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Significant figures in the measured value of a physical quantity tells us the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.

COMMON RULES FOR COUNTING

SIGNIFICANT FIGURES

Following are some of the common rules for counting significant figures in a given expression :

P                 All zeros occurring between two non zero digits are significant. For example: x = 5008 has four significant figures. Again x = 7.0102 has five significant figures.

P                 All non zero digits are significant.

For example: x = 7284 has four significant figures. Again x = 457 has only three significant digits.

P                 All zeros on the right of the last non zero digit in the decimal part are significant. For example                      x = 0.00400 has three significant figures.

P                 e.g., x = 0.00800, x=1.00;  The zeros before 8 are not significant.  1.00 has three significant figures.

P                 In a number less than one, all zeros to the right of decimal point and to the left of a non zero digit are NOT significant.

P                 For example: x = 0.0088 has only two significant digits. Again x = 1.0088 has five significant figures.

P                 All zeros on the right of the last non-zero digit become significant, when they come from a measurement.

For example, suppose distance between two stations is measured to be 3850 m. It has four significant figures. The same distance can be expressed as 3.850×10⁵ cm. In all these expressions, number of significant figures continues to be four.

P                 All zeros on the right of non-zero digit are NOT significant.

For example, x = 7000 has only one significant figure. Again x = 848000 has three significant figures.

ROUNDING OFF

While rounding off measurements, we use the following rules by convention:

P     If the digit to be dropped is more than 5, then the preceding digit is raised by one.

For example, x = 6.87 is rounded off to 6.9. Again x = 12.78 is rounded off to 12.8 .

P     If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

For example, x = 7.82 is rounded off to 7.8 . Again x = 3.94 is rounded off to 3.9 .

P     If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

For example, x = 16.351 is rounded off to 16.4. Again x = 6.758 is rounded off to 6.8 .

P     If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.

For example, x = 3.750 is rounded off to 3.8. Again x = 16.150 is rounded off to 16.2 .

If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is left unchanged, if it is even.

For example, x = 3.250 becomes 3.2 on rounding off, Again x = 12.650 becomes 12.6 on rounding off.

ARITHMENTICAL OPERATIONS WITH SIGNIFICANT FIGURES

(i) Addition and subtraction.

In addition or subtraction, the number of decimal places in the result should equal the smallest number of decimal places of terms in the operation. Suppose, in the measured values to be added or subtracted, the least number of significant digits after the decimal is N. Then in the sum or difference also, the number of significant digits after the decimal should be N.

For example, the sum of three measurements of length; 2.1 m, 1.78 m and 2.046 m is 5.926m, which is rounded off to 5.9 m (upto smallest number of decimal places).

In the subtraction of quantities of nearly equal magnitudes, accuracy is  almost destroyed. For example, if x = 42.87m and y = 12.86m, then

x – y = 12.87 – 12.86 = 0.01 m. The difference has only one significant figure, whereas x and y have four significant digits each.

(ii) Multiplication and Division

In multiplication and division, the number of significant figures in the product or in the quotient is the same as the smallest number of significant figures in any of the factors.

For example, suppose x = 3.8 and y = 0.125.  Therefore, xy = (3.8) (0.125) = 0.475. As least number of significant figures is 2 (in x = 3.8). Therefore, xy = 0.475 = 0.48 is rounded off to two significant figures.

Example

(i)      3.24 + 4.200018 + 5.0

= 12.440018≈  12.4

Here least number of significant digits after the decimal is one in 5.0. Same is the case with the sum

(ii)      6.21192 – 3.10   = 3.11192  ≈3.11

Here least number of significant digits after the decimal is two in 3.10. Same is the case with the difference.

Multiplication and division of measured values

ü     Suppose, in the measured values to be multiplied or divided, the least number of significant digits be N. Then in the product or quotient, the number of significant digits should be N.

