Study Notes 📝📓

Notes on Chemical Bonding

Chemical bond:-
Chemical bond is the attractive force which holds various constituents together in a molecule.

There are three types of chemical bonds: Ionic Bond, Covalent Bond, Co-ordinate Bond.

Octet Rule:
Atoms form chemical bonds in order to complete their octet i.e. eight electrons in their valence shell.

Lewis Structures:
Pair of bonded electrons is by means of a ‘dash’ (-) usually called a ‘bond’.

Lone pairs or ‘non-bonded’ electrons are represented by ‘dots’.

Electrons present in the last shell of atoms are called valence electrons.

Exceptions to the Octet Rule:
Species with odd number of electrons: NO, NO2,

Incomplete octet for the central atom: LiCl, BeH2 and BCl3

Expanded octet for the central atom: PF5, SF6 and H2SO4

Formal Charge:
Formal charge is the difference between the number of valence electrons in an isolated atom and number of electrons assigned to that atoms in Lewis structure.

Formal charge = [Total number of valence electrons in the free atom ) - (Total number of lone pairs of electrons) -1/2(Total number of shared electrons i.e. bonding electrons)]

Resonance:
For molecules and ions showing resonance it is not possible to draw a single Lewis structure.

All the properties of such species can only be explained by two or more Lewis structures. Example: Resonance of O3

Ionic Bonding:
Formation of Ionic Bond:
Formation of ionic bond takes place between a metal and a non-metal by transfer of electron.

Formation of gaseous cations

A(g) + I.E. → A+ (g) + e

Ionization Energy

Formation of gaseous anions

X(g) + e → X- (g) + E.A

Electron Affinity

Packing of ions of opposite charges to form ionic solids

A+ (g) + X- (g) →AX (s) +Energy

Lattice energy

Conditions required of formation of ionic bonds:

Low I.E of cation.

High E.A of anion.

High lattice energy.

Covalent Bonding:
Covalent bond is formed between two non-metals by sharing of electrons.

Electron pairs which participate in bonding are called bond pairs.

Electron pairs which do not participate in bonding are called lone pairs.

There could be single, double or triple covalent bonds between two elements depending on the number of electrons being shared.

VSEPR (Valence Shell Electron Pair Repulsion) Theory:
The shape of the molecule is determined by repulsions between all of the electron pairs present in the valence shell.

Order of the repulsion: Lone pair↔️ Lone pair > Lone pair↔️ Bond pair > Bond pair↔️ Bond pair.

Repulsion among the bond pairs is directly proportional to the bond order and electronegativity difference between the central atom and the other atoms.



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☑️Revision Notes on Biodiversity and Conservation


(1) The vast array of species of micro-organisms, algae, fungi, plants and animals occurring on the earth either in the terrestrial or aquatic habitats and the ecological complexes of which they are a part.
(2) Diversity ranges from macromolecules to biomes.
(3) Biodiversity on earth exists in three levels of organization:
(i) Genetic diversity

(ii) Species diversity

(4) Genetic diversity

(i) It is related to the variations of genes within species.

(ii) The variations may be in different variants of same genes (alleles), in entire genes or in chromosomal structures.
(ii) Greater the genetic diversity among organisms of a species, more sustenance it has against environmental perturbations.
(iii) Genetically uniform populations are highly prone to diseases.

(5) Species diversity
(i) it is related to the variety of species within a region.
(ii) Species richness refers to the number of species per unit area.
(iii) Species Evenness refers to the relative abundance with which each species is represented in an area.

Biodiversity in India
(1) Out of the twelve mega biodiversity counties, India is one.

(2) India has 10 biogeographical regions, 89 national parks, 500 wild life sanctuaries, 14 biosphere reserves, 6 westlands and 35 world heritage sites.

(3) There are about 45,000 species of plants and about 90,000-1,00,000 species of animals.



Patterns of Biodiversity
(1) Biodiversity changes with change in latitude or altitude.

(2) It is minimum at the poles and maximum near or at equator. Similarly, as one moves down from higher to lower altitudes, biodiversity is increased.

Loss of bio-diversity:
(1) Caused by three factors - Population, Urbanisation and Industrialisation.

(2) The colonisation of tropical Pacific Islands by human has led to the extinction of more than 2000 species of native birds.

(3) Loss of bio-diversity in a region leads to:

(i) decrease in plant production.

(ii) less resistance to environmental disturbances such as droughts.

