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Measurements
Measurement is the process by which we report on the existing amount of
matter and energy in a given system. It is very important to define the nature of the of
the system under consideration, so the expression of the language we use should
correspond to the real situation of this system. The measurement we carry out is to
report the real fact of the system in using numbers with specific units. The used
units are related to specific dimensions which are Mass (m), Length (L), Amount of
matter (n), Time (t) , Temperature and Energy (E). The use of one or more of
these dimensions depends on the nature of nature of the system and the required
information to report. For instance, if we talk about the population of a city or a
country we use numbers of people, not mass or length. This applies to any living
systems or individual structures such as cars, houses, atoms, molecules….In other
words we do not talk about masses in kilogram or lengths in meter of houses, cars but
about how many of these are present in a town or parking lot.
Scale of numbers of unit matter
Let us start by exactly defining the subject under investigation. Human being
is one of the living systems defined by a mass in kilogram and length in meter is
active part in doing the measurement on other units of matter. As mentioned
previously, the basic unit of matter is an atom. Atoms form molecules present in our
daily life in form of substances and compounds. As a matter of fact, atoms are
permanent structures and do not change as function of time, while humans as well as
other living systems are made of these atoms and have limited life time, then are
transformed to atoms and molecules. Based on this basic fact, it is essential to have a
clear idea about the dimension of an atom and a molecule as compared to our
dimensions. This question must be divided into two parts:
1. Atoms and molecules deal with each other as structures having the same
dimensions. For example, a hydrogen atom reacts with another hydrogen atom
to produce a hydrogen molecule. This operation consists of number 1 and 1
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atoms to yield 1 molecule, and so on. It does not deal with masses or
lengths…The same logic applies for humans. All relations and actions
,regardless of its kind, take place between individuals; numbers, not masses
or other dimensions.
2. The relation between basic units of matter: Atoms and molecules and human
being as individuals, especially that human being is the active part in
performing the measurement. In order to understand this process, we consider
the following example.
Example: i- The mass of on hydrogen atom is = 1 unit mass in its dimension =
1.67 x 10-24 g in our language of mass.
ii- If we have one gram of hydrogen atoms. This quantity is equal to
6.02 x 10 23 atoms
iii- Suppose that we want to count the number of these atoms which is the
right thing to do. If we count one atom per second non stop, then we need : The
number of seconds in one year = 365 day x 24 h/day x 3600 s/h
= 31536000 seconds or atoms counted per year.
Number of years = 6.02 x 10 23 atoms / (31536000 s/year) = 1.9 x 1016 years
=19000000000000000 years (long time)
In other words, it is impossible to account exactly the number of atoms in a small
mass of one gram. Since it is impossible to count all the atoms, then we measure
this quantity by weight. Even so, it is impossible to report exactly the real mass of
all the atoms present in this sample. Simply because we do not have a common
instrument of measuring the mass of all the atoms with an accuracy of one single
atom of the order of 10-24 g.
iii- From the above example in reporting the quantity of matter in a system,
we have to use different measuring devises which yield approximate numbers
about it. The accuracy of the reported data is related to the sensitivity of the
measuring instrument. In all cases, it will never by exact . The question is how
these measurements are close to the real number of unit matter in the system.
In conclusion, it is very important to be aware that no matter how good
our measurements are, it is always approximate. Consequently, the main
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question will always be how close the approximate measurement from the reality
and how to achieve it.
In reporting measurements, two information (terms) concerning these
measurements are encountered: Precision and Accuracy.
Precision: Signifies how different measurements using the same instrument
are close to each other.
Example: The reported weights of one sample using a balance are: 2.49 g, 2.48 g,
2.50 g, 2.49 g. These measurements are called precise.
Accuracy:
Signifies how much different measurements using different
instruments are close to the real value.
Example. Suppose that the real mass of a sample is 2.70 g. Different balances
were employed and the measured values are: 2.69 g, 2.71 g, 2.68 g, 2.70 g, 2.69 g.
Note. Precision does not necessary mean accurate. Suppose that one of the balances
was not calibrated correctly and gave the values: 2.49 g, 2.48 g, 2.50 g, 2.49 g. It
could be said in this case that the measurements are precise but not accurate.
In order to report measured data as close as possible to the real value,
different instruments should be used.
