining the powers of ten between them. 1) If you are “crossing the base” value (for instance moving between milli and kilo) you will take the absolute value of the exponents and add them together to get the power of ten. For the example I just gave, moving from milli to kilo, I would know that milli is 10 –3 and kilo is 10 3 . Taking the absolute value of both numbers, I would add 3 +3 = 6. There is a difference of 6 powers of 10. So then you could either move your decimal that many places or use the calculator to divide it out. 2) If you are NOT “crossing the base” (for instance converting from deci to micro) you use the same procedure of taking the absolute value of the two powers of ten, but instead of adding the exponents, you SUBTRACT them. For our example, deci is 10 –1 and micro is 10 –6 . Subtract 6-1=5. So there are 5 powers of ten between them. Now you can either move the decimal by that amount or use your calculator to multiply by the power of ten.

IV. Significant Figures/ Accuracy and Precision

A. Accuracy vs. Precision

1. accuracy=how close a measurement is to an accepted value.

Examples: water boils at 1000 Celsius. If you boil water and get a measurement of 99.90 Celsius, you are accurate, because you are extremely close to the accepted value. If you get a measurement of 75.00 you are NOT accurate, because it is far away from the accepted value.

2. Precision= the ability to repeat measurements that are very close to one another, but not necessarily close to an accepted value. For instance, if I am measuring a quantity of iron oxide, and I take 3 readings of 5.6 g, 5.59 g, and 5.61 g, I am precise, because those 3 measurements are very close to one another. However, if I took 3 readings that were 5.6, 4.2, and 6.0, those are NOT precise, because they are widely scattered.

3. Generally, good precision usually means the experimenter has good technique. However, it can also indicate that the measuring instrument is in good working order. Poor precision can result either from experimenter’s error or bad equipment. However, bad equipment usually affects accuracy rather than precision.

4. The illustration below shows a graphic representation of measurements with accuracy and/or precision.

[pic]

B. Significant Figures (Sig Figs) or Significant digits

1. In science, we acknowledge that measuring instruments can have inaccuracies. If we use inaccurate equipment, we might get inaccurate readings. To minimize the effects of this, science uses significant figures.

2. In any measurement, science allows all absolutely known numbers plus one “uncertain” number. This “uncertain” number is the last number in the measurement. It is considered an estimation.

3. Depending on the measuring instrument you use, you may have more significant figures or less. The more precise your measuring instrument, the more significant figures you can place in your measurement.

4. For example, beakers are not used for measuring in chemistry because there are very few “measurement lines” – called graduations -- on them. However, a graduated cylinder has many “measurement lines” – graduations – and is therefore more precise. We can get many more significant figures from a graduated cylinder than a beaker. Look at the comparisons below:

[pic] [pic]

The beaker in this picture only shows a graduation every 25 mL. So we cannot be very precise in our measurements. However, the graduated cylinder has a graduation for every 1 mL. We can be much more precise with our measurements in the graduated cylinder.

And even among graduated cylinders (or other devices such as rulers) there are differing levels of precision, and therefore, significant figures. Each of these rulers below has a different number of graduations. The one on top has the least, and is the least precise. The one on bottom has the most graduations and is therefore most precise. We can get more significant figures from the bottom reading.

[pic]

C. Rules for Significant Figures -- General

1. All non-zero numbers in a measurement are significant. Always. In a measurement of 59.4576 there are 6 significant figures (SF). In a measurement of 0.067 there are 2 SF.

2. For zeroes, there are 3 rules:

a. Leading zeroes are NEVER significant! Example: 0.0056 = 2 SF. All the zeroes are leading zeroes. They are place holders and attention getters. They are NOT actually measured, but tell us the place value of the first measured number.

b. Captive, or Trapped zeroes are ALWAYS significant. We assume that we measured those zeroes on the way from one non-zero number to the next one. Example: 1,001 = 4 SF 10.055 = 5 SF

c. Trailing zeroes are SOMETIMES significant. The ARE significant if they follow a decimal OR have a decimal right after them. Example: 23,000. =5 SF 15.00 = 4 SF

Trailing zeroes are NOT significant if there is no decimal involved. Examples: 1,000 = 1 SF 700 = 1 SF

D. Rules for Significant Figures – math operations

1. For multiplication/division – your answer cannot have more SFs than the measurement in the problem with the least number of SF. It must be rounded to that number. Example: 126/3 – the pure math answer is 42. However, looking at the two numbers in the problem, we see that 126=3 SF and 3=1 SF. The number 3 is therefore the measurement with the least number of SF, so we MUST round our math answer to only show 1 SF. This is done by rounding 42 down to 40, showing NO decimal after the zero. Example: 7,345 x 0.25 – the pure math answer is 1,836.25. However, examining the two measurements involved, 7,345 = 4 SF and 0.25 = 2 SF. Therefore, our answer must only have TWO SF in it. This is done by rounding 1,836.25 to 2 SF 1,800 with no decimal after the zeroes. You could also write it in scientific notation 1.8 x 103 .

2.For addition and subtraction – the answer cannot have more numbers in it than the measurement whose last SF is furthest to the LEFT. This is actually based on place value rather than number of SF. I usually find it is easier to figure out this value by always adding or subtracting in a vertical column. In the examples below, I’ll do that and underline/bold the LAST SF.

Example:

125.56

50.1

+ 1.25

______________

176.91

In this example, the pure math answer is 176.91. However, looking at the LAST SF of each measurement, you can see that the 1 in 50.1 is the last SF that is furthest to the left. It is sitting in the 10ths place. Therefore, our answer cannot go past the 10ths place! We would round it to 176.9.

Example:

77.35

- 11

__________

66.35

In this example, the pure math answer is 66.35. However, you can see that the 2nd 1 in 11 is the last SF furthest to the left. It is in the one’s place. Therefore, our answer must be rounded to the ones place, or simply 66.

