Uncertainty in measurement chemistry

UNCERTAINTY IN MEASUREMENT CHEMISTRY

What is meant by uncertainty in measurement chemistry? Let'us the study in the field of chemistry, one has to deal with experimental data as well as theoretical calculations. There are meaningful ways to handle the numbers conveniently and present the data realistically with certainty to the extent possible.how do you find the uncertainty of a measurement. These ideas  are discussed below in details.
1) Scientific Notation
As chemistry is the study of atoms and
molecules which have extremely low masses
and are present in extremely large numbers,
a chemist has to deal with numbers as large
as 602, 200,000,000,000,000,000,000 for the
molecules of 2 g of hydrogen gas or as small
as 0.00000000000000000000000166 gm mass of a H atom. Similarly other constants such as Planck’s constant, speed of light,charges on particles etc., involve numbers of the above magnitude.It may look funny for a moment to write or count numbers involving so many zeros but it offers a real challenge to do simple mathematical operations of addition subtractions,multiplication or division with such numbers. You can write any two numbers of the above type and try any one of the operations you like to accept the challenge and then you will really appreciate the difficulty in handling such numbers.This problem is solved by using scientific notation for such numbers, i.e., exponential notation in which any number can be represented in the form N × 10n
 where n is an exponent having positive or negative values and N is a number (called digit term) which varies between 1.000... and 9.999....
Thus, we can write 232.508 as 2.32508 ×10^2
 in scientific notation. Note that while writing it, the decimal had to be moved to the left by two places and same is the exponential (2) of 10 in the scientific notation.
Similarly, 0.00016 can be written as
1.6 × 10^–4. Here the decimal has to be moved
four places to the right and ( – 4) is the exponent
in the scientific notation.Now, for performing mathematical operations on numbers expressed in scientific notation s, the following points are to be kept in mind, measurement uncertainty calculations how we find out let's see below

Multiplication and Division
These two operations follow the same rules

which are there for exponential numbers, i.e
(5.6x10^5)X(6.9x10^8)=(5.6x6.9)(10^5+8)
                                        =(5.6x6.9)10^13
                                       =38.64 X 10^13
                                      =3.864 X 10^14
(9.8 X 10^-2) X (2.5x10^-6)=(9.8 X 2.5 )( 10^-2+{-6})
                                              =(9.8x2.5)10^{-2-6}
                                             = 24.50 X 10^-8
                                             =2.450 X 10^-7
(2.7x 10^-3)/(5.5x10^4)=(2.7x5.5)10^-3-4
                                        = 0.4909 X 10^-7
                                        = 4.909 X 10^-8                                             
Addition and Subtraction
These two operations follow the same rules
which are there for exponential numbers, i.e.
Addition and Subtraction
For these two operations, first the numbers are
written in such a way that they have same
exponent. After that, the coefficient are added
or subtracted as the case may be.
Thus, for adding 6.65 × 10^4 and 8.95 × 10^3
(6.65 × 10^4) + (0.895 × 10^4) exponent is made
same for both the numbers.
Then, these numbers can be added as follows
(6.65 + 0.895) × 10^4 = 7.545 × 10^4
Similarly, the subtraction of two numbers can
be done as shown below :
(2.5 × 10^–2) –( 4.8 × 10^–3)
= (2.5 × 10^–2) – (0.48 × 10^–2)
= (2.5 – 0.48) × 10^–2 = 2.02 × 10^–2

2) Significant Figures
Every experimental measurement has some
amount of uncertainty associated with it.
However, one would always like the results to
be precise and accurate. Precision and
accuracy are often referred to while we talk
about the measurement. Precision refers to the closeness of various measurements for the same quantity. However,accuracy is the agreement of a particular value to the true value of the result. For example, if the true value for a result is 2.00 g and a students ‘A’ takes two measurements and
reports the results as 1.95 g and 1.93 g. These
values are precise as they are close to each
other but are not accurate. Another student
repeats the experiment and obtains 1.94 g and
2.05 g as the results for two measurements.
These observations are neither precise nor
accurate. When a third student repeats these
measurements and reports 2.01g and 1.99g
as the result. These values are both precise and
accurate. This can be more clearly understood
from the data given see below
                               1.              2.                 AVG.(gm) 

Student A.          1.95         1.93                1.9940
Student B.           1.94        2.05               1.9950
Student C.           2.01        1.99               2.0000

