Regents Chemistry
Measurements and Calculations
(Chapter 5 Notes)
MEASUREMENT
SI System - “Le Systeme International d’Unites”
SI Base Units: (overhead)
Quantity
Name
Symbol
length
mass
time
amount of substance
temperature
meter
kilogram
second
mole
Kelvin
m
kg
s
mol
K
volume
density
concentration
force
pressure
energy
m3 (or L)
kg/m3 (or g/mL)
mol/m3 (or mol/L)
newton
N (m.kg/s2)
pascal
Pa (kg/ms2)
joule
J
(m2.kg/s2)
6.02 X 1023
SI Derived Units:
SI Prefixes:
Prefix
Symbol
kilo
deci
centi
milli
micro
nano
pico
k
d
c
m
u
n
p
Meaning
Example
1000X
0.1X
0.01X
0.001X
1 X 10-6X
1 X 10-9X
1 X 10-12X
1 m = .001 km
1 d = .1 m
1 cm = .01 m
1 mm = .001 m
1 u = 1 X 10-6 m
1 nm = 1 X 10-9m
1 pm = 1 X 10-12m
or
or
or
or
or
or
or
1 km = 1000m
1 m = 10 d
1 m = 100 cm
1 m = 1000 mm
1 m = 1 X 106 u
1 m = 1 X 109 nm
1 m = 1 X 1012 pm
Note: for water,
1000g = 1 kg = 1 dm3 = 1 L (liter)
or,
1 g = 1 cm3 = 1 mL (milliliter)
1
Using Significant Figures in Measurements
What are significant figures?
Here we are not talking about models in bathing suits. We are talking about the
proper way to record a measurement. In recording any measurement, the scale of the
measuring instrument determines the number of sig figs. Your job is to understand how
to correctly read a scale and to record the measurement with the proper number of sig
figs!
Significant figures are the ones we are certain of plus one estimated one. The
figures we are certain of are determined by the scale of our measuring instrument.
If the scale, for example, has its farthest division to the one place, then we are certain of
all digits to the one place and we estimate to the next place. Our estimate will then be in
the tenth place and we would express this as + 0.1 or + 0.5 depending on how precisely
we can estimate (i.e. can we visualize the space between the ones in tenths = + 0.1, or is
the best we can visualize halfway between the ones = + 0.5).
Let’s take an example. Measure the length of the rectangles below using the
rulers provided:
0
1
2
3
4
3
4
A
length = __________ + __________
0
1
2
B
length = __________ + __________
What is meant by the term precision?
Which of the above measurements is more precise? Why?
2
A student looked at the measurement in A and estimated it was one-quarter of the way
past 2 and recording the length in measurement as 2.25? Why is this incorrect?
Problem: Record the length indicated using each of the rulers. Include the uncertainty.
Which ruler is more precise? Why?
3
Significant Figures Review
The accuracy of any measurement made in a chemistry laboratory will depend on
the precision of the instrument used and the accuracy of the worker in using this
instrument. Therefore, when reporting a numerical measurement which has a limit of
accuracy, it is customary to retain one doubtful figure. For example, 7.68 is a number
which contains three significant figures. The first two figures are certain, while the last
one is uncertain. Unless otherwise given, the uncertainty is assumed to be to the 1 place.
Significant figures are all those figures which are certain plus one uncertain one.
7.69 (max)
7.68
assume uncertainty = + 0.01
7.67 (min)
The Box-and-Dot Strategy for Counting Sig Figs
The box-and-dot method consists of three simple steps for determining the significant
figures. The steps must be followed explicitly.
Step 1
Draw a box around all nonzero digits, beginning with the leftmost nonzero digit and
ending with the rightmost nonzero digit in the number.
For example, drawing a box around nonzero digits in the number 0.0123012300 gives
0.0 1230123 00 Any zero(s) trapped or “sandwiched” between nonzero digits will
necessarily be included in the box. For convenience, a digit or number surrounded by a
box may be referred to as a “boxed” digit or “boxed” number, respectively.
Step 2
If a dot (decimal point) is present, draw a box around any trailing zeros.
Continuing with the above example, a dot is present in the expression 0.0123012300;
therefore, trailing zeros are boxed, which gives 0.0 1230123 00
Step 2 uses the term dot in lieu of decimal point for reasons of brevity and ease of recall.