(iii) increase in variability in ecosystem processes like plant productivity, water use, pest and disease cycles etc.

Biodiversity Conservation
In situ conservation
(1) The most appropriate method to maintain species of wild animals and plants in their natural habitats. This approach includes conservation and protection of the total ecosystems and its biodiversity through a network of protected areas.

(2) The common natural habitats (protected areas) that have been set for in-situ conservation of wild animals and plants include:

(i) National parks

(ii) Wild life sanctuaries

(iii) Biosphere reserves

(iv) Several wetlands, mangroves and coral reefs.

(v) Sacred grooves and lakes.

(3) Hot spot of biodiversity are those regions of rich biodiversity which have been declared sensitive due to direct or indirect interference of human activities.

(4) There are 25 terrestrial hot spots in the world including two from India.

Ex situ conservation
(1) Threatened animals and plants are taken out from their natural habitat and placed in special setting where they can be protected and given special care.

(2) Ex situ conservation includes the following:

(i) Sacred plants and home gardens

(ii) Seed banks, field gene banks, cryopreservation.

(iii) Botanical gardens, Arborata, Zoological gardens, Aquaria.

Convention on Biodiversity:
(1) “The earth Summit” held in Rio de Jeneiro in 1992 called upon all nations to take appropriate measures for conservation of biodiversity and sustainable utilization of its benefits.

(2) Second international Conference on Sustainable development held in 2002 in Johannesburg, South Africa, 190 countries pledged their commitment to achieve by 2010 a significant reduction in the current rate of biodiversity loss at global, regional and local level.

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✅Notes on s-Block Elements:

Covalent Character:.

Small cation and large anion favors covalency.

Order: LiCl > NaCl > KCl > RbCl > CsCl & . LiI > LiBr > LiCl > LiF

Greater the charge on the cation greater is its polarizing power and hence larger is the covalent character: Na+CI- < Mg+2CI2 < AI+3 CI3

Greater the charge on the anion, more easily it gets polarized thereby imparting more covalent character to the compound formed eg covalent character increase in the order. NaCI < Na2SO4 < Na3PO4

c) Lattice Energies: Amount of energy required to separate one mole of solid ionic compound into its gaseous ions.

Greater the lattice energy, higher is the melting point of the alkali metals halide and lower is its solubility in water

d) Hydration Energy: Amount of energy released when one mole of gaseous ions combine with water to form hydrated ions.

M+ (g) + aq → M+ (aq) + hydration energy

X- (g) + aq → X- (aq) + hydration energy

Higher the hydration energy of the ions greater is the solubility of the compound in water.

The solubility of the most of alkali metal halides except those of fluorides decreases on descending the group since the decrease in hydration energy is more than the corresponding decrease in the lattice energy.

Due to high hydration energy of Li+ ion, Lithium halides are soluble in water except LiF which is sparingly soluble due to its high lattice energy.

For the same alkali metal the melting point decreases in the order

fluoride > chloride > bromide > iodide

For the same halide ion, the melting point of lithium halides are lower than those of the corresponding sodium halides and thereafter they decrease as we move down the group from Na to Cs.

The low melting point of LiCl (887 K) as compared to NaCl is probably because LiCl is covalent in nature and NaCl is ionic.

Anomalous Behavior of Lithium and diagonal relationship with Magnesium:

Li has anomalous properties due to

Very small size

High polarizing Power

Lithium show diagonal relationship with magnesium because both elements have almost same polarizing power.

The melting point and boiling point of lithium are comparatively high.

Lithium is much harder than the other alkali metals. Magnesium is also hard metal.

Lithium reacts with oxygen least readily to form normal oxide whereas other alkali metals form peroxides and superoxides.

LiOH like Mg (OH)2 is weak base. Hydroxides of other alkali metals are strong bases.

Due to their appreciable covalent nature, the halides and alkyls of lithum and magnesium are soluble in organic solvents.

Unlike elements of group 1 but like magnesium. Lithium forms nitride with nitrogen.6Li + N2 → 2Li3N

LiCl is deliquescent and crystallizes as a hydrate, LiCI2H2O. Other alkali metals do not form hydrates. also forms hydrate, MgCI2.8H2O .

Unlike other alkali metals lithium reacts directly with carbon to form an ionic carbide. Magnesium also forms a similar carbide.

