Notes on Plant Growth and Development

(d) Bolting and Flowering: Gibberellins induce stem elongation in ‘rosette plants’ e.g., cabbage, henbane, etc. Such plants show retarded internodal growth and profuse leaf development. In these plants just prior to the reproductive phase, the internodes elongate enormously causing a marked increase in stem height. This is called bolting.

(e) Enzyme formation: One of the most dramatic effects of GA is its induction of hydrolytic enzymes in the aleurone layer of endosperm of germinating barley seeds and cereal grains. GA stimulates the production of digestive enzymes like proteases, a-amylases, lipases which help to mobilise stored nutrients.

(f) Breaking of dormancy: Gibberellins overcome the natural dormancy of buds, tubers, seeds, etc. and allow then to grow. In this function, gibberellins act antagonistically to abscisic acid (ABA).

(g) Parthenocarpy: Gibberellins have been considered to be more effective than auxins for inducing parthenocarpy in fruits like apple, tomato and pear. GA application has also resulted in the production of large fruits and bunch length in seedless grapes.

(h) Sex expression: Gibberellins control sex expression in certain plants. In general, gibberellin promotes the formation of male flowers either in place of female flowers in monoecious plants such as cucurbits or in genetically female plants like Cannabis, Cucumis.

(3) Cytokinins (Phytokinins):
(i) Cytokinins are plant growth hormones which are basic in nature, either aminopurine or phenyl urea derivatives that promote cell division (cytokinesis) either alone or in conjugation with auxin.

(ii) Functions of cytokinins

(a) Cell division: Cytokinins are essential for cytokinesis and thus promote cell division. In presence of auxin, cytokinins stimulate cell division even in non-meristematic tissues.

(b) Cell enlargement and Differentiation: Under some conditions cytokinins enhance the expansion of leaf cells in leaf discs and cotyledons. These cells considered to be mature and under normal conditions do not expand.

(c) Delay in senescence: Cytokinin delay the senescence (ageing) of leaves and other organs by controlling protein synthesis and mobilization of resources (Disappearance of chlorophyll). It is called Richmond Lang effect.

(d) Counteraction of apical dominance: Auxins and cytokinins act antagonistically in the control of apical dominance. Auxins are responsible for stimulating growth of apical bud.

(e) Breaking of dormancy: Cytokinins breaks seeds dormancy of various types and thus help in their germination.

(f) Accumulation and Translocation of solutes

(4) Ethylene:
(i) Ethylene is a gaseous hormone which stimulates transverse growth but retards the longitudinal one.

(ii) Functions of ethylene

(a) Fruit growth and Ripening: Ethylene promotes fruit growth and its ripening. The harmone is used in the artificial ripening of climacteric fruits (e.g., Apple, Banana, and Mango).

(b) Transverse growth: Ethylene inhibits longitudinal growth but stimulates transverse growth so that stem looks swollen.

(c) Epinasty (leaf bending): Epinasty represents more growth on upper surface of leaf than on lower surface. Epinasty is said to be controlled by ethylene in many plants.

(d) Abscission: Ethylene stimulates formation of abscission zone in leaves, flowers and fruits.

(e) Apical dominance: Ethylene inhibits the growth of lateral buds and thus causes apical dominance (in pea). It is believed that auxin might be functioning partly through synthesis of ethylene in causing apical dominance.

(f) Root initiation: In low concentration, ethylene stimulates root initiation and growth of lateral roots and root hair.

(g) Flowering: Ethylene stimulates flowering in pineapple and related plants though in other cases, the hormone causes fading of flowers.

🔰Notes on Wave Optics

Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.

(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.

(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.

Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon

(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.

Reflection of plane wave at plane surface (Laws of reflection):-

a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.

So, ∠i = ∠r

This signifies angle of incidence is equal to the angle of reflection.

Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.

(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.

sin i/sin r = v1/v2 = 1µ2 = constant

Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.

(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.

Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.

Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.