In order to perform a measurement such as weight, temperature, pressure
length dimensions measuring instruments should be used. In order for the reported
measurement to be as close as possible to the real value in question, the measurement
should be precise and accurate as mentioned above. Despite that, there is always an
error in the conducted measurement. In this respect and in order to report
experimental data which cover the real value, the given number should reflect the
scale devise of the used instrument. For example if a balance has a scale in grams,
the reported number should be in gram (g). If the scale in milligram, then the reported
number should be in milligram (mg) and so on. In all these measurements, the last
digit is uncertain and should be included in the reported data. This is called
significant figures.
Example. Suppose that the real mass of a system is: 56.442324 g, and we have three
balances of different scales (sensitivity). The first one (balance 1) has a scale in
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grams, the second centi-gram (1/100 g) and the third one in milligram (mg). The
reported measures using the three balances should be:
Balance 1: 56 ± 1 g. This means that the real value is between (55-57) g, which is
true. The error in this case in units of gram
Balance 2:
56.44 ± 0.01 g. This means that the real value is between (56.43-56.45)
g, which is true. The error in this case in units of centi- gram.
Balance 3:
56.442 ± 0.001 g. This means that the real value is between (56.441-
56.443) g, which is true. The error in this case in units of milli- gram (mg).
From the above results, it is concluded that the reported data reflect the
sensitivity of the used instrument by considering the last digit is uncertain. This
underline the magnitude of the error (uncertainty) in the available data. In other
words, we can recognize the scale of the measuring instrument from the reported
number. Thus 56 ± 1 g signifies that the scale of the balance is in gram, while the
scale in the case of 56.442 ± 0.001 g, is in mg (more sensitive).
The number of significant figures (numbers) in approximate calculations are
counted as follows:
1. All numbers are counted as significant figures. However, if the first digit is
preceded by zeros on the left (mathematical transformation)*, these zeros are
not counted. The zeros on the right side (measurement unit)** are counted.
* suppose we have a value of 1 mg. This means that the scale (sensitivity ) of the
balance is in mg. This digit could written as 1 mg = 0.001 g . In both equality of
the equation, the sensitivity of the balance is in mg.
** In another measurement, the mass of a sample is 10 mg. In this case, the zero is
a digit of the measurement (significant).
Examples:
Number
Significant figures
2
1
2.0
2
6.0010
5
0.00212
3
0.002120
4
4
1.002120
7
2. The last digit in any number is always uncertain and the real number should be
written as follows:
6.0 ± 0.1
2.102 ± 0.001
5.00001 ± 0.00001
3. As the number of digits in any measurement is large, the measurement is more
sensitive ( more sensitive instrument).
Examples: If we take the three measurement of a given length using different rules:
Measurement (rule)
Value (m)
Error
1
0.6 ± 0.1
2
0.61± 0.01
1/61 = 0.016 (1.6 %)
3
0.610± 0.001
1/610 = 0.0016 (0.16 %)
1/6 = 0.17
(17%)
Obviously, the error in measurement 3 is the smallest because the rule is more
sensitive.
Numbers Approximation
In performing different computations on approximate (experimental data)
numbers, the numbers should be rounded if they are followed by a digit greater than
or equal to 5.
Examples:
1. Approximate the following numbers 5.16 , 4.256, 3.7445 to two significant
numbers each.
Solution:
5.16 = 5.2,
4.256 = 4.3 ,
2. Approximate the following numbers:
3.7445 = 3.7
7.0107 , 3.00454, 2.164436 to
four significant numbers each.
Solution:
7.0107 = 7.011,
3.00454 = 3.005 ,
5
2.164436 = 2.164
Scientific Notation
In order to avoid writing very large or very small numbers, it is more
convenient to write these numbers as a product of a number between 1 and 9
multiplied by 10 raised to some power. Numbers written in this manner are said to be
expressed in scientific notation.
Examples:
Number
Scientific notation
Significant figures
2000
2.000 x 103
4
310.789
3.10789 x 102
6
2001.56
2.000156 x 103
7
0.001
1 x 10-3
1
0.010030
1.0030 x 10-2
5
Computation using significant figures:
There are specific rules to apply when computations were carried out so the
results of these calculations well have the order of error to the component having the
largest error.
1. Addition and Subtraction
"The error (uncertainty) of the result of an addition or subtraction should have the
same error (uncertainty) as the component (term) having the largest error".