The position of a decimal point within a number is irrelevant – the only test for boxing
trailing zeros is the mere presence of a dot.
Because the method steps are followed explicitly, it is understood that trailing zeros
should not be boxed when a dot is not present. In other words, draw a box around trailing
zeros if and only if a dot is present.
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Step 3
Consider any and all boxed digits significant.
Boxed digits are significant, whereas digits that are not boxed are not significant.
Continuing the example from Step 2, the expression 0.0 1230123 00 reveals nine digits
surrounded by boxes. Therefore, there are nine significant figures. Specifically, the
significant figures are: 1, 2, 3, 0, 1, 2, 3, 0, and 0.
Note that the box-and-dot method does not expressly address leading zeros. There is no
need. Leading zeros, which are never significant, always lie to the left of the box drawn
in Step 1 and are therefore excluded from consideration as significant figures by virtue of
the explicitness of Steps 1 and 2. Those steps provide criteria for boxing (and thus
rendering significant) trapped and trailing zeros only, to the exclusion of all other zeros.
In general, at least one, but not more than two, boxes will be associated with any given
number. Step 1 always requires that a box be drawn. Step 2 only allows for a (second)
box to be drawn when two conditions are met: (i) a decimal point is present, and (ii) one
or more trailing zeros are present.
Let’s try some problems. How many sig figs are in each of the following numbers?
a) 123.01230?
b) 12301230?
c) 123.0123?
d) 10100?
e) 10000.0?
f) 0.00010?
5
The box-and-dot method also works for numbers written in exponential notation. In this
case we ignore the “ten-to-the-n-term”. For example, how many sig figs are in
Avogadro’s number, 6.022 X 1023?
An area of ambiguity involves numbers ending without a decimal point. How would you
write the number five thousand so that it would have four significant figures? Writing the
number as “5000” indicates the presence of one significant figure only. Writing the
number as “5000.” does not follow the accepted convention. The ambiguity can be
removed by expressing the number in scientific notation:
Getting back to the implied uncertainty in measured quantities and their min and max
values, the following serve as an example:
1700
1600
1601
1600.
1500
1600.01
1600.00
1599
1599.00
We will see later that the smaller the differences between the max and min of a
measurement, the more precise (or repeatable) the measurement is.
Numbers Rounded Off
In various calculations, the numerical answers can be rounded off to obtain the
proper number of significant figures. When the number dropped is 5 or greater, the last
significant figure is increased by one. For example, the number 5.166 is changed to 5.17,
so that it contains three significant figures. When the number dropped is less than 5, the
last significant figure remains unchanged. For example, the number 5.164 is changed to
5.16 so that it contains three significant figures.
When adding and subtracting the answer should be rounded off so as to contain
the least accurately known figure as the final one.
Example, add
16.2
111.51
+ 4.853
132.563
=
132.6
The least accurate figure, in the example above, is the first decimal place so that
the answer can have figures only to this place.
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When multiplying or dividing, the answer should be rounded of so as to contain
only as many significant figures as are contained in the least accurate number.
Example, multiply
4.123 x 5.12 = 21.10976 = 21.1 (3 sig figs)
Example, divide 6.1 by 3.16:
6.1/3.16 = 1.93 = 1.9 (2 sig figs)
Standard Notation for Numbers
The numbers obtained in some measurements or calculations may be extremely
small or very large. In order to facilitate the writing of these numbers, they are expressed
as powers of 10. For example, 1,000 can be written as 1 X 103. The exponent "3" means
that the decimal place has been moved 3 places to the left. Another example is 175,000
which can be written 1.75 X 105. In a similar manner, a very small number such as
0.0000560 can be written 5.60 X 10-5. What is done is to move the decimal 5 places to
the right with the exponent becoming negative. In any number so written, the digits
before the power of 10 are considered significant figures. For example, the number 1.03
X 105 contains three significant figures.
Mathematical Calculations with Exponent Numbers
Employ the following rules when using exponential numbers in mathematical operations:
1. If the original number contains a definite number of significant figures, the calculated
number shall also contain the same number of significant figures.
2. Numbers expressed in powers of 10 cannot be added or subtracted directly unless the
powers of 10 are the same. If they are not the same, the numbers can be rewritten so
that the powers of the numbers are the same.