The carbonates, hydroxides and nitrates of lithium as well as magnesium decompose on heating.
Li2CO3 → Li2O + CO2

MgCO3 → MgO + CO2

2LiOH → Li2O + H2O

Mg (OH)2 → MgO + H2O

4LiNO3 → 2Li2O + 4NO2 + O2

2Mg ( NO3)2 → 2Mg + 4NO2 +O2

The corresponding salts of other alkali metals are stable towards heat.
Lithium nitrate, on heating, decomposes to give lithium oxide, Li2O whereas other alkali metals nitrate decomposes to give the corresponding nitrite.

4LiNO3 → 2Li2O + 4NO2 + O2

2NaNO3 → 2NaNO2 + O2

2KNO3 → 2KNO2 + O2

Li2CO3, LiOH, LiF and Li3PO4 are the only alkali metal salts which are insoluble in water. The corresponding magnesium compounds are also insoluble in water.

Hydrogen carbonates of both lithium and magnesium can not be isolated in solid state. Hydrogen carbonates of other alkali metals can be isolated in solid state.

Sodium Hydroxide (NaOH):
a. Properties

NaOH is stable towards heat but is reduced to metal when heated with carbon

2NaOH + 2C → 2Na +2CO + H2

FeCl3 + 3NaOH →Fe(OH)3 + 3NaCl

NH4Cl + NaOH → NaCl + NH3 (pungent smell) + H2O

💠Properties of Solids and Liquids - Revision Notes on Liquids at Rest:💠

⛔️Force of cohesion:- It is force between two molecules of similar nature.

⛔️Force of adhesion:- It is the force between two molecules of different nature.

⛔️Molecular range:- The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.

⛔️Sphere of influence:- Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10-7 cm).

⛔️Surface film:- Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.

⛔️Surface Tension

Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.

⛔️Force of surface tension:- It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.

(a) T = Force/length = F/l

(b) T = Surface energy/Surface area = W/A

Units:- S.I – Nm-1

C.G.S- dyn cm-1

⛔️Additional force:-
(a) For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)

(b) For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)

(c) For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)

⛔️Surface energy:-
Potential energy per unit area of the surface is called surface energy.

(a) Expansion under isothermal condition:-

To do work against forces of surface tension:-

W= T×A (Here A is the total increase in surface area)

To supply energy for maintaining the temperature of the film:-

E = T+H

(b) Expansion under adiabatic conditions:-

E = T

Force of surface tension is numerically equal to the surface energy under adiabatic conditions.

⛔️Drops and Bubbles:-

(a) Drop:- Area of surface film of a spherical drop of radius R is given by, A = 4πR2

(b) Bubble:- The surface area of the surface films of a bubble of radius R is, A = 2×4πR2

⛔️Combination of n drops into one big drop:-

(a) R = n1/3r

(b) Ei = n (4πr2T), Ef =4πR2T

(c) Ef/ Ei = n -1/3

(d) ΔE/Ei = [1-(1/n1/3)]

(e) ΔE = 4πR2T (n1/3-1) = 4πR3T (1/r – 1/R)

⛔️Angle of contact:- Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.

(a) When θ < 90º (acute):-

Fa >Fc/√2

(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.

(ii) Liquid molecules will stick with the solid, thus making solid wet.

(iii) Such liquid is put in the solid tube; it will have meniscus concave upwards.

(b) When θ > 90º (obtuse):-FaTsa.

⛔️Capillarity:-

?Rise of Liquid in a Capillary Tube?Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.

Weight of liquid, W = Vρg = πr2[h+(r/3)]ρg (Here r is the radius meniscus)

If weight of meniscus is taken into account, the force of surface tension will be,

T = [r(h+(r/3)) ρg]/2 cosθ

For fine capillary, force of surface tension, T = rhρg/2 cosθ

So height, h = 2T cosθ/ rρg

✅Notes on s-Block Elements:

Physical Properties of Alkali Metals:

➖These are soft in nature and can be cut with the help of knife except the lithium.
➖The atoms of alkali metals have the largest size in their respective periods.
➖The first ionization energy of the alkali metals are the lowest as compared to the elements in the other group.
➖The alkali metals show +1 oxidation state.
➖The alkali metals have low values of reduction potential (as shown in table-I) and therefore have a strong tendency to lose electrons and act as good reducing agents.
➖The melting and boiling points of alkali metals are very low because the intermetallic bonds in them are quite weak.

Hydroxides of Alkali Metals:

a)All the alkali metals, their oxides, peroxides and superoxides readily dissolve in water to produce corresponding hydroxides which are strong alkalies.

