Scientific Notation, Significant Figures & Dimensional Analysis - Notes | Study Chemistry Class 11 - NEET

Introduction

Many times in the study 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 like:

• Scientific Notation
• Significant Figures
• Dimensional Analysis

Scientific Notation

• Scientific Notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
• In which any number can be represented in form N × 10ⁿ (where n is an exponent having positive or negative values and N can vary between 1 to 10).

Example: We can write 232.508 as 2.32508 × 10² in scientific notation. Similarly, 0.00016 can be written as 1.6 × 10^–4.
Thus, we can write 232.508 as 2.32508 × 10² in scientific notation. Note that while writing it, the decimal had to be moved to the left by two places and the same is the exponent (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.

Multiplication and Division for Exponential Numbers

• These two operations follow the same rules which are there for exponential numbers, i.e.
  

Addition and Subtraction for Exponential Numbers

• For these two operations, first, the numbers are written in such a way that they have the same exponent. After that, the coefficients (digit terms) are added or subtracted as the case may be.
• Thus, for adding 6.65 × 10⁴ and 8.95 × 10³, the exponent is made the same for both numbers. Thus, we get (6.65 × 10⁴) + (0.895 × 10⁴).

Examples:

Q.1. Which of the following options is not correct? 
(a) 8008 = 8.008 x 10³
(b) 208 = 3
(c) 5000 = 5.0 x 10³     
(d) 2.0034 = 4
Ans: (d)
Solution:
2.0034 = 4

Q.2. Exponential notation in which any number can be represented in the form, Nx 10ⁿ here N is termed as
(a) non –digit term    
(b) digit term    
(c) numeral     
(d) base term 
Ans: (b)
Solution:
In exponential notation N × 10ⁿ, N is a number called digit term which varies between 1.000 and 9.000….

Significant Figures

• The reliability of measurement is indicated by the number of digits used to represent it. 
• To express the measurement more accurately, we express it with digits that are known with certainty. These are called significant figures.
• They contain all the certain digits plus one doubtful digit in a number.

Rules for Determining the Number of Significant Figures

• All non-zero digits are significant.
  Example: 
  - 6.9 has two significant figures.
  - 2.16 has three significant figures.
• The decimal place does not determine the number of significant figures.
• A zero becomes significant in case it comes in between non-zero numbers.
  Example:
  - 2.003 has four significant figures.
  - 4.02 has three significant figures.
• Zeros at the beginning of a number are not significant.
  Example:
  - 0.002 has one significant figure.
  - 0.0045 has two significant figures.
• All zeros placed to the right of a number are significant.
  Example: 
  - 16.0 has three significant figures.
  - 16.00 has four significant figures.
• Zeros at the end of a number without a decimal point are ambiguous.
• In exponential notations, the numerical portion represents the number of significant figures.
  Example: 0.00045 is expressed as 4.5 × 10^-4 in terms of scientific notations. The number of significant figures in this number is 2, while in Avogadro's number (6.023 × 10²³) it is four.
• The decimal point does not count towards the number of significant figures.
  Example: The number 345601 has six significant figures but can be written in different ways, as 345.601 or 0.345601 or 3.45601, all having the same number of significant figures.

Math with Significant Figures

(i) 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
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.

(ii) Multiplication and Division of Significant Figures
The result must be reported in these operations with no more significant figures as 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.

Practice Questions

Q.1. How many significant figures in each term?
(a) 34.6209 = 6
(b) 0.003048 = 4
(c) 5010.0 = 5
(d) 4032.090 = 7

Q.2. Solve the following equations using the correct number of significant figures.
(a) 34.683 + 58.930 + 68.35112 = 161.964
(b) 45001 - 56.355 - 78.44 = 44866
(c) 0.003 + 3.5198 + 0.0118 = 3.535
(d) 36.01 - 0.4 - 15 = 21

Q.3. Solve the following equations using the correct number of significant figures.
(a) 98.1 × 0.03 = 3
(b) 57 × 7.368 = 4.2 × 10²
(c) 8.578/4.33821 = 1.977
(d) 6.90/2.8952 = 2.38

Q.4. How many significant figures in each term?
(a) 1.40 × 10³ = 3
(b) 6.01 = 3
(c) 02947.1 = 5
(d) 583.02 = 5

Precision and Accuracy

• 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 measurement.
• Accuracy refers to the closeness of a measured value to a standard or known value.
  Example: If in the lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg, then your measurement is not accurate. In this case, your measurement is not close to the known value.
• Precision refers to the closeness of two or more measurements to each other.
  Using the example above, if you weigh a given substance five times, and get 3.2 kg each time, then your measurement is very precise. Precision is independent of accuracy. 
• You can be very precise but inaccurate, as described above. You can also be accurate but imprecise.
  Example: If on average, your measurements for a given substance are close to the known value, but the measurements are far from each other, then you have accuracy without precision.

Example: If the true value for a result is 2.00 g.
(a) Student ‘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.

(b) 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.

(c) When a third student repeats these measurements and reports 2.01g and 1.99 g as the result.
These values are both precise and accurate.