Example, 2.76 X 105 + 2.54 X 106 must be rewritten so both numbers have the same
exponent:
= 2.76 X 105 + 25.4 X 105 = 28.2 X 105
Since the original number contained 3 significant figures, the final calculated value
must also contain 3 significant figures.
3. When numbers with the power of 10 are multiplied, the exponents are added; when
divided, the exponents are subtracted.
Example:
(2.76 X 105) X (2.54 X 106) = (2.76 X 2.54) X (105 + 106) = 7.01 X 1011
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Example:
2.76 X 105/2.54 X 106 = (2.76/2.54) X (105/106) = 1.09 X 10-1
4. When obtaining the square root of numbers with powers of 10, the exponent is divided
by 2; when obtaining the cube root, the exponent is divided by 3.
Example:
The square root of 16 X 108 is 4 X 104.
The cube root of 8 X 109 is 2 X 103.
Dimensional Analysis - Factor Label Method
Almost all the calculations we will be doing in chemistry involve using proportions to
convert from one unit/quantity to another unit/quantity. One proven method for doing
this is called the “factor label” method and it involves a fraction showing an equality
between two units. This fraction is called a conversion factor. For example, in
converting kilograms to grams, the equality is
1 kg = 1000g
and the conversion factor is:
1kg
1000 g
or
1000 g
1kg
Let’s try a simple one step conversion problem:
How many grams are in a 1.50 kilogram sample?
Step 1: write the given quantity and unit:
1.50 kg
Step 2: multiply the given by the conversion factor gives the unit converting to in the
numerator (top) and the unit converting from in the denominator (bottom):
1.50 kg X
1000 g
1kg
Cancel units (dimensional analysis) and do the math:
1.50 kg X
1000 g
1kg
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The equality 1 kg = 1000g is known as a defined equality and does not involve measured
numbers. These are called exact numbers and they do not have a limited number of
significant figures. Therefore, these exact numbers (defined equalities) are not
considered in our calculation of significant figures.
Problem: In France gasoline is sold by the liter. If you wanted 10.0 gallons of gasoline,
how many liters should you ask for? (1 gal = 3.78 L)? How much would this cost in
dollars?
Let’s try a problem involving converting two units.
Problem: The national speed limit is 55.0 mi/hr. What is the speed limit on km/hr? (1 mi
= 1.6 km)
Let’s try a multistep conversion problem.
Problem: The Sun is 92, 000, 000 miles from Earth.
a) How far away is this in centimeters? (use scientific notation)
b) Given that light travels 3.00 X 1010 cm/s, how many seconds foes it take for
light to travel from the Sun to the Earth?
Random and Systematic Error
Precision and Random Errors
Precision refers to how close repeated determinations done the same way are to
each other. We like to be as precise as possible, but this does not always insure accuracy.
We assume that the error between repeated measurements is due to random errors.
Random error (or indeterminate error) means that a measurement has an equal probability
of being low or high. This error occurs when estimating the value of the last digit of a
measurement.
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Let’s consider a series of measurements given below:
Trial
Measurement
1
2
3
4
5
6
4.93
4.83
4.98
4.88
4.96
4.90
What is the best estimate for the value of these measurements? Calculate the average
value:
Average = ____________
The average though, doesn't tell us anything about the uncertainty. The average of
one hundred measurements should have less uncertainty than the average of five
measurements. Therefore, a measure of uncertainty should take into account both the
spread of the data and the number of determinations. This method is called the standard
deviation and is given by s:
n
s
(X
i 1
i
 X )2
n 1
where s = standard deviation from the mean, Xi = members of the set, X = mean, and N =
number of members in the set of data. The symbol  means to sum over the members;
the symbol [ ] means absolute value, so all the differences are positive.
One way to make the calculation of standard deviation less cumbersome is to set up
a data table like the one below:
Trial
1
2
3
4
5
6
Measurement
4.93
4.83
4.98
4.88
4.96
4.90
Average = __________
Deviation from X
(Xi - X)
(Deviation)2
(Xi-X)2
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
Sum of Deviation2 = _______
Standard Deviation(s) =
10
A practical application of standard deviation is that if any individual
measurement deviates by more than two standard deviations from the mean
(average), than that result is not statistically valid and should be discarded. This is
known as the 2s rule.
Random error is only one source of error. The other source of error is called
systematic (or determinate) error. Systematic error causes a shift from the true value and
reduces the accuracy of the result. By making more measurements we can reduce the
random error and increase the precision of our results, but if systematic errors exist the
experimental result will still differ from the true (or accepted) value. Systematic errors
can result from miscalibration of the instruments or some error in the experimental
technique.
