MEASUREMENTS, UNITS
&
CALCULATIONS
Chemistry is a quantitative science:
The outcome of an experiment depends on amount of
substances taken, temperature, pressure, & duration of the
reaction.
QUALITATIVE vs. QUANTITATIVE Properties
Qualitative: color, smell, ability to undergo chemical transformation of a certain kind.
Quantitative: properties that are expressed using number & unit: mass, volume,
temperature, time, distance, etc.
Often properties previously considered as qualitative only, are then characterized as
quantitative, e.g. color is related to spectrum that is expressed in numbers & units.
Properties to be measured most often:
MASS, m
SIZE as volume, V
TEMPERATURE,
toC or T, K
TIME, t
To perform a measurement, one need an
INSTRUMENT & a STANDARD
e.g. for mass: a balance & a standard load of 1 kg (Pt cube kept in Sevres, in France, to
which all other standards are compared)
for distance: a ruler, caliper, etc. & a standard meter;
for time: a clock & a duration of some physical process for which we know
that it is always the same;
for temperature: a thermometer & some standard process, such as ice melting (taken
as 0oC) & water boiling (taken as 100oC).
The result of a measurement is expressed by a
NUMBER & a UNIT:
mass of body: 188 lb or 84 kg
Thus, we need to know how to deal with numbers & units, according to certain rules
providing reliable representation of the results of measurements & calculations.
SIGNIFICANT DIGITS
IN THE RESULTS OF DIRECT
MEASUREMENTS
Uncertainty in Measurements
- All scientific measurements are uncertain, i.e. subject to error.
- Due to errors, two successive measurements of the same quantity are
different.
- These errors are reflected in the number of figures (digits) reported for the
measurement.
SIGNIFICANT DIGITS IN THE RESULTS OF DIRECT MEASUREMENTS
All digits in which we are certain, & the next one, which is known tentatively
(estimated) are significant
Example: Measuring volume with a graded beaker.
Minor grade is 5 mL.
The volume may be
estimated as 47 mL,
with 2 sd
In 2 digits we are certain, the
3rd one is an estimation
Measuring volume with a graded cylinder _
Minor grade is 1 mL.
Estimated volume is 27.0 mL
3 s.d.
Here estimated
volume is 36.5 mL
30
25
20
15
10
5
SIGNIFICANT DIGITS in DATA GIVEN:
Non-zero digits are always significant.
22 has 2 significant digits,
0.2234 - 4 s. d.
22.3 has 3 s. d.
0.00234 – 3 s. d.
Numbers with zeroes:
Zeroes before all other digits are not significant;
0.046 has 2 s. d.
45.26 ft - 4 s.d.
Zeroes between other, non-zero, digits are always significant;
4009 kg 4 s. d.
35.054 5 s. d.
Zeroes after other digits but behind a decimal point are significant;
7.90 has 3 s. d.
8.20  103 has 3 s. d.
8.200  103 has 4 s. d.
8.2  103 has 2 s. d.
It is impossible to tell if zeroes are significant when they stand at the
end of a number with no decimal point. Apply your common sense.
8200: it is not clear if the zeroes are significant or not. The number of
significant digits in 8200 is at least two, but could be three or four.
USA population is estimated as 295 734 000. Obviously, it cannot be known exactly (due
to continuous migrations, birth & death). Most likely, there are 6 s.d.
Integer numbers & coefficients given by definition have infinite number of s. d.
e.g. 1 ft = 12 inches
1 km = 1000 m are considered definition, hence infinite
number of s.d.
SIGNIFICANT DIGITS IN CALCULATIONS
Basic principle: information cannot be created by calculations. If some quantity is
known not precisely, this lack of precision will propagate through calculations.
1. For multiplication & division, the number of SD in the result is the same as that in
the number with the smallest number of SD
27.9/3.0 = 9.3 (not 9.30 !)
160  2.16 = [345.6] = 350
Sometimes, to indicate that the last zero is significant, the number is terminated by a
dot:
160.  2.16 = [345.6] = 346
2. For addition & subtraction, the number of digits after decimal point in the result is
the number of decimal places (i.e. after decimal point) in the term with the smallest
number of decimal places:
2.345 + 33.11 + 210.2
=
245.7
2100 + 0.45
=
2100 (sic!)