Example

(i)   3.224 * 2.3 = 7.4152 ≈7.4

(ii)  46.64/2.3 =20.3=≈20

Here least number of significant digits is two in 2.3 and same should be the case with product or quotient

Þ  Change in the position of decimal point does not change the number of significant digits in the measured value.  For example, the number of significant digits both in 12.340 * 10² as well as 1234.0 is 5.

Þ The change in the units of measured value does not change the significant digits.

Rounding off a Digit

ü     We round off the number to obtain its value with a definite number of significant digits. Following are the rules for rounding off.

If the number lying to the right of cut off digits be

less than 5, then the cut off digit is retained as

such. However if it is more than 5, then the cut off

digit is increased by 1.

Example

Consider the number 324.1283. To round it off to 4 significant digits, we can write:

324.1283  ≈ 324.1. Again consider the number 324.1823. To round it off to 4 significant digits, we can write: 324.1823 ≈  324.2

Þ If the number to the right of cut off digit be 5, then we proceed as follows:

(a) Increase the cut off digit by 1 if it is odd.

(b) Retain the cut off digit as such if it is even

Example

324.1532  ≈324.2 and 324.2532  ≈324.2

Important

ü     While rounding off, the process should, in fact, be carried out from the last digit to the right. For example to round off 324.14821 to 4 significant digits, we should proceed as follows:

324.14821  ≈324.1482 ≈324.148 ≈324.15 ≈324.2

Þ In general no finally calculated value should have more significant figures than the least significant figures in the given data to be multiplied or divided. However, if multiple steps are involved, in the intermediate steps it is better to retain one significant figure more than the least number of significant figures in the given data.

The process of finding out the concentration of a solution by reacting it with another solution of known concentration is called volumetric analysis.

Volumetric analysis is done with the help of titrations. Suppose, we have a solution of unknown strength of a strong acid. We know that strong acids react with strong bases to give salts. We can prepare a standard solution (i.e. a solution of known strength) of base. Now a fix volume of solution is taken and base is slowly added to acid, in presence on an indicator (Phenolphthalein). After addition of a specific amount of base we find that pink colour appears in the reaction mixture which indicates that solution is completely neutralised.

In terms of m-equivalents same no. of (m-eq.) of reactants react and same no. of m-equivalent of product are formed. This is basic principle involved in volumetric analysis.

For the above reaction

Important

Þ    Acidic salts react with acid as well as base.

Þ      Neutral salts react with neither acid nor base.

Þ    Salts of strong acid and strong base do not react  with base.

Þ    Metal Oxide normally reacts with acid & non metal oxide reacts with base.

Þ    Metal normally reacts with acid and not with base at normal temperature.

Cathode rays and electrons

Electrical discharge through partially evacuated tubes produced radiation. This radiation originated from near the negative electrode, known as the cathode (thus, these rays were termed cathode rays).

• The “rays” traveled towards, or were attracted to the positive electrode (anode)
• Not directly visible but could be detected by their ability to cause other materials to glow, or fluoresce
• Traveled in straight line
• Their path could be “bent” by the influence of magnetic or electrical fields
• A metal plate in the path of the “cathode rays” aquired a negative charge
• The “cathode rays” produced by cathodes of different materials appeared to have the same properties

These observations indicated that the cathode ray were composed of negatively charged particles (now known as electrons).

J.J. Thompson (1897) measured the charge to mass ratio for a stream of electrons (using a cathode ray tube apparatus) at 1.76 x 10⁸ coulombs/gram.

• Charged particle stream can be deflected by both an electric and by a magnetic field
• An electric field can be used to compensate for the magnetic deflection – the resulting beam thus behaves as if it were neutral
• The field needed to “neutralize” the magnetic field indicates the charge of the beam

Thompson determined the charge to mass ratio for the electron, but was not able to determine the mass of the electron.

However, from his data, if the charge of a single electron could be determined, then the mass of a single electron could also be determined.

Robert Millikan (1909) was able to successfully measure the charge on a single electron (the “Milliken oil drop experiment”). This value was determined to be 1.60 x 10^-19 coulombs.