The uncertainty in the experimental or the
calculated values is indicated by mentioning
the number of significant figures. Significant
figures are meaningful digits which are known
with certainty. The uncertainty is indicated by
writing the certain digits and the last uncertain
digit. Thus, if we write a result as 11.2 mL, we
say the 11 is certain and 2 is uncertain and
the uncertainty would be +1 in the last digit.
Unless otherwise stated, an uncertainty of +1
in the last digit is always understood.
There are certain rules for determining the
number of significant figures. These are stated
below:
(1) All non-zero digits are significant. For
example in 285 cm, there are three
significant figures and in 0.25 mL, there
are two significant figures.
(2) Zeros preceding to first non-zero digit are
not significant. Such zero indicates the
position of decimal point.
Thus, 0.03 has one significant figure and
0.0052 has two significant figures.
(3) Zeros between two non-zero digits are
significant. Thus, 2.005 has four significant
figures.
(4) Zeros at the end or right of a number are
significant provided they are on the right
side of the decimal point. For example,
0.200 g has three significant figures.
But, if otherwise, the terminal zeros are not
significant if there is no decimal point. For
example, 100 has only one significant
figure, but 100. has three significant
figures and 100.0 has four significant
figures. Such numbers are better
represented in scientific notation. We can
express the number 100 as 1×10^2
 for one significant figure, 1.0×10^2 for two
significant figures and 1.00×10^2 for three
significant figures.
(5) Counting numbers of objects, for example,
2 balls or 20 eggs, have infinite significant
figures as these are exact numbers and can
be represented by writing infinite number
of zeros after placing a decimal i.e.,
2 = 2.000000 or 20 = 20.000000
In numbers written in scientific notation,
all digits are significant e.g., 4.01×10^2
 has three significant figures, and 8.256 × 10^–3 has four significant figures.
Addition and Subtraction of Significant
Figures
The result cannot have more digits to the right
of the decimal point than either of the original
numbers.
                         12.11
                         18.0
                     +    1.012
 Total               31.122
Here, 18.0 has only one digit after the decimal
point and the result should be reported only
up to one digit after the decimal point which
is 31.1

Multiplication and Division of Significant
Figures
In these operations, the result must be reported
with no more significant figures as are there in
the measurement with the few significant
figures.
2.5×1.25 = 3.125
Since 2.5 has two significant figures, the
result should not have more than two
significant figures, thus, it is 3.1.
While limiting the result to the required
number of significant figures as done in the
above mathematical operation, one has to keep
in mind the following points for rounding off
the numbers
1). If the rightmost digit to be removed is more
than 5, the preceding number is increased
by one. for example, 1.386
If we have to remove 6, we have to round it
to 1.39
2). If the rightmost digit to be removed is less
than 5, the preceding number is not changed.
For example, 4.334 if 4 is to be removed,
then the result is rounded upto 4.33.
3). If the rightmost digit to be removed is 5,
then the preceding number is not changed
if it is an even number but it is increased
by one if it is an odd number. For example,
if 6.35 is to be rounded by removing 5, we
have to increase 3 to 4 giving 6.4 as the
result. However, if 6.25 is to be rounded
off it is rounded off to 6.2.

3) Dimensional Analysis
Often while calculating, there is a need to
convert units from one system to other. The
method used to accomplish this is called factor
label method or unit factor method or
dimensional analysis. This is illustrated
below.
Example
A piece of metal is 3 inch (represented by in)
long. What is its length in cm?
We know that 1 in = 2.54 cm
From this equivalence, we can write
(1 in / 2.54cm) =1=(2.54cm /1 in)
thus
(1 in / 2.54cm) equals 1 and (2.54cm /1 in) also
equals 1. Both of these are called unit factors.
If some number is multiplied by these unit
factors (i.e. 1), it will not be affected otherwise.
Say, the 3 in given above is multiplied by
the unit factor. So,
3 in = 3 in × (2.54cm/1 in) = 3 × 2.54 cm = 7.62 cm
Now the unit factor by which multiplication
is to be done is that unit factor [(2.54cm/1 in) in
the above case] which gives the desired units
i.e., the numerator should have that part which
is required in the desired result.It should also be noted in the above example that units can be handled just like other numerical part. It can be cancelled,divided, multiplied, squared etc. Let us study one more example for it.
Example
A jug contains 2L of milk. Calculate the volume
of the milk in m3
Since 1 L = 1000 cm3
and 1m = 100 cm which gives
(1 m / 100cm)=1=(100 cm / 1m )
To get m3
 from the above unit factors, the
first unit factor is taken and it is cubed.
 ( 1m / 100cm )^3=1m3 / 10^6 cm3
 = (1)^3=1
Now 2 L = 2×1000 cm3
The above is multiplied by the unit factor
(2 x 1000 cm )^3 X ( 1m3/10^6 cm3 )
=(2 m3 / 10 m3 )  = 2 x 10^-3 m3