3
2.100  10 + 1.56 = [2101.56] = 2102
ROUND OFF
If the digit to be removed is <5 (midway pnt), the preceding digit stays the same.
If it is >5 (even in the next digit), the preceding digit is increased by 1.
If it is = 5, the agreement is to go to the closest even number (up or down):
to 3 SD:
9.374 = 9.37;
4.345 = 4.34
4.3451 = 4.35;
4.355 = 4.36
in math, always, 2000 + 1 = 2001
in science, sometimes it is still 2000!
SCIENTIFIC NOTATION
is based on powers of the base number 10, i.e.
100 =1
(true for any number, not only 10: 20 =1, 450 =1, etc.)
101 = 10
10-1 = 0.1
102 = 100
10-2 = 0.01
103 = 1000
10-3 = 0.001
……………
……………..
Sci. notation is used to present very large (approx. >1000) & very small (approx
<0.001) numbers.
The number 123,000,000,000 in scientific notation is written as:
1.231011
The first number 1.23 is the coefficient. It must be >1 but < 10.
The second number, 10, is the base.
In the number 1.23  1011 the number 11 is referred to as the exponent or power of
ten.
Other notations: The number 123,000,000,000 can also be written as:
1.23E+11
0.00000683 = 6.83  10-6
or as
6.83E-6
1.23  10^11
6.83  10^(-6)
Rules for Multiplication/Division in Scientific Notation:
- Multiply the coefficients
- Add the exponents (subtract if negative)
(3  104) (2  105) = 6  109
Examples:
If the coefficient is > 10, it must be converted:
(6  104) (2  105) = 12  109 = 1.2 101  109 = 1.2  1010
The number of particles in a mole is NAv = 6.0231023;
in 2 moles: 26.0231023 = 12.051023= 1.2051024.
Mass of one hydrogen atom in gram is:
1/NAv = 1/6.0231023 = 0.1660 10-23 = 1.660  10-1  10-23 = 1.660  10-24 g
(6 x 106) (2x 103) (2x 103)
________________________________
______4 x
10 4
6x2x2
=
________________
106+3+3-4 = 6 x 108
4
Rules for Addition/Subtraction in Scientific Notation:
Before adding/subtracting, the numbers must be presented as the same
power of 10. Then, the coefficients are added/subtracted, & the power of 10
does not change.
Example:
1.73104 + 9.83103 =17.3103 +9.83103 = (17.3+9.83)103 = 27.13103= 2.7  104
Note:
if a very small number is added to a very large one, the large one does not change:
1.73 104 + 9.83 10-3 = 1.73 104
Power to power in scientific notations: (ax) y = axy
(102)3 = 106
Example: what is the volume (in cm3) of a sodium atom with its atomic radius of
1.9  10-8cm?
V = (4/3)r3 = (4/3)  3.14 1.9 10-8)3 = 1.33  3.14  (1.9)310-8)3 =
= 28.64  10-24 = 2.9  10-23 cm3
UNI T S
In science, metric International System of
units is used (SI).
Its basic
units are those of:
Length (meter)
Mass (kilogram)
Time (second)
Temperature (K)
SI is a metric system.
Other units of length, mass & time are
n
produced using decimal factors, 10 ,
integer n, & are named using Greek
numeral prefixes.
deci for 10-1:
1dm = 10-1m
centi 10-2:
1cm = 10-2m
milli 10-3:
1mg = 10-3g
micro 10-6:
1s = 10-6s
nano 10-9:
1nm = 10-9m
kilo
103:
1km = 103m
mega 106:
1MW=106 W (Watt)
pico
10-12:
1ps = 10-12s
femto
10-15: 1fs = 10-15s
Length, mass, temperature & time are
considered as basic properties, & their
units are considered as basic, or simple.
Others are DERIVATIVE or complex units:
Volume of a cube is l3, hence V[m3].
In chemistry, smaller unit is used:
1 L = 1 dm3 = 1000 mL,
1 mL = 1cm3
Example: what is the volume (in cm3) of a sodium atom with its atomic radius of
1.9  10-8cm?