Displacement equation:- y = R sin 2π/λ (vt+x/2)

Amplitude:- R = 2a cos πx/λ

Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]

Maxima:- A point having maximum intensity is called maxima.

x = 2n (λ/2)

A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.

Imax = 4Ka2 = 4i (Here, i = Ka2)

Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)

A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.

Imin = K. 4a2×0 = 0

Condition for constructive interference:-
Path difference = (2n)λ/2

Phase difference = (2n)π

Condition for destructive interference:-
Path difference = (2n+1)λ/2

Phase difference = (2n+1)π

Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.

(b) The amplitudes of the two waves should be either or nearly equal

(c) The two sources should be narrow.

(d) The sources should be close to each other.

(e) The two sources should be coherent one.

Young’s double slit experiment:-
Path difference, x = yd/D

Maxima, y = nλD/d

Here, n = 0,1,2,3….

Minima, y = (2n+1) λD/d

Here, n = 0,1,2,3….

Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d

Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-

?y = (β/λ) (µ-1)t

(b) Shift across a particular point of observation:-

µ = (mλ/t) +1

Lloyd’s single mirror:-
?λ = β .2a/D

Power of lens:- P = 100/f

📚Notes on Trigonometric Equations and Identities📚


A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).

In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).

Periods of various trigonometric functions are listed below:

1) sin x has period 2π

2) cos x has period 2π

3) tan x has period π

4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a

5) tan (ax+b) and cot (ax+b) have π/a as their period

6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a

7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period

➖Sum and Difference Formulae of Trigonometric Ratios

1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)

2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)

3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)

4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)

5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]

6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]

7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)

8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)

9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]

10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]

➖Double or Triple -Angle Identities

1) sin 2x = 2sin x cos x

2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1

3) tan 2x = 2 tan x / (1-tan 2x)

4) sin 3x = 3 sin x – 4 sin3x

5) cos3x = 4 cos3x – 3 cosx

6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)

➖For angles A, B and C, we have

1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC

2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC

3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C

4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]


➖List of some other trigonometric formulas:

1) 2sinAcosB = sin(A + B) + sin (A - B)

2) 2cosAsinB = sin(A + B) - sin (A - B)

3) 2cosAcosB = cos(A + B) + cos(A - B)

4) 2sinAsinB = cos(A - B) - cos (A + B)

5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]

6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]

7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]

8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]

9) tanA ± tanB = sin (A ± B)/ cos A cos B

10)cot A ± cot B = sin (B ± A)/ sin A sin B

➖Method of solving a trigonometric equation:

1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.

2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.

3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.

➖Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a

2) sin θ = 0 ⇒ θ = nπ

3) cosθ = 0 ⇒ θ = (2n + 1)π/2

4) tan θ = 0 ⇒ θ = nπ

5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]

6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]

7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]

8) sinθ = 1 ⇒ θ= (4n + 1)π/2

9) sin θ = -1 ⇒ θ = (4n - 1) π /2

10) sin θ = -1 ⇒ θ = (2n +1) π /2

11) |sinθ| = 1⇒ θ =2nπ

12) cosθ = 1 ⇒ θ =(2n + 1)

13) |cosθ| = 1⇒ θ =nπ

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–

(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.

🔋Chemistry Mnemonics🔋

ACTIVITY SERIES OF METALS

🍎Potassium>Sodium>Calcium>Magnesium>Aluminium>(Carbon)>Zinc>Iron>Tin>Lead>(Hydrogen)>Copper>Mercury>Silver>Gold>Platinum🍎

Mnemonics

👌Please Stop Calling Me A Carless Zebra Instead Try Learning How Copper Miners Save Gold Pit

🧣Note that Carbon and Hydrogen are non-metals, used as a baseline.

Notes on Plant Growth and Development

(1) The analysis of growth curve shows that it can be differentiated into three phases:

(i) Lag phase: It represents initial stages of growth. The rate of growth is very slow in lag phase. More time is needed for little growth in this phase.

(ii) Log phase (Exponential phase): The growth rate becomes maximum and more rapid. Physiological activities of cells are at their maximum. The log phase is also referred to as grand period of growth.

