|
Uncertainty: make sure that you read between the lines, there is always one more possible decimal place than is labeled by the actual measuring device.
Length: measure of distance : done in meters
Mass: measure of the amount of matter possessed by an object : measured in grams
Volume: Space occupied by a solid, a liquid, or a gas.
Significant figures (digits):
Relates to uncertainty in a measurement
How sure are you of each number in a measurement
Limited by the ability of the equipment to measure accurately
Fewer significant figures, the more uncertainty there is in the equipment
The Basic Rules:
Determining the Number of Significant Digits
Rule 1: Count the number of digits in a measurement from left to right:
(a) Start with the first nonzero digit
(b) Do not count place-holder zeros (0.011, 0.00011, and 11,000 each have two significant digits)
Rule 2: The rules for significant digits apply only to measurements and not to exact
numbers.
Rounding Off Non-significant Digits
Rule 1: If the first non-significant digit is less than 5, drop all non-significant digits.
Rule 2: If the first non-significant digit is 5, or greater than 5, increase the last significant
digit by 1 and drop all non-significant digits
Rule 3: If calculation has two or more operations, retain the non-significant digits until
the final operation. Not only is it much more convenient, but it is also more accurate to round off the final answer.
Corwin. Introductory Chemistry. Pages 21 & 22
Significant figures only apply to measurements, not to exact numbers because only measurements have uncertainty. Exact numbers do not have uncertainty, therefore significant figures do not apply.
Examples:
How many significant figures are there in:
a) 1.10202
b) 0.001650
c) 1000
d) 100010.2
e) 10.
f) 2.31201000
g) 23.632
Round to the given significant figure:
a) 9.7210 (3 significant figures)
b) 6.21471292 (8 significant figures)
c) 0.00103 (2 significant figures)
d) 1.0001492 (5 significant figures)
e) 10001.98 (6 significant figures)
More on Significant Figures:
The uncertainty of an instrument describes the degree to which you can accurately describe a measurement. The last number in any measurement should be the uncertain measure (read between the lines). If the measurement is 5.234 the four is the uncertain number. That means that 4 could be either a 3 or a 5. So, the measurement may have been either 5.235 or 5.233. To get to 5.235, .001 was added. To get to 5.233, .001 was subtracted. Therefore, the uncertainty of the measurement is 5.234 + or - .001.
What would be the uncertainty of 4.3 L
3.44 m
3 g
1232 K
Uncertainty is very important when considering addition and subtraction of measurements:
Rule for addition and subtraction of measurements:
Find the measurement with the most uncertainty (fewest number of decimal places) and round the answer to that uncertainty.
Ex. 4.5 g + 3.221 g + 4.3332 g = ?
4.5 has an uncertainty of .1
3.221 has an uncertainty of .001
4.3332 has an uncertainty of .0001
So, the number with the most uncertainty is 4.5 with an uncertainty of .1, therefore, the answer must be rounded off to that uncertainty (in this case, one decimal place)
The Math:
12.0542 is the answer so far, but remember to correct for uncertainty the answer may only have one decimal place. In this case, 0 should be the last significant figure. Look to the right of zero, notice that it is a five. Since it is a five, we need to round that five up. The zero now becomes a 1. So the answer to this question would become: 12.1 because 12.1 has an uncertainty of +/- .1
Rule for multiplication or division of measurements:
Find the measurement with the fewest number of significant figures and round the answer to that number of significant figures.
Lets remember some rules for significant figures first:
Zeros to the right of the decimal place before a whole number are not significant. .001 has only one significant figure. .0000001 also only has one significant figure.
Zeros between two whole numbers are significant. 1.01, now has 3 significant figures. .0101 also has 3 significant figures.
Numbers following a whole number:
If a decimal place is not present, the zeros to the right of the number are not significant unless marked with a bar. So, 1000 has only one significant figure.
If a decimal place is present, the zeros to the right of the number are significant. So 1000. has 4 significant figures.