V = (4/3)r3 = (4/3)  3.14 1.9 10-8)3 = 1.33  3.14  (1.9)310-8)3 =
= 28.64  10-24 = 2.9  10-23 cm3
The choice of units is a matter of
convenience:
Density, D = m/V,
hence, D[kg/L] = g/mL
-used for condensed matter
or D[g/L], for gases
transp
Densities of some materials:
Liquids & Solids, D[g/mL]
Dry Wood
Ethanol
Vegetable oil
Water
Sugar
Magnesium
Salt
Lead
Gases D[g/L] (standard temp & pressure)
Hydrogen
Helium
Methane
Neon
CO2
Air
Oxygen
HCl
0.512
0.789
0.91
1.00
1.59
1.74
2.16
11.34
Speed, l/t [m/s] or [km/hr]
Three temperature scales: Celsius, Kelvin
& Fahrenheit.
temperature
kelvins degrees Celsius degrees Fahrenheit
symbol
K
°C
°F
boiling point of water
373
100.
212.
melting point of ice
273
0.
32.
absolute zero
0
-273
-459
Temperature reflects the internal motion of particles. T = 0 K is temperature at
which all motion stops. This is the absolute zero of T, K (it cannot be negative!)
The value of |1K| = |1°C|, but these two scales are shifted against each other by 273:
T, K = t°C + 273; or
t°C = T K – 273
The interval between water freezing & boiling is 100°C, but 212-32 = 180°F,
Hence the value of |1°C| = |1.8°F|, or 9/5 °F & conversion rule is:
t°C = (5/9)(t°F - 32) [roughly: ½ (t°F - 32)] & t°F = (9/5) t°C + 32 [roughly: 2t°F +
32)]
Example: At what temperature its values in °F & in °C coincide?
t°F = (9/5) t°C + 32
t = (9/5)t + 32
t = -40, i.e. t = -40°F = -40°C
UNIT CONVERSION
Example:
Speed 110. km/hr. What is this speed in mi/hr & m/s?
From the table: 1 mi = 1.61 km
Also, we know: 1 hr = 60 min = 6060 =3600 s.
It is always possible to 1
(UNIT FACTOR METHOD).
from 1 mi = 1.61km, it follows:
1.61km /1mi = 1
Speed(mi/hr) = (110 km/hr)  (1mi/1.61km) = 68.3mi/hr
Speed(m/s)=(110 km/hr)(103m/km)(1hr/3600s) = 30.6m/s
At unit conversion, the number of s.d. does not change.
“rails” may help in formatting those calculations:
110 km 103m
1 hr
Speed(m/s) =
= 30.6 m/s
hr
1 km 3600 s
ACCURACY & PRECISION
ACCURACY
refers to the closeness of a measurement to the true or accepted value of the
quantity measured.
PRECISION
refers to the agreement between the numerical values of a set of measurements of
the same quantity made in the same way.
PERCENT ERROR =
|Value accepted - Value obtained| 100%
Value accepted
Not accurate
not precise
Accurate
but not precise
Precise
but not accurate
Accurate
& precise
Example:
Aluminum density, from a handbook, is D = 2.71 g/mL
In an experiment, you got 2.63 g/mL. What is the percent error?
(2.71 - 2.63) / 2.71  100% = 3.0%
This is the measure of your accuracy.
The result is accurate within 3%.
The results my be accurate but not precise, & vice versa.
Example: In 4 trials, the aluminum density was found as 2.63, 2.82, 2.79 & 2.51.
What is the precision of these measurements?
1.
Calculate the mean, Dm:
Dm = (2.63+2.82+2.79+2.51)/4  2.68 g/mL
2. Deviations from the Dm are:
D1 = |2.63-2.68| = 0.05
[absolute values are taken]
D2 = |2.82-2.68| = 0.14
D3 = |2.79-2.68| = 0.11
D4 = |2.51-2.68| = 0.17
3. Mean deviation: Dm = (0.05+0.14+0.11+0.17)/4  0.12
The result may be presented as:
D = Dm ± Dm = (2.68 ± 0.12) g/mL
Dm is the measure of precision.
It may be presented as percent deviation:
(Dm/Dm).100% = (0.12/2.68)  100% = 4.5%
What is the accuracy of the above set of data?
Handbook data: D(Al) = 2.71 g/mL
The mean value of 4 measurements was Dm = 2.68 g/mL
Percent error = |2.68-2.71|/2.71100% = 1.1%