A. Significant Figures
The rules for determining which numbers in a measurement are significant are: Every nonzero number is significant. Ex. 24.7 g and 0.247 g have three sig figs. Zeros that are in between nonzero numbers are significant Ex. 7003 mL and 40.79 mL have four sig figs. Zeros to the left of all nonzero numbers are not significant. (They are acting as placeholders). (Check to see if zeros are necessary by putting the number into scientific notation.)
Zeros to the right of nonzero numbers and to the right of a decimal point are significant. Ex. 4.50 mL and 40.0 mL have three sig figs. Zeros at the end of a number are not significant unless they are followed by a decimal point, or are already after a decimal point. Ex. 450 g and 9800000 g have two sig. figs. (the zeros are placeholders, put the number into scientific notation to check)
4.5 x 102 g and 9.8 x 106 g 2 sig. figs. In each
Ex. 450. g has 3 sig. figs (4.50 x 102 g) the decimal point makes the ending zero significant.
SF in calculations – Addition and Subtraction
The answer to an addition or subtraction problem should have the same number of decimal places as the number in the problem with the least number of decimal places.
Ex.: 20. g + 20.5 g + 20.00 g = 60.5 g 61 g
Since the number 20. g has no numbers past the decimal (the number in the problem with the least number of decimal
SF in calculations – Multiplication and Division
The answer to a multiplication or division problem can only have as many total significant figures as the number in the problem with the least number of significant figures.
(Position of the decimal does not matter.)
Ex.: 25 cm x 25.5 cm x 25.55 cm = 16288.125 cm3 16000 cm3
Since the number 25 cm has only 2 significant figures (the
B. Scientific Notation
Format of a number in scientific notation:
Coefficient
x 10
exponent
Coefficient
: a number that is greater than or equal to 1,
or less than (but not equal to) 10. (1 – 9.999999…)
Exponent
: a whole number that represents the number
Rules for converting a number into scientific notation The coefficient
Find the first significant number and place the decimal after it. Include only significant numbers after the decimal
The power / exponent
Scientific Notation
Numbers are converted to scientific notation by moving the
decimal point of the original number to get a number between 1 and 10.
Keep track of the number of places the decimal has been moved
and the direction to get the exponent.
Example: 16,400 = 1.64 x 104
The exponent is equal to the number of places the decimal is
moved:
Calculations in Scientific Notation
Multiplication
Multiply the coefficients to get the new coefficient.
Add the exponents to get the new exponent.
Ex. (2.0 x 10
2) x (3.0 x 10
5) = (2.0 x 3.0) x 10
2+5= 6.0 x 10
7Division
Divide the coefficients to get the new coefficient.
Subtract the exponents to get the new exponent.
Calculations in Scientific Notation
Addition and subtraction
Numbers must be a multiple of the same power of 10;
The coefficient can then be added or subtracted;
Precision
How close the measurements are to each other
◦ Repeatable – if one person takes all readings ◦ Reproducible – of many people take readings
Accuracy
How close the measurement is to the true value of
Example – Accuracy
Who is more accurate when measuring a book that has a
true length of 22.0 cm?
Claire
22.0 cm, 21.0 cm, 23.0 cm, 20.0 cm
Mariam
Example – Precision
Who is more precise when measuring the same 22.0 cm
book?
Claire
22.0 cm, 21.0 cm, 23.0 cm, 20.0 cm
Mariam
Good accuracy Good precision
Poor accuracy Good precision
Poor accuracy Poor precision
Systematic errors: reduce accuracy
Systematic error
Errors due to "incorrect" use of equipment or poor
experimental design/setup.
A systematic error makes the measured value always smaller or larger than the true value, but not both.
Cannot be reduced by repetition
Can be reduced by careful experimental design (will increase accuracy)
Sources of Systematic Errors and how to eliminate them
Reading top of meniscus instead of bottom (volumes too high)
Overshooting the volume of a liquid delivered in a titration
(volumes too high)
Heat loss in an exothermic reaction (temperature changes too
low)
Not allowing a reaction to react completely
Random Error
Errors that have equal probability of the reading being high or
low form one reading to the next.
Can be reduced through repeated measurements, or more precise instruments
Reading the scales of a balance, graduated cylinder,
thermometer, etc. produces random errors. In other words, you can weigh a dish on a balance and get a different answer each time simply due to random errors.
The real measure of how far off a value is, is the percentage error.
It is a comparison of the experimental value and the literature
(true/accepted) value
Experimental value - is what you recorded/calculated based on your own
experiment in the lab
Literature value - True value - accepted value based on references
(textbook,…)
% error = |experimental – literature|/literature x 100%
If the % error is positive (+), then errors existed that resulted in the
experimental value being higher than the literature
If the % error is negative (-), then errors existed that resulted in the
It is defined as the mass of a unit volume of a substance.
A measurement of how much mass is contained in a given volume
If, for example, you have 1.00 gram of water, it has a volume of
1.00 cm3. Its density is 1.00 g/cm3.
The proper units would be kg/m3 but g/cm3 are frequently used.
If 2.0 grams of sulfur has a volume of 1.0 cm3, the density of
sulfur is 2.0 g/cm3.
Solids
The densities of solids vary over a wide range. A few are less dense than water and thus will
float.
The density of ice is 0.92 g/cm3. When you