Zeros following a whole number after a decimal are significant. So, in the number 14.000 the three zeros are significant so there are 5 significant figures. .0002012300 has 7 significant figures. Remember, those first three zeros are not significant, but the ones following the three are significant.
Ex. 5.32 g x .01 g
5.32 has 3 significant figures and .01 has 1 significant figure, so the answer to the problem must have only one significant figure.
The answer is .0532. Remember, now you need to correct for significant figures. Since the answer can only have one significant figure, anything after the 5 must be rounded off. Since 3 is less than five, it is rounded down, and the five does not change. The answer to this problem then is .05 (one significant figure).
Fire: Most of Class Canceled
Quiz is still tomorrow (Friday) on Significant Figures, Rounding, and Mathematical manipulation of numbers.
Rules for Addition, Subtraction, Multiplication, and Division of Unit Exponents.
In addition and subtraction, the exponent must be the same before the numbers may be added or subtracted from one another.
For example:
8 m + 16 m2 cannot be added together. The units are quite different. The same thing goes for scientific notation. (we will cover this after the quiz on Friday).
Before you can add or subtract numbers in scientific notation, the exponents must be to the same power.
So 6 x 102 + 5 x 105 cannot be added until the exponent is the same.
In multiplication and division, the rule is slightly different:
Multiplication: when variables with exponents are multiplied, the exponents are added together.
Division: when variables with exponents are divided, the exponents are subtracted (numerator denominator).
The rules for multiplication and division of exponents holds true for scientific notation as well. So (6 x 102) x (5 x 105) can be multiplied by simply multiplying the 6 and the 5 to get 30, then adding the exponents to get 107
A couple final notes before the Quiz:
In cases where parentheses are found in a mathematical calculation of measurements, do not round to the end. When rounding, choose the sign that is found between the parenthesis and round according to those rules.
For example: (4.321 g + 6.4 g) + (4.10 g x 2.342 g)
You would carry out the mathematics within the parenthesis, but round using the rules for addition because that is the operation found between. In this case, the most uncertainty would be +/- .1
One other note before the quiz:
10 has an uncertainty of +/- 10
10. has an uncertainty of +/- 1
remember, it is necessary to find the last significant number in the measurement before figuring out the uncertainty.
So, 100 would have an uncertainty of +/- 100
Quiz Time!
Post Quiz Notes:
Exponential notation:
Number taken to a power (how many times that number is multiplied by itself to get an answer:
2 x 2 x 2 x 2 is 24
because, 2 is multiplied by itself 4 times. Both answers would be 16.
10 x 10 x 10 is 103 since ten is multiplied by itself 3 times.
Since the metric system is based on a multiple of 10, exponential powers of 10 are quite prevalent in the scientific literature.
Scientific notation:
Scientific measurements usually appear in the form of scientific notation, written in the general form as:
t.tt x 10n where t is any integer and n is the some whole number of times that 10 has been multiplied to reach a given answer.
Rule # 1
Find the first non-zero significant figure in a number and place the decimal place immediately to the right of that number.
Rule # 2
IF the decimal was moved to the left, then the power of 10 is a positive number
IF the decimal was moved to the right, then the power of 10 is a negative number
Remove all non-significant numbers (place holding zeros)
Example:
Write .00923 in scientific notation:
Ok, since the 2 zeros are place holders, they will not appear in the final answer.
The fist non-zero sig fig is the 9, so place the decimal after the 9
9.23
Now, since the decimal was moved to the right 3 places, the power of ten is a number
9.23 x 10-3
Write 5600 in scientific notation:
Ok, the first sig fig is the 5, so place the decimal immediately to the right of the 5
5.600
Since the two zeros are not significant, they should not appear in the final answer:
5.6
Since the decimal was moved to the left three place, the power of 10 is a positive number:
5.6 x 103 (to check to see if you have done this correctly simply revert to the ordinary number)
Since 103 is 10 x 10 x 10, 5.6 is being multiplied by 1000. when 5.6 is multiplied by 1000, the answer becomes 5600
|