➖2Na + 2H2O → 2NaOH + H2

➖Na2O + 2H2O 2NaOH

➖Na2O2 + 2H2O → 2NaOH + H2O2

➖2KO2 + 2H2O → 2KOH + H2O2 + O2

b) The basic strength of these hydroxides increases as we move down the group Li to Cs.
c) All these hydroxides are highly soluble in water and thermally stable except lithium hydroxide.
d) Alkali metals hydroxides being strongly basic react with all acids forming salts.

➖NaOH + HCI → NacI + H2O
➖2NaOH + H2 SO4 → Na2SO4 + 2H2O

Halides of Alkali metals:
➖M2O + 2HX → 2MX + H2O
➖MOH + HX → MX + H2O
➖M2CO3 + 2HX → 2MX + CO2 + H2O (M = Li, Na, K, Rb or Cs)

(X = F, Cl, Br or I)
➖ll the alkali metals form ionic (electrovalent) compounds.
➖The alkali metals are good conductors of heat and electricity.
➖Alkali metals (except Li) exhibit photoelectric effect
➖The alkali metals and their salts impart a characteristic colour to flame.

✅Notes on s-Block Elements:

Physical Properties of Alkali Metals:

➖These are soft in nature and can be cut with the help of knife except the lithium.
➖The atoms of alkali metals have the largest size in their respective periods.
➖The first ionization energy of the alkali metals are the lowest as compared to the elements in the other group.
➖The alkali metals show +1 oxidation state.
➖The alkali metals have low values of reduction potential (as shown in table-I) and therefore have a strong tendency to lose electrons and act as good reducing agents.
➖The melting and boiling points of alkali metals are very low because the intermetallic bonds in them are quite weak.

Hydroxides of Alkali Metals:

a)All the alkali metals, their oxides, peroxides and superoxides readily dissolve in water to produce corresponding hydroxides which are strong alkalies.

➖2Na + 2H2O → 2NaOH + H2

➖Na2O + 2H2O 2NaOH

➖Na2O2 + 2H2O → 2NaOH + H2O2

➖2KO2 + 2H2O → 2KOH + H2O2 + O2

b) The basic strength of these hydroxides increases as we move down the group Li to Cs.
c) All these hydroxides are highly soluble in water and thermally stable except lithium hydroxide.
d) Alkali metals hydroxides being strongly basic react with all acids forming salts.

➖NaOH + HCI → NacI + H2O
➖2NaOH + H2 SO4 → Na2SO4 + 2H2O

Halides of Alkali metals:
➖M2O + 2HX → 2MX + H2O
➖MOH + HX → MX + H2O
➖M2CO3 + 2HX → 2MX + CO2 + H2O (M = Li, Na, K, Rb or Cs)

(X = F, Cl, Br or I)
➖ll the alkali metals form ionic (electrovalent) compounds.
➖The alkali metals are good conductors of heat and electricity.
➖Alkali metals (except Li) exhibit photoelectric effect
➖The alkali metals and their salts impart a characteristic colour to flame.

✍️Notes on Biotechnology

Steps of recombinant DNA technology

(1) Isolating a useful DNA segment from the donor organism.
(2) Splicing it into a suitable vector under conditions to ensure that each vector receives no more than one DNA fragment.
(3) Producing of multiple copies of his recombinant DNA.
(4) Inserting this altered DNA into a recipient organism.
(5) Screening of the transformed cells.

Vectors:
Vector in genetic engineering is usually a DNA segment used as a carrier for transferring selected DNA into living cells. These are as follows:

(1) Plasmid: Plasmid is extra chromosomal, closed circular double stranded molecules of DNA present in most eukaryotes. All plasmid carry replicons pieces of DNA that have the genetic information required to replicate. Plasmid pBR 322 was one of the first widely used cloning vectors, it contain both ampicillin and tetracycline resistance genes.

(2) Phage: It is constructed from the phage l chromosomes and acts as bacteriophage cloning vectors.

(3) Cosmid: The hybrids between plasmid and the phage l chromosome give rise to cosmid vectors.

(4) Beside all these there are artificial chromosomes like
(i) BACs (Bacterial Artificial chromosomes)
(ii) YACs (Yeast Artificial chromosomes)
(iii) MACs (Mammalian Artificial chromosomes) are very efficient vectors for eukaryotic gene transfers.