(iii) Final steady state (Stationary phase) or Adult phase: When the nutrients become limiting, growth slows down, so physiological activities of cells also slows down. This phase is indicated by the maturity of growth system. The rate of growth can be measured by an increase in size or area of an organ of plant like leaf, flower, fruit etc. The rate of growth is called efficiency index.

(2) Phytohormones:-

(i) Growth hormones also called phytohormones

(ii) Term given by Thimann (1948),

(iii) It can be defined as ‘the organic substances which are synthesized in minute quantities in one part of the plant body and transported to another part where they influence specific physiological processes’.

Growth Hormones and Growth Regulators
(1) Auxins:
(i) Auxins (Gk. auxein = to grow) are weakly acidic growth hormones having an unsaturated ring structure and capable of promoting cell elongation, especially of shoots (more pronounced in decapitated shoots and shoot segments) at a concentration of less than 100 ppm which is inhibitory to the roots. Among the growth regulators, auxins were the first to be discovered.

(ii) Types of auxins: There are two major categories of auxins natural auxins and synthetic auxins:

(a) Natural auxins: These are naturally occurring auxins in plants and therefore, regarded as phytohormones. Indole 3-acetic acid (IAA) is the best known and universal auxin. It is found in all plants and fungi.

(b) Synthetic auxins: These are synthetic compounds which cause various physiological responses common to IAA. Some of the important synthetic auxins are 2, 4-D (2, 4-dichlorophenoxy acetic acid) is the weedicide. IBA is both natural and synthetic auxin.

(iii) Functions of auxins: Auxins control several kinds of plant growth processes. These are as follows:

(a) Cell elongation: Auxins promote elongations and growth of stems and roots and enlargement of many fruits by stimulating elongation of cells in all directions.

(b) Apical dominance: In many plants, the apical bud grows and the lower axillary buds are suppressed. Removal of apical bud results in the growth of lower buds. The auxin (IAA) of the terminal bud inhibits the growth of lateral buds. This phenomenon is known as apical dominance.

(c) Weed control: Weeds are undesirable in a field with a crop. By the spray of 2, 4-D, broad-leaved weeds can be destroyed but 2, 4-D does not affect mature monocotyledonous plants.

(d) Root differentiation

(e) Control of lodging

(f) Parthenocarpy: Parthenocarpy can be induced by application of IAA in a paste form to the stigma of a flower or by spraying the flowers with a dilute solution of IAA.

(2) Gibberellins:
(i) Gibberellins are weakly acidic hormones having gibbane ring structure which cause cell elongation of intact plants in general and increased internodal length of genetically dwarfed plants (i.e., corn, pea) in particular.

(ii) Functions of gibberellin

(a) Stem elongation: The gibberellins induce elongation of the internodes.

(b) Leaf expansion: In many plants leaves become broader and elongated when treated with gibberellic acid.

(c) Reversal of dwarfism: One of the most striking effects of gibberellins is the elongation of genetic dwarf (mutant) varieties of plants like corn and pea.

❄️Notes on Quadratic Equations⛄️


In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.

The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a

If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:

α + β = -b/a

α.β = c/a

α – β = √D/a

It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.

Nature of Roots:

Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:

D > 0 iff the roots are real and distinct i.e. the roots are unequal

D = 0 iff the roots are real and coincident i.e. equal

D < 0 iffthe roots are imaginary

The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.

Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then

If D > 0 and is also a perfect square then the roots are rational and unequal.

If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.

If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.

So if α and β are the roots of equation then the quadratic equation is

x2 – (α + β)x + α β = 0

For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.

If a > 0, then the shape of the parabola is concave upwards

If a < 0, then the shape of the parabola is concave upwards

Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.

The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

y ∈[(4ac-b2) / 4a, ∞] if a > 0

If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]

The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0

In general, if α1,α2, α3, …… ,αn are the roots of the equation

f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then

1.Σα1 = - a1/a0

2.Σ α1α2 = a2/a0

3.Σ α1α2α3 = - a3/a0

……… ……….