In other words, it have the same digits after the point as the least one in the
calculation.
Examples:
1. addition
4.15 + 0.1012 = 4.2512 = 4.25
± 0.01
2. Subtraction
Uncertainty
± 0.0001 the result should be of the largest error ± 0.01
25.0102 - 10.0 = 15.0102 = 15.0
± 0.0001
± 0.1
± 0.1
2. Multiplication and Division
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"For division or multiplication, the quotient or product should not contain more
significant figures than the least term in the calculation".
Examples:
1.
2.56 x 4.0 = 10.24 = 10
Significant figs.
2.
3
2
2
3.456 x 7.8 = 26.9568 = 27
Significant figs.
4
2
2
41.7 ÷ 22.124 = 1.88483 = 1.88
3.
Significant figs.
3
5
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Exact Numbers
In performing calculations, if it is required to transform numbers from one type of
units to another such as gram to kilogram or meter to kilometer these are called exact
numbers and do not follow rules of approximate numbers and significant numbers.
Examples:
1. Transform the following numbers from grams to kilograms:
i- 5 g
ii- 500000 g iii- 10 g
solution: i- 5 g ÷ 1000g /kg = 0.005 kg = 5 x 10 -3 kg
ii- 500000 g ÷ 1000g /kg = 500 kg = 5 x 102 kg
iii- 10 g ÷ 1000g /kg = 0.01 kg = 1 x 10-2 kg
2. Transform the following numbers from kilograms to grams :
i- 5 kg
ii- 1.01 kg
iii- 2.0785 kg
solution: i- 5 kg x 1000 g /kg = 5000 g
ii- 1.01 kg x 1000 g /kg = 1010 g
iii- 2.0785 kg x 1000 g /kg = 2078.5 g
Mathematical Operations
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1. Powers
(Xa)b = Xab
Example.
(102)3 = 106
2. Roots
(Xa)1/b = Xa/b
Example
(102)1/3 = 102/3
Example
103/102 = 103-2 = 10
Xa / Xb = Xa-b
3. Division
4. Multiplication
Xa . Xb = Xa+b
Example
103 x 102= 103+2 =105
5. Addition and subtraction:
In the case of additions or subtractions operations of terms having exponential
numbers, the exponents (raised powers) should be unified and similar to the largest
exponent.
Examples:
1. Perform the following calculation: 5 x 103 + 4 x 104
Solution. Before doing the calculations, the exponents should be unified:
0.5 x 104 + 4 x 104 = 4.5 x 104
2. Perform the following calculation: 2 x 10-5 + 3 x 10-3
Solution.
0.02 x 10-3 + 3 x 10-3 = 3.02 x 10-3
Measurements Dimensional Units: International Symbols for Units
A coherent system of unites is a system based on a certain set of "basic
units" from all "derived unites" are obtained by multiplication or division
with out introducing numerical factors: The three metric systems used in
science are:
1. The cgs system. This stands for: centimeter (cm),gram(g),second(s).
This is a coherent system of units based on three basic units for the
basic quantities: length (cm), mass (g) and time (s).
2. The mks system. This stands for meter (m), kilogram (kg), second(s).
Here the three basic units are again length (m), mass (kg) and time(s).
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3. SI-International system of units. This is basically the mks system
with certain improvements.
IN 1960, The international conference of weights and measures
adopted and recommended the use of the SI system of units by all
scientists.
Table (2-1) The SI system of unites
Physical
SI units
quantity
1-Basic units
Name of unit
Symbol
Length
meter
m
Mass
kilogram
kg
Time
second
s
temperature
degree Kelvin
K
mole
mol
Amount of
substance
2-Derived Units
Force
Newton (N)
kg m s-2
Area
square metre
m2
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Volume
cubic metre
m3
Density
kilogram per
kg m-3
cubic metre
Velocity
metre per
m s-1
second
Acceleration
metre per
m s-2
second square
Pressure
Newton per
N m-2 =
square meter;
Kg m-1 s-2
pascal (Pa)
Work, energy
joule ( J )
Nm=
kg m2 s-2
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Derived Units
Quite often, units which are of the dimension atoms and molecules (Atomic scale)
are employed. This is practical in a sense that small numbers are used.