Application of recombinant DNA technology:
The technique of recombinant DNA can be employed in the following ways.

(1) It can be used to elucidate molecular events in the biological process such as cellular differentiation and ageing. The same can be used for making gene maps with precision.
(2) In biochemical and pharmaceutical industry, by engineering genes, useful chemical compounds can be produced cheaply and efficiently which is shown in table.

♻️Important Notes - Electrochemical Cells♻️


► An electrochemical cell can convert electrical energy to chemical energy and can also convert electrical energy to chemical energy. There are two types of electrochemical cells- Galvanic cell and Electrolytic cell.

► Cathodes are usually metal electrodes. It is the electrode where reduction takes place. The cathode is the positive electrode in a galvanic cell and a negative electrode in an electrolytic cell. Electrons move into the cathode.

► A half-cell is half of an electrochemical cell (electrolytic or galvanic), where either oxidation or reduction occurs. At equilibrium, there is no transfer of electrons across the half cells. Therefore, the potential difference between them is nil.

► A salt bridge is a device used to connect the oxidation and reduction half-cells of a galvanic cell (a type of electrochemical cell). Strong electrolytes are generally used to make the salt bridges in electrochemical cells. Since ZnSO4 is not a strong electrolyte, it is not used to make salt bridges.

► Emf of a cell is equal to the maximum potential difference across its electrodes, which occurs when no current is drawn through the cell. It can also be defined as the net voltage between the oxidation and reduction half-reactions.

► Cell potential is an intensive property as it is independent of the amount of material present. Gibbs free energy is defined for an electrochemical cell and is an extensive property as it depends on the quantity of the material.

► Electrode potential is the tendency of an electrode to accept or to lose electrons. Electrode potential depends on the nature of the electrode, temperature of the solution and the concentration of metal ions in the solution. It doesn’t depend on the size of the electrode.

► The salt bridge connects the two half-cell solutions to complete the circuit of the electrochemical cell. The electrolytes of the salt bridge are generally prepared in agar-agar or gelatin so that the electrolytes are kept in a semi-solid phase and do not mix with the half-cell solutions and interfere with the electrochemical reaction.

► A salt bridge is a junction that connects the anodic and cathodic compartments in a cell or electrolytic solution. It maintains electrical neutrality within the internal circuit, preventing the cell from rapidly running its reaction to equilibrium.

► A Voltaic or Galvanic cell is a type of electrochemical cell that converts chemical energy into electrical energy. Photovoltaic cells are used to convert light energy into electrical energy. An Electrolytic cell is a type of electrochemical cell that converts electrical energy into chemical energy. A fuel cell is an electrochemical cell that converts the chemical energy of a fuel and an oxidizing agent into electricity.

► For all spontaneous chemical reactions, the change in Gibbs free energy (ΔG°) is always negative. For a spontaneous reaction in an electrolytic cell, the cell potential (E°cell) should be positive.

► In an electrochemical cell, when an opposing externally potential is applied and increased slowly, the reaction continues to take place. When the external potential is equal to the potential of the cell, the reaction stops. Once the externally applied potential is greater than the potential of the cell, the reaction goes in the opposite direction and the cell behaves like an electrolytic cell.

► Primary cells cannot be used again and again. Since there is no fluid inside, these cells are also known as dry cells. The internal resistance is high and the chemical reaction is irreversible. Their initial cost is cheap.

► A secondary battery (a series of cells) is one which can be charged, discharged into a load, and recharged many times. Nickel-cadmium cell, Lead storage cell and Mercury cell are examples of secondary cells. Leclanche cell is an example of a primary cell.

🧩Algebra- Revision Notes on Probability🧩
➖➖➖➖➖➖➖➖➖➖➖➖

The sum of all the probabilities in the sample space is 1.

The probability of an event which cannot occur is 0.

The probability of any event which is not in the sample space is zero.

The probability of an event which must occur is 1.

The probability of the sample space is 1.

The probability of an event not occurring is one minus the probability of it occurring.

The complement of an event E is denoted as E' and is written as P (E') = 1 - P (E)

P (A∪B) is written as P (A + B) and P (A ∩ B) is written as P (AB).

If A and B are mutually exclusive events, P(A or B) = P (A) + P (B)

When two events A and B are independent i.e. when event A has no effect on the probability of event B, the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P (A and B) = P (A) P (B|A).

A and B are independent if P (B/A) = P(B) and P(A/B) = P(A).