Σ α1α2α3 ……αn= (-1)n an/a0

Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.

If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.

Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.

💠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):-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) In this case, liquids do not wet the solids.

(iii) Such liquids when put in the solid tube will have a meniscus convex upwards.

(c) When θ = 90º:-?

Fa=Fc/√2

The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.

(d) cosθc = Tsa – Tsl/Tla

Here, Tsa,Tsl and Tla represent solid-air, solid-liquid and liquid-air surface tension respectively). Here θc is acute if Tsl < Tsa while θc is obtuse if Tsl >Tsa.

⛔️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

🧩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).

♦️Revision Notes on Arithmetic Progression♦️
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d

The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]

If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)

If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]

If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.

If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.

If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.

Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.

If nth term of a series is tn = An + B, then the series is in A.P.

If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.

In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.

Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d

Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.

In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.

Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.

The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.

The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.

Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d

Hence, this gives ‘d’ = (b-a)/(n+1)

✅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.

Some results which are useful for solving trigonometric equations:

1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a

2) sin θ = 0 ⇒ θ = nπ

3) cosθ = 0 ⇒ θ = (2n + 1)π/2

4) tan θ = 0 ⇒ θ = nπ

5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]

6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]

7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]

8) sinθ = 1 ⇒ θ= (4n + 1)π/2

9) sin θ = -1 ⇒ θ = (4n - 1) π /2

10) sin θ = -1 ⇒ θ = (2n +1) π /2

11) |sinθ| = 1⇒ θ =2nπ

12) cosθ = 1 ⇒ θ =(2n + 1)

13) |cosθ| = 1⇒ θ =nπ

Tricks for Conversions in Organic Chemistry

✅Haloalkanes can give you every possible functional group. They can also help you in increasing the chain size. So you might want to convert your starting material to haloalkanes and then go to the desired product

✅Remember the series of oxidation/reduction: Hydrocarbons (with various substituents) can be oxidised to alcohols, then appropriate carbonyl compounds and lastly carboxylic acids (or their derivatives)

✅Strong oxidizing agents are KMnO4 and K2Cr2O7

✅Mild oxidising agents depending on the situation are CrO4, Ammoniacal AgNO3, Benedicts Solution, Fehling’s solution, Cu or CuO at 573K, Bromine water etc.

✅Any carboxylic acid derivative can be converted back to the carboxylic acid by hydrolysis.

✅There are possibilities of hydride and methyl to get the most stable intermediate (carbocation/free radical)

✅A most common reaction, if your starting material is an alkane, is free-radical halogenation.

✅Saytzeff/Markovnikov's rules must be kept in mind while dealing with alkenes (they are based on the electron displacement effects only)

✅Decarboxylation and ozonolysis could go-to methods for decreasing the number of carbon atoms

✅Grignard reagent gives you the much-needed Alkyl nucleophile, which can be used at appropriate places.

🎓🎓🧩Mathematics formulas🧩🎯🎯

1. (α+в)²= α²+2αв+в²
2. (α+в)²= (α-в)²+4αв
3. (α-в)²= α²-2αв+в²
4. (α-в)²= (α+в)²-4αв
5. α² + в²= (α+в)² - 2αв.
6. α² + в²= (α-в)² + 2αв.
7. α²-в² =(α + в)(α - в)
8. 2(α² + в²) = (α+ в)² + (α - в)²
9. 4αв = (α + в)² -(α-в)²
10. αв =1. (α + в + ¢)² = α² + в² + ¢² + 2(αв + в¢ + ¢α)
12. (α + в)³ = α³ + 3α²в + 3αв² + в³
13. (α + в)³ = α³ + в³ + 3αв(α + в)
14. (α-в)³=α³-3α²в+3αв²-в³
15. α³ + в³ = (α + в) (α² -αв + в²)
16. α³ + в³ = (α+ в)³ -3αв(α+ в)
17. α³ -в³ = (α -в) (α² + αв + в²)
18. α³ -в³ = (α-в)³ + 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α+в)+ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α+в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α+в)+¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α+в)
ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
» ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв +ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢ = 2я
» α = в ¢σѕ¢ + ¢ ¢σѕв
» в = α ¢σѕ¢ + ¢ ¢σѕα
» ¢ = α ¢σѕв + в ¢σѕα
» ¢σѕα = (в² + ¢²− α²) / 2в¢
» ¢σѕв = (¢² + α²− в²) / 2¢α
» ¢σѕ¢ = (α² + в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η + 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα

1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 + ѕιη2α = (ѕιηα + ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 + тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 + тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα + ¢σѕ3α

» ѕιη²Θ+¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ

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

This signifies, height of liquid risen (or depressed) in a capillary tube varies inversely as the radius of tube. Smaller the diameter of capillary tube, greater is the rise of liquid in it.

Tube of insufficient length:-
Rh = 2T/ρg

As, T, ρ and g are all constant, Rh = Constant

Smaller the value of h, greater will be the value of R. But liquid will never flow.

Effect of temperature affecting surface tension of liquids:-

Surface tension of a liquid decreases with an increase in its temperature.

Tθ = K (θc-θ)

Here Tθ is the surface tension at a particular temperature θ while θc is the critical temperature of the liquid and K is constant.

⛔️Effect of density:- Density of liquid also affects its surface tension. Surface tension of a liquid is given by,

T = A (ρ - ρ')n

Here, ρ is the density of liquid, ρ' is the density of saturated vapors of liquid and A is the constant depending on the nature of liquid.

⛔️Pressure difference across a liquid surface:-

(a) Plane surface:- There is no difference of pressure on the two sides of the film.

(b) Convex surface:-Pressure below the surface film must be greater than that just above it.

(c) Concave surface:- Pressure on the upper side is greater than that just below it.

⛔️General formula for excess pressure:-

Pexcess =T[1/R1 + 1/R2]

⛔️Excess pressure in liquid drop:-

Pexcess = 2T/R, Here R is the radius of liquid drop.

⛔️Excess pressure for an air bubble in liquid drop:-

Pexcess = 2T/R

⛔️Excess pressure in soap bubble:-

Pexcess = 4T/R, Here R is the radius of soap bubble.

⛔️Pressure inside an air bubble at a depth h in a liquid:- Pin = Patm+ hdg + (2T/R)

⛔️Forces between two plates with thin water film separating them:-

(a) ΔP = T (1/r – 1/R)

(b) F = AT (1/r – 1/R)

(c) If separation between plate is d, then ΔP = 2T/d and F = 2AT/d

⛔️Radius of curvature of common film:- Rcomon = rR/R-r

⛔️Capillary depression, h = 2T cos (π-θ)/rdg

⛔️Shape of liquid surface:-

(a) Plane surface (as for water – silver) if Fadhesive > Fcohesive/√2

(b) Concave surface (as for water – glass) if Fadhesive > Fcohesive/√2

(c) Convex surface (as for mercury-glass) if Fadhesive < Fcohesive/√2

⛔️Increase in temperature:-

Δθ = 3T/ρs (1/r – 1/R) or Δθ = 3T/ρsJ (1/r – 1/R)

✅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

✅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):
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

🔥Increasings or Decreasing Order🔥

🔴 01. Melting point=
Li > Na > K > Rb > Cs

🔴 02. Colour of the flame=
Li-Red, Na-Golden, K-Violet, Rb-Red, Cs-Blue, Ca-Brick red, Sr-Blood red, Ba-Apple green