1 Angstrom (Å) = 1x10-10 m
1 electron volt (eV) = 1.602 x10-19 J
Electron volt is the energy received by an electron when it is accelerated by one volt.
The charge of the electron =1.602 x10-19 Coulomb
So, 1eV = 1.602 x10-19 C x 1V = 1.602 x10-19 J
Coulomb x Volt = Joule
EXERCISES
1. How many significant figures are there in the following numbers?
A. 0.0100210, B. 1.00120, C. 0.00020, D. 100.002, E. 10.000, F. 0.00300
2. Express the following measurement numbers in scientific notation. How many
significant figures does each number contain?
a. 0.00066
b. 0.000660
c. 0.00102030
e. 114.809
f. 896500000 g. 17000.12
3. Perform the following computations:
a. 325.12 - 7.8432
b. 0.015 - 0.009
c. 0.029 - 0.01
d. 14.2854 - 7.815
e. 4.005 - 2.010
4. Perform the following computations:
a. 6.050 + 14.001
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d. 0.0000020
b. 0.0213 + 0.40
c. 789.0 + 1.5987
d. 432 + 1.3980
e. 0.0010 + 2.01
5. Perform the following computations
a. 18.910 ÷ 7.8
b. 2.01 ÷ 1.5
c. 979 ÷ 9.08
d. 8543 ÷ 1.2
e. 17.000 ÷ 1.53241
6. Perform the following computations
a. 3.4 x 4.05
b. 18.9 x 0.016
c. 7.020 x 3.01
d. 24.6 x 695
e. 0.146 x 0.0132
7. Perform the following computations
a. (8.05 x106) + (9.65 x 106)
b. (9.77 x 104) + (3.01 x 105)
c. (8.09 x 105) - (7.01 x 104)
d. (1.645 x 107) - (8.9 x 106)
e. (9.98 x 103) - (7.85 x 102)
f. (4.05 x 10-3) - (5.1 x 10-4)
g. (2.010 x 10-4) + (5.10 x 10-3)
8. Perform the following computations
a. (315.2 + 41.527) + (92.346 - 78.05)
b. (7.081 ÷ 3.02) + (6.001 x 3.0)
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c. (5.003 x 6.01) ÷ (18.04 x 0.0010)
d. (7.00030 x 3.101) ÷ (295.008 – 227.15)
e. (5.02 x 10-3 ÷ 3.0 x 10-4) x (2.097 x 105 + 9.811 x 104)
9. Which of the following operations concerning the weight of a sample is:
Precise, Accurate, Precise and Accurate, meaningless
a. The following masses were obtained using one balance: 5.3 g, 4.8 g,
6.2 g
(
)
b. The following masses were obtained using one balance: 5.3 g, 5.1 g,
5. 2 g, 5.3 g
c.
(
The following masses were obtained using two balances: 4.9 g, 5.3 g,
5.1 g, 4.8 g, 5.4 g
d.
)
(
)
The following masses were obtained using one balance: 5.1 g, 5.2 g,
5.1 g, 5.3 g, 5.0 g
(
)
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ANSWERS
1. A. (6), B. (6), C. (2), D. (6), E. (5), F. (3)
2.
Question
Number
Scientific Notation
Significant Figures
a
0.00066
6.6 x 10-4
2
b
0.000660
6.60 x 10-4
3
c
0.00102030
1.02030 x 10-6
6
d
0.0000020
2.0 x 10-6
2
e
114.809
1.14809 x 102
6
f
896500000
8.9650000 x 108
9
g
17000.12
1.700012 x 104
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3. a. 317.28
b. 0.006 (6 x 10-3)
4. a. 20.051
b. 0.42
c. 790.6
c. 0.02
d. 433
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d. 6.470
e. 2.01
e. 1.995
5. a. 2.4
b. 1.3
c. 108
6. a. 1.4
b. 0.3
c. 21.1
7. a. 1.77 x 107
8. a. 371.1
b. 3.99 x 105
b. 20
9. a. meaningless
d. 7.1 x 103
d. 1.71 x 104
c. 7.39 x 105
c. 1.7 x 103
b. precise
e. 11.094
e. 1.93 x 10-3
d. 7.6 x 106
d. 0.3199
c. accurate
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e. 9.2 x 103
e. 5.2 x 106
d. precise and accurate