If E1, E2, ......... En are n independent events then P (E1 ∩ E2 ∩ ... ∩ En) = P (E1) P (E2) P (E3)...P (En).

Events E1, E2, E3, ......... En will be pairwise independent if P(Ai ∩ Aj) = P(Ai) P(Aj) i ≠ j.

P(Hi | A) = P(A | Hi) P(Hi) / ∑i P(A | Hi) P(Hi).

If A1, A2, ……An are exhaustive events and S is the sample space, then A1 U A2 U A3 U ............... U An = S

If E1, E2,….., En are mutually exclusive events, then P(E1 U E2 U ...... U En) = ∑P(Ei)

If the events are not mutually exclusive then P (A or B) = P (A) +P (B) – P (A and B)

Three events A, B and C are said to be mutually independent if P(A∩B) = P(A).P(B), P(B∩C) = P(B).P(C), P(A∩C) = P(A).P(C), P(A∩B∩C) = P(A).P(B).P(C)

The concept of mutually exclusive events is set theoretic in nature while the concept of independent events is probabilistic in nature.

If two events A and B are mutually exclusive,

P (A ∩ B) = 0 but P(A) P(B) ≠ 0 (In general)

⇒ P(A ∩ B) ≠ P(A) P(B)

⇒ Mutually exclusive events will not be independent.

The probability distribution of a count variable X is said to be the binomial distribution with parameters n and abbreviated B (n,p) if it satisfies the following conditions:

The total number of observations is fixed

The observations are independent.

Each outcome represents either a success or a failure.

The probability of success i.e. p is same for every outcome.

Some important facts related to binomial distribution:

(p + q)n = C0Pn + C1Pn-1q +...... Crpn-rqr +...+ Cnqn

The probability of getting at least k successes out of n trials is

P(x > k) = Σnx = k nCxpxqn-x

Σnx = k nCxqn-xpx = (q + p)n = 1

Mean of binomial distribution is np

Variance is npq

Standard deviation is given by (npq)1/2, where n

Sum of binomials is also binomial i.e. if X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable with distribution X + Y ~ B(n + m, p).

If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distributionY ~ B( n, pq).

The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B (1, p) has the same meaning as X ~ Bern (p).

If an experiment has only two possible outcomes, then it is said to be a Bernoulli trial. The two outcomes are success and failure.

Any binomial distribution, B (n, p), is the distribution of the sum of n independent Bernoulli trials Bern (p), each with the same probability p.

The binomial distribution is a special case of the Poisson Binomial Distribution which is a sum of n independent non-identical Bernoulli trials Bern(pi). If X has the Poisson binomial distribution with p1 = … = pn = p then X ~ B(n, p).

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

Differential Calculus: Notes on Maxima and Minima

Local Maximum: A function f(x) is said to have a local maximum at x = a if the value of f(a) is greater than all the values of f(x) in a small neighbourhood of x = a. Mathematically, f (a) > f (a – h) and f (a) > f (a + h) where h > 0, then a is called the point of local maximum.


Local Minimum: A function f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighboring points of x = a. Mathematically, f (a) < f (a – h) and f (a) < f (a + h) where h > 0, then a is called the point of local minimum.


A point of local maximum or a local minimum is also called a point of local extremum.

A point where the graph of function is continuous and has a tangent line and where the concavity changes is called point of inflexion.

At the point of inflexion, either y” = 0 and changes sign or y” fails to exist.

At the point of inflexion, the curve crosses its tangent at that point.

A function cannot have point of inflexion and extrema at the same point.

Working rules to find points of local maxima and local minima:

1. First Derivative Test:

If f'(a) = 0 and f'(x) changes its sign while passing through the point x = a, then

f(x) would have a local maximum at x = a if f'(a – 0) > 0 and f'(a + 0) < 0. It means that f'(x) should change its sign from positive to negative.

f(x) would have local minimum at x = a if f'(a – 0) < 0 and f'(a + 0) > 0 . It means that f'(x) should change its sign from negative to positive.

If f(x) doesn’t change its sign while passing through x = a, then f (x) would have neither a maximum nor minimum at x = a. e.g. f (x) = x3 doesn’t have any local maxima or minima at x = 0.

2. Second Derivative Test:

Let f(x) be a differentiable function on a given interval and let f'' be continuous at stationary point. Find f'(x) and solve the equation f'(x) = 0 given let x = a, b, … be solutions.

There can be two cases:

Case (i): If f''(a) 0 then f(a) is minimum.