Uncertainty in Measurements
Some definitions:
absolute uncertainty (au) - the uncertainty in the measurement (i.e. the + )
relative uncertainty (ru) - (the absolute uncertainty/the measurement) X 100
When adding or subtracting measured quantities, the uncertainty of the sum or difference
is determined by adding the absolute uncertainties of each measurement.
Example:
26.60 + .01 cm
- 20.00 + .01 cm
6.60 + .02 cm
When multiplying or dividing measured quantities, the uncertainty in the product or
quotient is determined by adding the relative uncertainties of each measurement.
Example
12.42 + .01 X 6.35 + .05 = 78.86700 = 78.9 + ?
ru in 12.42 =
.01
X 100 = .08%
12.42
.05
ru in 6.35 = 6.35 X 100 = .8%
.08 + .8 = .88 = .9%
therefore, final answer = 78.9 cm + .9%
and .9% of 78.9 cm = 0.7 cm
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so the answer can also be expressed as 78.9 cm + .7 cm, which is a way of
expressing the maximum and minimum possible values discussed below.
Min & Max Calculations
Another way to express uncertainty in manipulating measured quantities is to
calculate a minimum value and a maximum value based on the absolute uncertainties of
the measurements. The average of the min and max is thus the final answer and the
uncertainty is how much the average differs from the min or the max (the difference
should be the same).
Example: Calculate the density of an object having the following mass and volume:
mass:
volume:
Min density =
=
min mass
max vol
24.62 g + 0.01g
12.7 mL + .1 mL
Max density =
24.61g
12.8mL
max mass
min vol
=
= 1.92 g/mL
24.63 g
12.6mL
= 1.96 g/mL
AVG = 1.94 g/mL + 0.02 g/mL = final answer
Measuring and Calculating the Perimeter of the Index Card
Uncertainty in ruler: + _____ cm
l = __________ + _____
l = __________ + _____
h = _________ + _____
h = _________ + _____
perimeter = _______________ + __________
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Now do a Min and Max calculation:
Min = __________
__________
__________
__________
Max = __________
__________
__________
__________
 =__________
AVG (average) =
=
__________
min  max
, and the + is the difference between the average and the
2
min and the max)
AVG = _______________ + __________
Calculating the Area of the Index Card
l=
__________ + _____
calculation for ru in length measurement:
(from above)
ru = __________ %
h=
__________ + _____
calculation for ru in height measurement:
(from above)
ru = __________ %
area = __________ + _____ %
Now do Min and Max for the area calculation:
min length:
__________
max length:
__________
min height:
__________
max height:
__________
min area:
__________
max area:
__________
AVG = __________ + _____
How does the + from the min/max calculation compare to the relative uncertainty?
13
Error in Measurements – A Summary
Random Error
Systematic Error
this is the +
(or absolute uncertainty)
this is error that
consistently made in
the same direction
equal chance of being
high (+) or low (-)
associated with precision how close measurements are
to each other (i.e. reproducibility)
we measure precision by
calculating standard deviation (s) this takes into account both the
number of measurements and the
spread (high-low) of the data
n
s
(X
i 1
i
may be due to a miscalibrated instrument
associated with accuracy how close your measurement
is to an accepted value
we measure accuracy by
calculating percent error % error = obs - acc X 100
acc
 X )2
n 1
where Xi = individual measurement
X = average
N = number of measurements
Temperature

a measure of the average kinetic energy of the atoms/molecules in a body

the property or condition of a body that determines the direction of heat flow
between it and another body (law: heat flows from hot to cold until thermal
equilibrium)
How do we measure temperature?
14
All thermometers have two fixed points:
How do we convert from one temperature scale to another?
15
A couple of density problems:
1. Given the following data:
Mass
21.58 g + 0.01 g
Volume water in cylinder
Volume water and object
10.0 mL + 0.5 mL
21.5 mL + 0.5 mL
Calculate the density of the object and include the uncertainty.
2. Given the following data:
Mass
Length
Width
Height
9.55 g + 0.01 g
2.00 cm + 0.01 cm
1.00 cm + 0.01 cm
3.60 cm + 0.01 cm
Calculate the density and include the uncertainty.
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