E. General hints for significant figures – 1) Numbers that are universally accepted as correct values (such as the value of pi) or conversion factors (such as 12 inches to 1 foot) are considered to have an infinite number of significant figures. They are IGNORED in measurement calculations. 2) When you have a problem to solve involving measurements, always use your initial given quantity to figure out the limit on your significant figures. 3) When doing a problem with mixed math (addition and then multiplication, for instance) we generally use the rules for multiplication/division.

F. Rounding rules: In this class, we will use “normal” rounding rules for rounding numbers. Numbers 1-4 round down, and numbers 5-9 round up. However, there is a rule sometimes used in chemistry and physics known as the “even-odd” rule, where 5’s are rounded up if the number preceding it is odd, and rounded down if the number preceding it was even. We will not use this rule unless I hear from AP teachers that I need to do so. Subject to change, so listen for instruction!

V. Dimensional Analysis

A. Dimensional analysis is a very important tool in the world of chemistry. By using it, you can convert between unlike quantities with ease. Unlike quantities refers to items like converting feet to meters, moles to grams, or even milligrams to megagrams. The key to dimensional analysis is the ability to cancel units. By treating units as algebraic entities, they may be canceled, just like variables in algebra. In fact, when using dimensional analysis, the units are more important than the numbers – at least in setting up the problem. We always want to get the correct numerical answers, but if you set up the problem correctly using units, the numbers usually take care of themselves.

B. Dimensional analysis uses conversion factors to equate unlike quantities that are actually equal in value.

C. Conversion factors are ratios between unlike quantities to show how they are actually equal. For our purposes, these ratios are written in the form of a fraction. Here is a simple example: we all know that 1 dollar is the same equivalence in money as 4 quarters. Quarters and dollars are different items, yet 4 quarters and 1 dollar are equal in value. We can say accurately “1 dollar contains 4 quarters” and we can also say that “4 quarters equals 1 dollar.” It does not matter which way we say it, it is still true. Similarly, we can write that ratio as follows:

1 dollar OR 4 quarters Either way we write it, it is a true statement, and a correct

4 quarters 1 dollar ratio. The way we write it will depend on the problem.

Here is another example of a conversion factor: 1 dozen OR 12 items

12 items 1 dozen

D. The point of using dimensional analysis is to set up the problem so that we cancel units we don’t want and wind up with the unit we do want. Here is the first example problem, which will be worked out step by step – How many dollars are in 345 quarters?

E. How to work the problems

1. Step 1 – place your given quantity over 1, in the form of a ratio/fraction.

345 quarters

1

2.Step 2 – looking at the problem, determine which unit you need to cancel and choose a conversion factor that will allow that to happen. In this problem, we need to cancel quarters and be left with dollars. Therefore, we need to choose the conversion factor 1 dollar because in our

4 quarters

given, quarters is in the numerator. The only way to cancel that is to place quarters in the denominator of our conversion factor. Now the problem will look like this:

345 quarters x 1 dollar Notice the use of “x” as a multiply factor. We will be multiplying

1 4 quarters and dividing in this problem.

3. Step 3 – multiply all numerators together, then multiply all denominators together, cancelling the appropriate unit(s). Divide the denominator answer into the numerator answer, and bring down the remaining unit.

345 quarters x 1 dollar = 88.5 dollars

1. 4 quarters

4. Step 4 – check your work and round answer to the correct number of significant figures. You should physically mark out the cancelled units as I have done here. You should use the GIVEN quantity for your SF. Our given has 3 SF, so our answer must only have 3 SF also, which it does.

Here is a second simple example: How many minutes are equivalent to 5.6 x 109 seconds?

5.6 x 109 seconds x 1 minute = 9.3 x 107 minutes Our SF in the answer (2) matches the SF in the

1 60 seconds given (2).

F. More complicated dimensional analysis – it is possible to have to use more than 1 conversion factor in a problem. In fact, you could string together an infinite number of conversion factors, as long as they were equivalent and true ratios. We’ll work two now that has more than 1 conversion factor. Don’t worry about knowing all the conversion factors. Most you will know from experience (like seconds and minutes or feet and inches) or you will be able to look them up in a reference material , or have them given to you on a test. The only conversion factors you need to memorize are the metric prefixes.

Example: How many seconds are in 14 hours?

We need two conversion factors for this problem. I don’t know how many seconds are in a whole hour (actually, I DO, because when you take physics you memorize that!) but I do know there are 60 seconds in a minute and 60 minutes in an hour. So we set the problem up like this:

14 hours x 60 minutes x 60 seconds = 50,400 seconds or 50,000 seconds

1 1 hour 1 minute 1

Notice that I put an “intermediate answer” before the final one. For multi-step problems, this is a WISE thing to do. It cuts down on errors. Also, notice that our given had 2 SF. Our original answer had 3 SF, so we had to round to 2. I wrote it as 50,000, but it would be MUCH BETTER to write it in scientific notation as 5.0 x 104 seconds. Remember, once you get to 3 zeroes, you need to put it in scientific notation! Also, putting it in scientific notation usually solves your problem with how many SF. LASTLY, do NOT round at each step of a multistep problem. Round at the END ONLY or you will get incorrect answers.

Example: how many feet are in 1 km? This problem is a conversion between English and metric (SI) quantities. We also need to convert between two metric quantities. We need conversion factors between feet and meters, kilometers and meters, and I am even going to use feet and inches.

1 km x 1000 m x 39.3 inches x 1 foot = 39,300 feet = 3,275 feet BUT!! SF!!!

1 1 km 1 m 12 inches 12

Our given only has 1 SF! We must round to 1 SF – use scientific notation 3 x 103 feet

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