Units & Dimensional Analysis

Conversion of Units

• The simplest way to carry out calculations that involve different units is to use dimensional analysis. The simplest way to carry out calculations that involve different units is to use dimensional analysis.
• As there is a need to convert units from one system to the other. The method used to accomplish this is called the factor label method or unit factor method or dimensional analysis.
• In this method a quantity expressed in one unit is converted into an equivalent quantity with a different unit by using a conversion factor that expresses the relationship between units:
  
• This is based on the fact that the ratio of each fundamental quantity in one unit with their equivalent quantity in another unit is equal to one For example, in the case of mass
  1 kg = 2.205 pound = 1000 gm
• In this way, any derived unit first expressed in dimension and each fundamental quantity like mass length time is converted in another system of the desired unit to work out the conversion factor.

Example: How unit of work / energy i.e. joule, in S.I. system is related with unit erg in C.G.S system?
Dimension of work = force x displacement = MLT^-2 × L = ML²T^-2
1 joule = 1 kg (1 metre)² × (1sec)^-2
⇒ 1 × Kg × 1000 gm/1Kg × [1 metre × 100 cm/1 metre] × [1 sec]^-2
⇒ 100 gm × (100)² × 1 em² × (1 sec)^-2
⇒ 1000 × 10000 × 1 gm × 1 cm² × 1 sec^-1
⇒ 1 joule = 10⁷ erg

Similarly, we can deduce other conversion factors for other quantities in different units by the dimensional analysis method.
Another interesting example is the conversion of litre – atmosphere to joule (the SI unit of energy) by multiplying with two successive unit factors.
Thus,
1 L atm × (10^-3 m³/1L) × 101.325 Pa/1 atm 
= 101. 325 Pa M³ 
Knowing that Pa = N/m²,
we can write
101.325 Pa m³ = 101.325 (N/m²) m³ 
= 101.325 N m = 101.325 J
Hence, 1 L atm = 101.325 J  

Mass and Weight

• Mass of a substance is the amount of matter present in it while weight is the force exerted by gravity on an object. 
• The mass of a substance is constant whereas its weight may vary from one place to another due to a change in gravity. The mass of a substance can be determined very accurately by using an analytical balance.

Volume

• Volume has units of (length)³. So, the volume has units of m³ or cm³ or dm³. A common unit, litre (L) is not an SI unit, is used for the measurement of the volume of liquids.
  1 L = 1000 mL, 1000 cm³ = 1 dm³

Some Volume Measuring Devices

Density

• The density of a substance is its amount of mass per unit volume. SI unit of density = SI unit of mass/SI unit of volume = kg/m³ or kg m^–3. 
• This unit is quite large and a chemist often expresses density in g cm^–3.

Relationship between Density, Mass and Volume

Temperature

There are three common scales to measure temperature:

1. °C (degree Celsius)

2. °F (degree Fahrenheit)

3. K (kelvin)

Here, K is the SI unit.

Conversion Formulas: 

• K = °C + 273.15
• °F = (9/5) °C +32

Note: Temperature below 0 °C (i.e. negative values) are possible in the Celsius scale but in the Kelvin scale, the negative temperature is not possible.

Temperature Kelvin scale

Example 1. What is the mass of 1 L of mercury in grams and in kilograms if the density of liquid mercury is 13.6 g cm⁻³?
Solution. We know the relationship, 1 L = 1000 cm³ and
Also, density = mass/volume
We can write, mass = (volume) (density)
Therefore, the mass of 1 L of mercury is equal to

 
The mass in kilograms can be calculated as

(Remember,

are conversion factors with which we have to multiply for getting our answer in appropriate units).

Example 2. How unit of velocity i.e. kilometer/hour is related to unit meter/second.
Solution. We know the relationship, velocity = total displacement/total time 

Prefixes in S.I. system

Solved Examples: 
Q.1. In scientific notation,0.00016 can be written as 
(a) 1.6 x 10^-4 
(b) 1.6 x 10^-3  
(c) 1.6 x 10^-2  
(d)  1.6 x 10^-1
Ans: (a)
Solution:
0.00016 can be written as 1.6 × 10^-4  in scientific notation

Q.2. Addition of 6.65 x 10⁴ and 8.95 x 10³, in terms of scientific notation will be 
(a) 7.545×10⁴ 
(b) 75.45×10³ 
(c) 754.5×10² 
(d) 75.45×10⁰
Ans: (a)
Solution:
6.65 × 10⁴ + 8.95 × 10³
= (6.65 + 0.895) × 10⁴ = 7.545 × 10⁴

Q. 3. The substraction of two numbers 
2.5 x 10^-2 -4.8 x 10 ^-3 gives the following value.
(a) 2.02 x 10^-3 
(b) 2.02 x 10^-2 
(c) 2.02 x 10^-1  
(d) 2.02 x 10⁰
Ans: (b)
Solution:
2.5 × 10^-2 - 4.8 × 10^-3
= 2.5 × 10^-2 — (0.48 × 10^-2) = 2.02 × 10^-2

Q.4. A refers to the closeness of various measurements for the same quantity. B is the agreement of a particular value to the true value of the result. A and B respective are
(a) A  → Significant figures, B → accuracy
(b) A → accuracy, B → precision
(c) A → Precision, B → accuracy
(d) A → significant figures,  → precision 
Ans: (c)