In case, f''(a) = 0 the second derivatives test fails and then one has to go back and apply the first derivative test.

If f''(a) = 0 and a is neither a point of local maximum nor local minimum then a is a point of inflection.

3. nth Derivative Test for Maxima and Minima: Also termed as the generalization of the second derivative test, it states that if the n derivatives i.e. f '(a) = f''(a) = f'''(a) =………. = f n(a) = 0 and fn+1(a) ≠ 0 (all derivatives of the function up to order ‘n’ vanish and (n + 1)th order derivative does not vanish at x = a), then f (x) would have a local maximum or minimum at x = a iff n is odd natural number and that x = a would be a point of local maxima if fn+1 (a) < 0 and would be a point of local minima if fn+1 (a) > 0.

In some questions involving determination of maxima and minima, it might become difficult to decide whether f(x) actually changes its sign while passing through x = a and here, nth derivative test can be applied.

Global Minima & Maxima of f(x) in [a, b] is the least or the greatest value of the function f(x) in interval [a, b].

1. The function f(x) has a global maximum at the point ‘a’ in the interval I if f (a) ≥ f(x), for all x ∈ I.

2. Function f(x) has a global minimum at the point ‘a’ if f (a) ≤ f (x), for all x ∈ I.

Global Maxima Minima always occur either at the critical points of f(x) within [a, b] or at the end points of the interval.

Computation of Global Maxima and minima in maxima minima problems:

1. Compute the critical points of f(x) in (a, b). Let the various critical points be C1, C2, …. , Cn.

2. Next, compute the value of the function at these critical points along with the end points of the domain. Let us denote these values by f(C1), f(C2)………..f(Cn).

3. Now, compute M* = max{f(a), f(C1), f(C2)………..f(Cn), f(b)} and M** = min{f(a), f(C1), f(C2)………..f(Cn), f(b)}.Now M* is the maximum value of f(x) in [a, b] and M** is the minimum value of f(x) in [a, b].

(a) Lactic acid: Lactic acid is commercially produced from pasteurized whey (the watery part of milk) through fermentation caused by Lactobacilus bulgaricus and L. delbrueckii.

(b) Curd: Curd is prepared from pasteurized milk by the process called curdling. It is initiated by adding a starter culture of Lactobacillus bulgaricus and Streptococcus thermophillus, into the milk at 40°C. Lactobacillus converts lactose to lactic acid whereas Streptococcus causes coagulation of casein due to acidity.

(c) Cheese: Preparation of cheese from the milk involves two main steps – first curdling of milk, and second the subsequent ripening of solid curd by the use of different bacterial strains.

(d) Butter: It is prepared by churning of sweet or sour cream. The microorganisms responsible for preparation of butter cream are – Streptococcus lactis and Leuconostoc citrivorumare. The characteristic butter aroma develops due to a volatile substance – diacetyl. It is produced by the action of streptococcus on pasteurized milk.

(e) Retting process: Fibres of flax, hemp and jute are separated by the process called retting. During this process the stems of the plants are submerged in water, where the bacterial activity results in the rotting of softer parts. The tough bast fibres become loosened and easily separated from each other. These fibres are spun and woven into various articles.

(f) Vinegar: Country made vinegar is a fermentation product of cane juice, molasses or fruit juices. It is produced in two steps – first conversion of sugars into alcohols by alcholic fermentation carried by yeast, and the second, conversion of alcohol to acetic acid by the action of bacteria Acetobacter (A. orieansis, A. acetic, A. schuizenbachi, etc.). Vinegar is used in the preparation of pickles or in place of acetic acid. It is used as preservative of meats and vegetables.

(v) Role of bacteria in human being: E.coli (gram-ve) bacteria live in colon region of intestine of man and other animals and play an important role in digestion process.

(vi) Medicinal uses

(a) Vitamins: Production of riboflavin (vitamin B2) involves the activity of bacterium – Clostridium butyticum. The well known vitamin C (ascorbic acid) is produced from sorbital by the action of Acetobactor spp.

(b) Serum and vaccines: Many bacteria are used in the preparation of serums and vaccines. These substances induce immunity to various diseases in man. Serums are effective against certain diseases like diphtheria, pneumonia, etc., whereas the vaccines are effective against typhoid, smallpox, cholera, etc.