Thus, the mass of a single electron was determined to be:

(1 gram/1.76 x 10⁸ coulombs)*(1.60 x 10^-19 coulombs) = 9.10 x 10^-28 grams

Note: the currently accepted value for the mass of the electron is 9.10939 x 10^{- 28} grams.

The tiny constituent of an element is an atom. The word atom is a Greek word meaning indivisible, i.e., an ultimate particle which cannot be further subdivided. The idea that all matter ultimately consists of extremely small particles was conceived by ancient Indian and Greek philosophers. The old concept was put on a firm footing by John Dalton in the form of atomic theory which he developed in the years 1803-1808. This theory was a landmark in the history of chemistry.

OBJECTIVE

Towards the end of the nineteenth century, it began to appear that the atom itself might be composed of even smaller particles. In 1833, Michael Faraday showed that there is a relationship between matter and electricity. This was the first major breakthrough to suggest that an atom was not a simple indivisible particle yet smaller but was made up of  particles. On the basis of Faraday’s work, Stoney proposed that units of electrical charge are associated with atoms. In 1891, he suggested that these units be called electrons. Electron is a Greek word meaning amber, a material which becomes electrically charged when rubbed with wool or silk.

It is now believed that the atom consists of several particles called sub-atomic particles like electron, proton, neutron, etc. the electron, the proton and the neutron are called Fundamental particles and are building blocks of the atoms about which we shall deal with in this chapter.

PRE-REQUISITE

Law of conservation of Mass

In a chemical reaction the weight of products is equal to the weight of reactants.

Law of definite proportions

If a compound is analysed from various sources, its elemental composition remains the same i.e., analysis of water from a river or ditch or pond either in India or in USA would always give H:O ratio as 2:1. (atom ratio)

Law of Multiple Proportions

Elements combine in simple whole number ratios to form various types of compounds e.g. The ratio of N:O is 1:1, 1:2 and 2:1 in NO, NO₂ and N₂O, respectively.

Atomic Number (Z)

The total number of protons present in the nucleus of an atom is called as atomic number of that atom.

Mass Number (A)

Total number of nucleons (protons + neutrons) in the nucleus of an atom is called it mass number.

Isotopes

Atoms with the same atomic number but different mass numbers are called isotopes of each other. For example the isotopes of hydrogen atom are: ₁H¹, ₁H², ₁H³.

Isobars

Atoms with same mass number but different atomic number are called as isobars of each other. For example ₁₅P³² and ₁₆S³² are isobars of each other.

Isotones

Atoms having the same number of neutrons but different number of protons are called isotones. For example ₆C¹⁴, ₈O¹⁶, ₇N¹⁵ are isotones as they all have 8 neutrons.

Isodiaphers

Atoms having the same value of (A – 2Z) but different value of A or Z are called isodiapheres. For example ₁₁Na²³, ₉F¹⁹, ₇N¹⁵ are isodiapheres as (A – 2Z) for all these three atoms is 1.

Nuclear Isomers

Nuclear isomers (isomeric nuclei) are the atoms with the same atomic number and same mass number but different radioactive properties are called nuclear isomers. This type of isomerism is due to the different energy states of the two isomeric nuclei. For example ₃₀Zn⁶⁹ and ₃₀Zn⁶⁹ are two atoms with their half life periods 13.8 hours and 57 minutes respectively.

Isoelectronic

Atoms, molecules or ions with same number of electrons are called isoelectronic. For example N₂, CO, CN^- are isoelctronic.

When an atom/ion/molecule loses electrons, oxidation of the species takes place, such a molecule is termed as reductant.

or

When oxidation number of an atom increases in a reaction, it is said to be oxidised.

Oxidation Number / Oxidation State

The real or hypothetical charge present over an element is called oxidation number. Whereas oxidation state defines charge on one atom but oxidation number refers to charge present on all atoms of one element in a compound.

