please watch the attachment and answer the question in the last page
Worksheet 1 – Chapter 2
You must show your work for Credit
Part 1: Significant Figures
Let’s start off with scientific notation
…
Large numbers (numbers for which the absolute value is greater than 1) will always have a positive exponent when in scientific notation. When converting to scientific notation, you move the decimal point until there is a single digit to the left. The number of places that the decimal spot moved becomes the exponent and the “x10”.
Example: –450000 –4.5×105. The decimal point was moved 5 times to the left, so the exponent is 5.
Example: 106709001 1.06709001×108. The decimal was moved to the left by 8 spots so the exponent is 8.
Example: 57293.264 5.7293264×104 since the decimal was moved 4 times to the left.
Small numbers (numbers between 1 and –1) will always have a negative exponent when in scientific notation. When converting to scientific notation, you move the decimal point until there is a single digit to the left. The number of places that the decimal spot moved becomes the exponent and the “x10–”.
Example: 0.0003528 3.528×10–4. The decimal moved 4 times to the right, so the exponent become –4.
Example: –0.0000000000000058500 –5.8500×10–15. The decimal point was moved 15 times to the right, so the exponent became –15.
Example: 0.002 2×10–3 since the decimal was moved 3 times to the right.
You try:
1a) 54,670,000,000
1b) –5526.7
1c) 0.03289
1d) 100.00
How many significant figures in a number
:
First and foremost, you need to be able to tell how many sig. figs. are in a number. Here are three rules that you can use:
1) If the number is in scientific notation:
The number of digits shown, excluding the order of magnitude, is equal to the number of sig. figs.
Examples: 6.626×10–34 has
4 significant figures (6.626×10–34)
8.30×104 has
3 significant figures (8.30×104)
3.0×101 has
2 sig. figs. (3.0×101)
2) If the number has a decimal in it:
Start at the RIGHT of the number and count to the left until you get to the last NONZERO number, this is the number of sig. figs. Any additional zeros to the left are NOT significant.
Examples: 195.3040 has
7 sig. figs. (195.3040)
0.003081 has
4 sig. Figs. (0.003081)
180048.00 has
8 sig. figs. (180048.00)
0.0000002 has
1 sig. fig. (0.0000002)
10. has
2 sig. fig. (10.)
3) If the number does NOT have a decimal in it:
Start at the LEFT of the number and count to the right until you get to the last NONZERO number, this is the number of sig. figs.
Examples: 160 has
2 sig. figs. (160)
20000 has
1 sig. figs (20000)
704 has
3 sig. figs. (704)
49003100 has
6 sig. figs. (49003100)
10 has
1 sig. fig. (10)
You try:
2a) 6200 2e) 0.0006000 2h) 23.4400
2b) 1.032 2f) 1×104 2i) 100.0003
2c) 420. 2g) 35000000 2j) 100
2d) 3.750×10–6
Precision
There are a lot of different ways of thinking about precision. The definition of precision is “how close a series of measurements are
to one another
or how reproducible they are”. Huh? Basically, this means that as you use an instrument to make measurements, the closer those measurements are to each other, the more precise the instrument is. Wait… huh?? Try this. You can think about precision in terms of the size of the graduations on the instrument. The smaller the graduations are, the more precise the instrument is. This is directly related to the above definition because the smaller the graduations are, the smaller the space is that you have to “guess” in and the smaller the space that you have to “guess” in, the more likely that your “guesses” will be the same. BAM, definition of precision.
When we are trying to determine the precision of a measurement, we can use significant figures. (In fact, another definition of precision is related to significant figures.) In order to determine the precision of a measurement, we will need to find the
last
significant figure in the measurement. We learned above how to count
how many sig figs there are in a value and this is related, but slightly different. The last sig fig in a measurement is always the one that is
furthest to the right, regardless of the presence or absence of a decimal place.
Example: 6200
6200 has two significant figures (the 6 and the 2), but we want to know the precision in this case. Again, the precision of this measurement lies in the
last sig fig (the sig fig that lies furthest to the right). For 6
200, the last sig fig is the 2, which lies in the hundreds–place. Therefore this measurement is precise to the hundreds–place.