(c) Enzymes: Some bacteria live in the alimentary canal of herbivorous animals like cow, horse, goat, etc. and help in the production of certain enzymes which digest the cellulose. The enzymes proteases are produced by bacteria Bacillus subtilis. Similarly, the enzyme pectinase is produced by Clostridium sp, which is used in retting of flax.

(d) Antibiotics: These are the chemical substances produces by living microorganisms capable of inhibiting or destroying other microbes. These are the products of secondary and minor metabolic pathways, mostly secreted extracellularly by the microorganisms. These are used in controlling various infectious diseases.

At present more than 5000 antibiotic substances are known and approximately 100 are available for medicinal use. The most important bacterium which produces maximum number of antibiotics is Streptomyces.


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🔰 Role of Bacteria in nitrogen cycle 🔰

Nitrogen fixation:
(1) Many free-living soil inhabiting bacteria such as, Azotobacter (aerobic), Clostridium (anaerobic), etc. have ability to fix atmospheric nitrogen into ammonia.

(2) The other group of nitrogen fixing bacteria lives in symbiotic association with other plants.

(3) The most important symbiotic nitrogen fixing bacteria is Rhizobium spp.

(4) The various species of Rhizobium inhabit different leguminous plants. For example, R. leguminosarium infects soyabeans, etc.

(5) They develop root nodules and fix atmospheric nitrogen into ammonia in symbiotic association with leguminous plants.

(6) The fixed nitrogen is partly taken up by the leguminous plants and metabolised.

(7) A part of fixed nitrogen is diffused out into the surrounding soil.

Ammonification:
(1) The nitrogenous compounds of the dead remains of plants, animals and their excretory products are decomposed into ammonia by a number of bacteria and other microorganisms.

(2) The conversion of nitrogenous organic compounds into ammonia is termed as ammonification.

(3) It is carried by many ammonifying bacteria such as Bacillus ramosus, B. vulgaris, B. mycoides, etc.

Nitrification:
(1) Many bacteria enhance the nitrogen fertility of soil by converting ammonium compounds to nitrites (e.g., Nitrosomonas) and nitrites into nitrates (e.g., Nitrobacter).

(2) The Nitrosomonas group oxidizes ammonia into nitrite –

(3)The Nitrobacter group oxidizes nitrite to nitrates –



Denitrification:
The nitrates and ammonia are converted to nitrous oxide and finally to nitrogen gas by several denitrifying bacteria, e.g., Pseudomonas fluorescence, P. denitrificans, Bacillus subtilis, Thiobacillus denitirficans, etc.

Useful activities
(i) Decay of organic wastes: Many saprotrophic bacteria act as natural scavengers by continuously removing the harmful organic wastes (i.e., dead remains of animals and plants) from man's environment. They decompose the organic matter by putrifaction and decay. The simple compounds produced as a result of decomposition and decay (viz., carbon dioxide, carbon monoxide, nitrates, sulphates, phosphates, ammonia, etc.) are either released back into the environment for recycling or absorbed by the plants as food. Thus, the bacteria play duel role by disposing of the dead bodies and wastes of organisms and by increasing the fertility of soil.

(ii) Role in improving soil fertility: Saprotrophic bacteria present in soil perform various activities for their survival. Some of these activities improve the fertility of soil by formation of humus, manure, etc.

(a) Humus: The microbial decomposition of organic matter and mineralization results in the formation of complex amorphous substance called humus. The humus improves the aeration, water holding capacity, solubility of soil minerals, oxidation-reduction potential and buffering capacity of the soil.

(b) Composting: It is conversion of farm refuse, dung and other organic wastes into manure by the activity of saprotrophic bacteria (e.g., Bacillus stearothermophilus, Clostridium thermocellum, Thermomonospora spp, etc.)

(c) Adding sulphates: A few sulphur bacteria (e.g., Beggiatoa) add sulphur into the soil by converting H2S into sulphates.

(iii) Sewage, disposal: Ability of anaerobic bacteria to purify the organic matter is used in the the sewage disposal system of cities. The faeces are stored in covered reservoirs and allowed to purify. The solid matter is decomposed into liquidy sludge which is passed through coarse filters. The effluent is finally purified and drained out into the river or used as fertilizer in the fields. The common bacteria involved in sewage disposal are – Coliforms (E. coli), Streptococci, Clostridium, Micrococcus, Proteus, Pseudomonas, Lactobacillus, etc.

(iv) Role in Industry: Useful activities of various bacteria are employed in the production of a number of industrial products. Some of these are given below–

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