🔴 03. Stability of hydrides =
LiH > NaH > KH > RbH> CsH

🔴 04. Basic nature of hydroxides=
LIOH < NaOH < KOH < RbOH < CsOH

🔴 05. Hydration energy=
Li> Na > K> Rb > Cs

🔴 06. Reducing character=
Li > Cs > Rb > K > Na

🔴 07. Stability of +3 oxidation state=
B> Al > Ga > In > T1

🔴 08. Stability of +1 oxidation state= Ga < In < TI

🔴 09. Basic nature of the oxides and hydroxides=
B< Al< Ga < In < TI

🔴 10. Relative strength of Lewis acid= BF3 < BCl3 < BBr3 < BI3

🔴 11. Ionisation energy=
B> Al <Ga > In <TI

🔴 12. Reactivity=
C<Si< Ge < Sn <Pb

🔴 13. Metallic character=
C< Si < Ge < Sn < Pb

🔴 14. Acidic character of the oxides=
Co2 > SiO2 > Ge02 > SnO2 > PbO2

🔴 15. Reducing nature of hydrides=
CH4 < SiH4 < GeH4 < SnH4 < PbH4

🔴 16. Thermal stability of tetrahalides=
CCl4> SiCl4> GeCl4> SnCl4 > PbCl4

🔴 17. Oxidising character of M+4 species=
GeCl4 < SnCl4 < PbCl4

🔴 18. Ease of hydrolysis of tetrahalides=
SiCl4 < GeCl4 < SnCl4 < PbCI4

🔴 19. Acidic strength of trioxides=
N203 > P2O3 > As2O3

🔴 20. Acidic strength of pentoxides=
N2O2 > P2O2> As202 > Sb2O2 > Bi̟202

🔴 21. Acidic strength of oxides of nitrogen=
N2O < NO <N2O3 <N2O4 < N2O5

🔴 22. Basic nature/ bond angle/ thermal stability and dipole moment of hydrides=
NH3 > PH3 > AsH3 > SbH3 > BiH3

🔴 23. Stability of trihalides of nitrogen=
NF3 > NCl3 > NBr3

🔴 24.Lewis base strength=
NF3 <NCl3 <NBr3 < NI3

🔴 25. Ease of hydrolysis of trichlorides=
NCl3 > PCI3 > AsCl3 > SbCl3 > BiCl3

🔴 26. Lewis acid strength of trihalides of P, As, and Sb=
PCl3 > ASCl3 > SbCl3

🔴 27. Lewis acid strength among phosphorus trihalides
PF3 > PCl3 > PBr3 > PI3

🔴 28. Melting and boiling point of hydrides=
H2O > H2Te > H2Se >H2S

🔴 29. Volatility of hydrides=
H2O < H2Te < H2Se < H2S

🔴 30. Reducing nature of hydrides=
H2S < H2Se < H2Te

🔴 31. Covalent character of hydrides=
H2O < H2S < H2Se < H2Te

🔴 32. The acidic character of oxides (elements in the same oxidation state)=
SO2 > SeO2 > TeO2 > PoO2
SO3 > SeO3 > TeO3

🔴 33. Acidic character of oxide of a particular element (e.g. S)=
SO < SO2 < SO3
SO2 > TeO2 > SeO2 > PoO2

🔴 34. Bond energy of halogens=
Cl2 > Br2 > F2 > I2

🔴 35. Solubility of halogen in water =
F2 > Cl2 > Br2 > I2

🔴 36. Oxidising power=
F2 > Cl2 > Br2 > I2

🔴 37. Enthalpy of hydration of X ion=
F- > Cl- > Br- >I-

🔴 38. Reactivity of halogens:=
F> Cl> Br > I

🔴 39. Ionic character of M-X bond in halides
= M-F > M-Cl > MBr > M-I

🔴 40. Reducing character of X ion:=
I- > Br- > Cl- > F-

🔴 41. Acidic strength of halogen acids=
HI > HBr > HCI > HF

🔴 42. Reducing property of hydrogen halides
= HF < HCL < HBr < HI

🔴 43. Oxidising power of oxides of chlorine
= Cl2O > ClO2 > Cl206 > Cl2O7

🔴 44. Decreasing ionic size=
02- > F- > Na+ > Mg2+

🔴 45. Increasing acidic property=
Na2O3 < MgO < ZnO< P205

🔴 46. Increasing bond length=
N2 <02 < F2 < CL2

🔴 47. Increasing size=
Ca2+ < Cl- < S2-

🔴 48. Increasing acid strength=
HClO < HClO2 < HClO3 < HClO4

🔴 49. Increasing oxidation number of iodine=
HI< I2 <ICl <HIO4

🔴 50. Increasing thermal stability=
HOCl < HOClO < HOClO2 < HOClO3

🔰 Notes on Kinetic Theory of Gases 🔰

Kinetic Theory of Matter:-

(a) Solids:- It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.
(b) Liquids:- It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.
(c) Gases:- It is the type of matter does not have any fixed shape or any fixed volume.