Example: 1.032
1.032 has four significant figures (all of the digits in this one happen to be significant). Again, the precision of this measurement lies in the
last sig fig (the sig fig that lies furthest to the right). For 1.03
2, the last sig fig is the 2, which happens to lie in the 3rd decimal place. Therefore this measurement is precise to the 3rd decimal place (or the thousandths place).
Example: 420.
420. has three significant figures (the 0 is significant in this case because of the decimal place). For 42
0., the last sig fig is the 0, therefore this measurement is precise to the ones–place.
Example: 3.750×10–6
This one is actually a bit tricky because it is written in scientific notation. If you’re not careful, you will report the precision of this measurement as the 3rd decimal place, which is incorrect. To find the precision of this value, you will need to convert it from scientific notation to decimal notation.
3.750×10–6 = 0.000003750 has four significant figures, the last of which is the 0 at the right of the value. For 0.00000375
0, the precision of this measurement is in the 9th decimal place.
Example: 1×104
Again, watch out for the scientific notation, you will need to convert it from scientific notation to decimal notation.
1×104 = 10000 which has only one significant figure (the 1). For
10000, the precision of this measurement is in the ten–thousands–place.
Your turn:
3a) 280100 3e) 3.20×105
3b) 6.341×10–4 3f) 9900.
3c) 90 3g) 43.0
3d) 0.00301 3h) 7800000
Significant figures in calculations
One of the most missed rules in significant figures is that you are not allowed to round a value until you are
completely finished
with a calculation. This is especially important in situations where you are mixing operations or when the answer to one question is then used in the next question. In both of these situations, you will be tempted to round values before you get to the answer to the questions you are working on. We will see how to avoid this mistake with mixed operations later in this exercise, so for now, let’s focus on the second of these two mistakes.
Let’s say that you worked through “part a” of a questions and your calculator spit out the answer 52.48612119 and you knew that your answer had to have three significant figures. What would happen if you simply wrote down 52.5 as your answer and got halfway down the page before realizing that you needed to use that answer in “part f”? Three options. One, just use the rounded value of 52.5 to start the calculation in “part f”. This is completely wrong. Two, put the calculation from “part a” back into your calculator so that you have the unrounded value to work with in “part f”. This is a complete waste of time. Three, write both the unrounded
and
rounded values when you complete “part a”. This is by far the best option as it takes next to no time and it provides easy access to the unrounded value that you might need later. Now, you do NOT need to write down every digit that your calculator displays. A general rule of thumb is to keep two digits more than sig figs would allow.
calculation
calculator
write
and
write
Example: 845.0265512 / 16.1 = 52.48612116 52.486 52.5
Rules:
There are two distinct rules that you need to be able to use and keep straight.
Addition and/or subtraction:
The rule for addition and subtraction is based on the precision of the values being added and/or subtracted. When adding and/or subtracting values, the resulting answer MUST have the same precision as the LEAST precise value used in the calculation. For each value that you are adding or subtracting, you will need to determine the precision of the measurement. The value that is least precise (has its precision furthest to the left) dictates the precision of your answer.
(Note: Your book puts this rule in terms of decimal places, which works fine as long as all values have decimal places. This [incomplete] version of the addition/subtraction rule requires that you count the number of decimal places in each of the values. The answer must have the same number of decimal places as the value in the problem that has the
FEWEST
DECIMAL PLACES
. What does that mean? If you add 5.12345 (5 decimal places), 12.123 (3 decimal places), and 0.12 (2 decimal places), your answer must have 2 decimal places.)
Example:
First, write the digits vertically with the decimal points
lined up and find the number of decimal places for each value (this will help until you get more comfortable with the process). The answer must have the same precision as the
least precise value in the problem.
2500.0 is precise to the 1st decimal place, 1.236 to the 3rd decimal place, and 367.01 to the 2nd decimal place.
The precision allowed in the answer is dictated by the first value (because that value is the
least precise measurement), so you must round to that digit (the 2 in 2868.246 here).