Certain features of oxidation number

• The pure oxidation number is always an integer, but the mathematical average may be in fraction.
• Oxidation number may be positive as well as negative.
• Oxidation number of I(A) group elements is +1, II(A) group element +2, and in III(A) group Al & B have +3 oxidation number and rest are variable.
• The molecules which exist in free state, always have zero oxidation number (NH₃, H₂O etc).
• Oxidation state of hydrogen is always +1, but when hydrogen is directly attached to metal (metal hydride) then its oxidation no. is always –1.
• The oxidation No. of oxygen is –2 but in peroxide compounds its oxidation No. is –1.
• Oxidation No. of oxygen is +ve when it is directly attached to fluorine.
• ^· In superoxide compounds oxidation number of oxygen  –¹/₂.

• The sum of oxidation No. in neutral species is zero.

• In halogens oxidation number of F is ¬–1 because F has maximum electronegative value in periodic table.

Let us now focus our attention on the calculation of the oxidation number

• Break a molecule into its atoms

• Now valencies of all the atom’s are added

• This is than equated to the total formal charge on the molecule

• The total formal charge on a neutral molecule is taken as zero. For a charged molecule, total formal charge is taken equal to charge on cation/anion.

Important
For finding out formal charge on an atom, hypothetically break all bonds to that atom. The e-pair of bond goes to more electronegative atom. After this exercise total charge left on central atom would be called the formal charge on that atom.

Illustrations

Introduction:

The branch of chemistry which deals with mass relationship in chemical reactions is called stoichiometry. Stoichiometry is the quantitative analysis of various types of chemical reactions. Most of these calculations are done on the basis of mole concept. The term ‘mole’ was first introduced by ‘Ostwald’. It is a Latin version of the term ‘heap’ or ‘pile’ or ‘weight’, which refers the amount of a substance containing a fixed number of its elementary particles equivalent to the Avogadro’s number                (6.023 × 10²³). In modern practice, it is easy to express the mole of substance in terms of its weight or its volume. The analysis based on weight is called Gravimetric analysis whereas the analysis based on volume is known as volumetric analysis.
Core Concepts
Equivalent weight
The minimum weight of any chemical species, which reacts (completely) or liberates 1 g hydrogen (11.2 litre), 8 g Oxygen (5.6 litre), 35.5 g Chlorine (11.2 litre), 80 g Bromine (11.2 litre), 127 g Iodine (11.2 litre) is called Equivalent weight of that particular chemical species.

The above definition for equivalent weight is not sufficient. For example if an acid is given then the equivalent weight of an acid is defined as the ratio of Molecular weight of acid to its basicity.
Basicity means number of acidic hydrogens present in the molecule.
Acidic hydrogen means hydrogen atoms directly attached to electronegative element.

Chemistry syllabus:

General topics: Concept of atoms and molecules; Dalton’s atomic theory; Mole concept; Chemical formulae; Balanced chemical equations; Calculations (based on mole concept) involving common oxidation-reduction, neutralisation, and displacement reactions; Concentration in terms of mole fraction, molarity, molality and normality.

Gaseous and liquid states: Absolute scale of temperature, ideal gas equation; Deviation from ideality, van der Waals equation; Kinetic theory of gases, average, root mean square and most probable velocities and their relation with temperature; Law of partial pressures; Vapour pressure; Diffusion of gases.

Atomic structure and chemical bonding: Bohr model, spectrum of hydrogen atom, quantum numbers; Wave-particle duality, de Broglie hypothesis; Uncertainty principle; Qualitative quantum mechanical picture of hydrogen atom, shapes of s, p and d orbitals; Electronic configurations of elements (up to atomic number 36); Aufbau principle; Pauli’s exclusion principle and Hund’s rule; Orbital overlap and covalent bond; Hybridisation involving s, p and d orbitals only; Orbital energy diagrams for homonuclear diatomic species; Hydrogen bond; Polarity in molecules, dipole moment (qualitative aspects only); VSEPR model and shapes of molecules (linear, angular, triangular, square planar, pyramidal, square pyramidal, trigonal bipyramidal, tetrahedral and octahedral).