Ideal Gas:- A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.

Ideal Gas Equation:- A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.
PV/T = Constant or PV = nRT
Here, n is the number of moles and R is the universal gas constant.

Gas Constant:-
(a) Universal gas constant (R):-
R= P0 V0/T0
=8.311 J mol-1K-1

(b) Specific gas constant (r):-
PV= (R/M) T = rT,
Here, r = R/M

Real Gas:-The gases which show deviation from the ideal gas behavior are called real gas.

Vander wall’s equation of state for a real gas:-
[P+(na/V)2?][V-nb] = nRT
Here n is the number of moles of gas.

Avogadro’s number (N):- Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.

N = 6.023×10^23

(a) To calculate the mass of an atom/molecule:-
Mass of one atom = atomic weight (in gram)/N
Mass of one molecule = molecular weight (in gram)/N

(b) To calculate the number of atoms/molecules in a certain amount of substance:-
Number of atoms in m gram = (N/atomic weight)×m
Number of molecules in m gram = (N/molecular weight)×m

(c) Size of an atom:-
Volume of the atom, V = (4/3)πr3
Mass of the atom, m = A/N
Here, A is the atomic weight and N is the Avogadro’s number.
Radius, r =[3A/4πNρ]1/3\
Here ρ is the density.

Gas laws:-

Graph Between Pressure and Volume for Boyle's Law(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.

PV = Constant

(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.

Graph Between Volume and Temperature for Charles LawV/T= Constant

V–V0/V0t = 1/273 = γp

Here γp (=1/273) is called volume coefficient of gas at constant pressure.

Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.

(c) Gay Lussac’s law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.

P/T = P0/T0 or P – P0/P0t = 1/273 = γp

Here γp (=1/273) is called pressure coefficient of the gas at constant volume.

Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.

(d) Dalton’s law of partial pressures:-

Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.

P= p1+p2+p3+……..

nRT/V = p1+p2+p3+……..

(e) Grahm’s law of diffusion:- Grahm’s law of diffusion states that the rate of diffusion of gases varies inversely as the square root of the density of gases.

R∝1/√ρ or R1/R2 =√ρ2/ ρ1

So, a lighter gas gets diffused quickly.

(f) Avogadro’s law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.

For m gram of gas, PV/T = nR = (m/M) R

📖Notes on Thermal Properties of Matter

Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.

Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.

β = St-S0/S0t

Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .

γ = Vt-V0/V0t

Relation between expansion coefficients:-

(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3

Thermal expansion of liquids:-

(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.

γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)

(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.

γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)

Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.

W∝H or W = JH

Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.

J = W/H

Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1

Specific heat capacity or specific heat (c):-

Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.

c = Q/mΔT

Unit:- kcal kg-1K-1 or J kg-1K-1

Dimension:- M0L2T-2K-1

Molar specific heat capacity(C):-

Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)

Heat Capacity or Thermal Capacity:-

It is defined as the amount of heat required to raise the temperature of body through 1ºC.

Q = mcΔT

If ΔT = 1ºC, Q = heat capacity = mc

Unit:- kcal K-1 or JK-1

Water Equivalent:-

Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.

w = mc

Water equivalent of a body is equal to the product of its mass and its specific heat.

Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.

(a)Specific latent heat of fusion (Lf):-

Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.

(b) Specific latent heat of vaporization (Lv):-

Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.

Dimensional formula:- M0L2T-2

Unit:- kg cal kg-1 or J kg-1

Triple point of water = 273.16 K

Absolute zero = 0 K = -273.15ºC

For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)

For a resistance thermometer, Re = R0[1+αθ]