The answer is
2868.2
Example:
So the answer is
0.0209
Example:
The precision of the answer can only be to the 2nd decimal place.
The answer is
0.00
Example:
Again, your answer must have the same precision as the least precise value in the question; (3rd decimal place in this case).
The answer is
1.000
Example:
When adding or subtracting, you need to first convert values in scientific notation into decimal notation, so this problem is actually 7215900 – 28000
Again, your answer must have the same precision as the least precise value in the question; (the thousands–place for 28000 in this case), so the answer is
7188000 or 7.188×106
Multiplication and/or division:
The rule for multiplication and division is all about how many sig. figs. a number has. The value in the calculation that has the
FEWEST
number of
SIGNIFICANT FIGURES
determines the number of sig. figs. in your answer. If you are multiplying 3 different numbers, one has 4 s.f., one has
2 s.f. and one has 7 s.f., your answer can only have
2 s.f.
Example: 0.01116 x 23.44600 = 0.26165736
0.01116 has 4 s.f. and 23.44600 has 7 s.f. Therefore the answer is limited to 4 s.f.
The answer would be rounded to
0.2617
Example: 26.375 x 3790 = 99961.25
26.375 has 5 s.f. and 3790 has 3 s.f., so the answer is again limited to 3 s.f. This is a fairly large number, so put it into scientific notation before rounding. It becomes 9.996125×104. Now do your rounding and you get 10.0×104. There can only be one digit to the left of the decimal, so the final answer is
1.00×105.
Example:
3.14159 has 6 s.f. and 502000 has 3 s.f. so the answer can only have 3 s.f. The answer is
0.00000626 or
6.26×10–6
Examples:
536 has 3 s.f., 0.3301 has 4 s.f., 60.002 has 5 s.f., 0.0048 has
2 s.f., and 12.1 has 3 s.f., so the answer can only have
2 s.f. This is a large number, so put it into scientific notation
BEFORE rounding 1.8278873738×105. Since you can only keep 2 s.f., the answer is
1.8×105.
You try:
4a) 4e)
4b) 4f)
4c) 4g)
4d) 4h)
Mixed operations – multiplication/division AND addition/subtraction in the same problem:
When working with significant figures where there is a mixture of operations, the rules for the individual operations do not change, but the order in which those operations are performed is important. The order in which you perform the calculations follows the “order of operations” which you may remember from algebra. That order is:
parentheses,
exponents,
multiplication,
division,
addition, and
subtraction (
please
excuse
my
dear
aunt
sally). After each of these steps, you need to mark the last significant figure you are allowed in that step (usually with a line over that digit) so that you can keep track of what the limiting significant figure is in each step.
YOU ARE NOT ALLOWED TO ROUND YOUR ANSWER AFTER EACH STEP, but rather you should wait to do the rounding at the end of the entire problem and this is why it is important to mark the last sig. fig allowed in each step. I am going to start these examples with something we have already seen this semester: isotopic abundance calculations.
Example: Gallium has two stable isotopes, gallium–69 and galium–71. If the mass of gallium–69 is 68.926
amu and the mass of galium–71 is 70.9247
amu, then what are the percent abundances of each isotope?
The beginning equation is:
According to the order of operations, we need to clear the parentheses first, but since we don’t know what X is, there is nothing we can do here. The first operation we are actually going to do is the multiplication step. The equation becomes:
Because you are multiplying by 1 (an exact number), there is no change in sig. figs. to worry about in this step. Now that all of the multiplication is taken care of, we will deal with subtraction.
Notice the line over the top of the zero on the right hand side of the equation. Since we are subtracting, we base our answer on the number of
decimal places in the values we are subtracting. 69.72 has two decimal places and 70.9247 has 4. This means that my answer must have 2 decimal places and I indicate that with the line over the second decimal place in the –1.2047. The next step is to perform the subtraction on the left side of the equation.
Again following the rules for addition/subtraction, I have placed a line over 8 in the value on the left because we are only allowed three decimal places after performing this subtraction. Also note that I have NOT done any rounding yet! The next step is to divide both sides by –1.9987 in order to get X by itself.