Energetics: First law of thermodynamics; Internal energy, work and heat, pressure-volume work; Enthalpy, Hess’s law; Heat of reaction, fusion and vapourization; Second law of thermodynamics; Entropy; Free energy; Criterion of spontaneity.

Chemical equilibrium: Law of mass action; Equilibrium constant, Le Chatelier’s principle (effect of concentration, temperature and pressure); Significance of DG and DGo in chemical equilibrium; Solubility product, common ion effect, pH and buffer solutions; Acids and bases (Bronsted and Lewis concepts); Hydrolysis of salts.

Electrochemistry: Electrochemical cells and cell reactions; Standard electrode potentials; Nernst equation and its relation to DG; Electrochemical series, emf of galvanic cells; Faraday’s laws of electrolysis; Electrolytic conductance, specific, equivalent and molar conductivity, Kohlrausch’s law; Concentration cells.

Chemical kinetics: Rates of chemical reactions; Order of reactions; Rate constant; First order reactions; Temperature dependence of rate constant (Arrhenius equation).
Solid state: Classification of solids, crystalline state, seven crystal systems (cell parameters a, b, c, alpha, beta, gamma), close packed structure of solids (cubic), packing in fcc, bcc and hcp lattices; Nearest neighbours, ionic radii,
Solutions: Raoult’s law; Molecular weight determination from lowering of vapour pressure, elevation of boiling point and depression of freezing point.

Surface chemistry: Elementary concepts of adsorption (excluding adsorption isotherms); Colloids: types, methods of preparation and general properties; Elementary ideas of emulsions, surfactants and micelles (only definitions and examples).

Nuclear chemistry: Radioactivity: isotopes and isobars; Properties of alpha, beta and gamma rays; Kinetics of radioactive decay (decay series excluded), carbon dating; Stability of nuclei with respect to proton-neutron ratio; Brief discussion on fission and fusion reactions.

Inorganic Chemistry: Isolation/preparation and properties of the following non-metals: Boron, silicon, nitrogen, phosphorus, oxygen, sulphur and halogens; Properties of allotropes of carbon (only diamond and graphite), phosphorus and sulphur.

Preparation and properties of the following compounds: Oxides, peroxides, hydroxides, carbonates, bicarbonates, chlorides and sulphates of sodium, potassium, magnesium and calcium; Boron: diborane, boric acid and borax; Aluminium: alumina, aluminium chloride and alums; Carbon: oxides and oxyacid (carbonic acid); Silicon: silicones, silicates and silicon carbide; Nitrogen: oxides, oxyacids and ammonia; Phosphorus: oxides, oxyacids (phosphorus acid, phosphoric acid) and phosphine; Oxygen: ozone and hydrogen peroxide; Sulphur: hydrogen sulphide, oxides, sulphurous acid, sulphuric acid and sodium thiosulphate; Halogens: hydrohalic acids, oxides and oxyacids of chlorine, bleaching powder; Xenon fluorides.

Transition elements (3d series): Definition, general characteristics, oxidation states and their stabilities, colour (excluding the details of electronic transitions) and calculation of spin-only magnetic moment; Coordination compounds: nomenclature of mononuclear coordination compounds, cis-trans and ionisation isomerisms, hybridization and geometries of mononuclear coordination compounds (linear, tetrahedral, square planar and octahedral).

Preparation and properties of the following compounds: Oxides and chlorides of tin and lead; Oxides, chlorides and sulphates of Fe2+, Cu2+ and Zn2+; Potassium permanganate, potassium dichromate, silver oxide, silver nitrate, silver thiosulphate.

Ores and minerals: Commonly occurring ores and minerals of iron, copper, tin, lead, magnesium, aluminium, zinc and silver.