→
Following the rules for multiplication/division of sig. figs., we must base the sig. figs. in our answer on the number of
significant figures in values we are dividing. Looking at the lines that we have been placing above our values as we have proceeded, we see that –1.2047 has 3 sig. figs. and –1.9987 has 4 sig. figs. Because of this, the answer is
0.603
Example:
We start with the parentheses and because the operation with the parentheses is addition, we will follow that rule and base our intermediate answer on
decimal places. The first value has five decimal places and the second has four, so our answer must have four and we will denote that by putting a line over the top of the 4th decimal place in the intermediate answer.
The next step is multiplication, so the answer will be based on the number of
significant figures in the two values. The first has 5 sig. figs. (we know that because of the line) and the second has 4, so our answer will have 4.
0.5327 which is the answer
Example:
Remember your order of operations!! We must do the multiplication step first which means the sig. figs in the intermediate answer will be determined by the number of
sig. figs. in the values being multiplied (3 in 0.0777 and 4 in 1.430×103). We’ll put a line over the last sig. fig. we are allowed to keep.
The next step is subtraction which means that the number of sig. figs. in the answer is based on the number of
decimal places in the values being subtracted (4 in the first value and 0 in the second, look for the line!!)
–83 which is the answer
Example:
Do each set of parentheses first making sure to mark the last sig. fig. you are allowed to keep (for this question, based of course on the addition/subtraction rules)
The final step is a division, so follow that rule. The top value as 4 sig. figs. and the bottom has 3.
–55.4 which is the answer
Example:
Remember to put large numbers into scientific notation BEFORE rounding
which is the answer
Your turn:
5a)
5b)
5c)
5d)
5e)
5f)
5g)
5h)
Significant figures and defined values versus measured values
All measured values limit the sig figs (due to uncertainty) that your answer can have, while defined values do not. The questions is, how can you tell measured versus defined values with conversion factors? Conversions that are within a system of measure (i.e. metric to metric or standard to standard)
AND
within the same type of unit (i.e. volume to volume or mass to mass) are defined values! Conversions between different systems (i.e. metric to standard) or conversions between different types of units (i.e. time to length or mass to volume) are considered measured values. What does all of this mean? Here are some examples:
This is defined because inches and feet are both in the same system of measure (standard) and are both length units. This conversion would have infinite sig figs. (i.e. not affect the sig figs in the answer).
This is measured because miles are standard units of length while meters are metric units of length. This conversion WOULD affect the sig figs in the answer. (4 sig figs)
This is a measured value because grams are a unit of mass and mL are a unit of volume. This conversion WOULD affect the sig figs in your answer. (3 sig figs.)
This is defined because L and mL are both in the same system of measure (metric) and are both length volume. This conversion would not affect the sig figs in the answer.
Part 2: Dimensional analysis
YOU
MUST
SHOW ALL OF YOUR WORK TO GET CREDIT!!!! USE THE PROVIDED CONVERSIONS AND YOUR CONVERSION SHEET TO ANSWER THESE QUESTIONS.
Dimensional analysis is the process of cancelling units, no matter what those units mean. The point of this assignment is NOT to make you learn these units. The point is for you to realize that you NEED TO CANCEL OUT the unwanted units.
1) The mass of a proton is 1.673×10-27 kg. What is the mass of a proton in grains? (1 grain = 64.79891 mg)
2) The volume of an average gas chromatography injection is 1.5×10-6 L. What is that volume in drops? (1 drop = 0.0500 mL)
3) Jules Verne wrote the book Twenty Thousand Leagues Under the Sea. If 1 league = 5.556 km and 1 furlong = 660.0 feet, how many furlongs did the Nautilus travel.
4) The maximum depth of Lake Tahoe is 99.6 rod. How deep is Lake Tahoe in fathoms if 1 fathom = 0.364 rod?
5) Red light has a wavelength of roughly 700 nm. What is the wavelength in m?
6) An epoch is a period of time that is related to the alignment of the sun and the moon. If 1 epoch = 19 years, how long is our 125 minute class period in epochs?
7) It is 90.123 km from here to Stockton. Wha
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