Extractive metallurgy: Chemical principles and reactions only (industrial details excluded); Carbon reduction method (iron and tin); Self reduction method (copper and lead); Electrolytic reduction method (magnesium and aluminium); Cyanide process (silver and gold).
Organic Chemistry

Concepts: Hybridisation of carbon; Sigma and pi-bonds; Shapes of simple organic molecules; Structural and geometrical isomerism; Optical isomerism of compounds containing up to two asymmetric centres, (R,S and E,Z nomenclature excluded); IUPAC nomenclature of simple organic compounds (only hydrocarbons, mono-functional and bi-functional compounds); Conformations of ethane and butane (Newman projections); Resonance and hyperconjugation; Keto-enol tautomerism; Determination of empirical and molecular formulae of simple compounds (only combustion method); Hydrogen bonds: definition and their effects on physical properties of alcohols and carboxylic acids; Inductive and resonance effects on acidity and basicity of organic acids and bases; Polarity and inductive effects in alkyl halides; Reactive intermediates produced during homolytic and heterolytic bond cleavage; Formation, structure and stability of carbocations, carbanions and free radicals.

Preparation, properties and reactions of alkanes: Homologous series, physical properties of alkanes (melting points, boiling points and density); Combustion and halogenation of alkanes; Preparation of alkanes by Wurtz reaction and decarboxylation reactions.

Preparation, properties and reactions of alkenes and alkynes: Physical properties of alkenes and alkynes (boiling points, density and dipole moments); Acidity of alkynes; Acid catalysed hydration of alkenes and alkynes (excluding the stereochemistry of addition and elimination); Reactions of alkenes with KMnO4 and ozone; Reduction of alkenes and alkynes; Preparation of alkenes and alkynes by elimination reactions; Electrophilic addition reactions of alkenes with X2, HX, HOX and H2O (X=halogen); Addition reactions of alkynes; Metal acetylides.

Reactions of benzene: Structure and aromaticity; Electrophilic substitution reactions: halogenation, nitration, sulphonation, Friedel-Crafts alkylation and acylation; Effect of o-, m- and p-directing groups in monosubstituted benzenes.

Phenols: Acidity, electrophilic substitution reactions (halogenation, nitration and sulphonation); Reimer-Tieman reaction, Kolbe reaction.

Characteristic reactions of the following (including those mentioned above): Alkyl halides: rearrangement reactions of alkyl carbocation, Grignard reactions, nucleophilic substitution reactions; Alcohols: esterification, dehydration and oxidation, reaction with sodium, phosphorus halides, ZnCl2/concentrated HCl, conversion of alcohols into aldehydes and ketones;

Characteristic reactions of the following (including those mentioned above): Alkyl halides: rearrangement reactions of alkyl carbocation, Grignard reactions, nucleophilic substitution reactions; Alcohols: esterification, dehydration and oxidation, reaction with sodium, phosphorus halides, ZnCl2/concentrated HCl, conversion of alcohols into aldehydes and ketones; Ethers:Preparation by Williamson’s Synthesis; Aldehydes and Ketones: oxidation, reduction, oxime and hydrazone formation; aldol condensation, Perkin reaction; Cannizzaro reaction; haloform reaction and nucleophilic addition reactions (Grignard addition); Carboxylic acids: formation of esters, acid chlorides and amides, ester hydrolysis; Amines: basicity of substituted anilines and aliphatic amines, preparation from nitro compounds, reaction with nitrous acid, azo coupling reaction of diazonium salts of aromatic amines, Sandmeyer and related reactions of diazonium salts; carbylamine reaction; Haloarenes: nucleophilic aromatic substitution in haloarenes and substituted haloarenes (excluding Benzyne mechanism and Cine substitution).

Carbohydrates: Classification; mono- and di-saccharides (glucose and sucrose); Oxidation, reduction, glycoside formation and hydrolysis of sucrose.

Amino acids and peptides: General structure (only primary structure for peptides) and physical properties.

Properties and uses of some important polymers: Natural rubber, cellulose, nylon, teflon and PVC.

Practical organic chemistry: Detection of elements (N, S, halogens); Detection and identification of the following functional groups: hydroxyl (alcoholic and phenolic), carbonyl (aldehyde and ketone), carboxyl, amino and nitro; Chemical methods of separation of mono-functional organic compounds from binary mixtures.