Functional group :
A functional group is an'atom or a specific combination of bonded atoms that react in a characteristic way, irrespective of the organic molecule in which it is present.
oxalic acid - COOH

(i) 2-ethyl-but-4-ene-oic acid
(ii) 2, 2-dimethyl-hexanoic acid

σ- bon:
When two atomic orbitals overlap linearly along the axis, the resultant bond is called a sigma (σ) bond.

(i) The temperature below which a gas can be liquefied by application of pressure is known as critical temperature.
(ii) The critical temperature of CO₂ gas is 303.98 K.

Shape of s-orbital - sphere
Shape of p-orbital- dumb bell
Shape of d-orbital- clover leaf

The rate of effusion or diffusion of a gas is inversely proportional to the square root of molar mass at constant temperature and pressure.

(i) By bombarding lithium with slow neutrons
(ii) ₃Li⁶+₀n¹ ➝ ₁T³+₂He⁴

Mg + I.E₁ ⟶ Mg⁺ +1e^-     .....(i)
Mg⁺ + I.E₂ ⟶ Mg²⁺ + 1e^-  ....(ii)
Mg²⁺ + I.E₃ ⟶ Mg³⁺ + 1e^-  ....(iii)
(i) The step (iii) which involves the formation of Mg³⁺ requires higher ionisation energy.
(ii) Mg²⁺ consist of 10 electrons (2, 8) attaining the stable noble gas configuration of argon (Z = 10).
(iii) Since the valence orbital is completely filled, more energy will be required to remove electrons.

(i) \(\underset { n-pentane }{ { CH }_{ 3 }-{ CH }_{ 2 }-{ CH }_{ 2 }-{ CH }_{ 2 }-{ CH }_{ 3 } } \) 
(ii) \(\underset { Isopentane }{ { CH }_{ 3 }-\underset { \overset { | }{ { CH }_{ 3 } } }{ CH } -{ CH }_{ 2 }-{ CH }_{ 3 } } \) 
(iii) \(\underset { Neopentane }{ { CH }_{ 3 }-\overset { \underset { | }{ { CH }_{ 3 } } }{ \underset { \overset { | }{ { CH }_{ 3 } } }{ C } } -{ CH }_{ 3 } } \)
(i), (ii) and (iii) are chain isomers.

The knowledge of equilibrium constant helps us to
1. Predict the direction in which the net reaction will take place
2. Predict the extent of the reaction and
3. Calculate the equilibrium concentrations of the reactants and products.
It is to be noted that these constants do not provide any information regarding the rates of the forward or reverse reactions.

A carbon atom whose tetra valency is satisfied by four different substituents (atoms or groups) is called asymmetric carbon or chiral carbon. It is indicated by an asterisk as C*. A molecule possessing chiral carbon atom and non-super impossible to its own mirror image is said to be a chiral molecule or asymmetric, and the property is called chirality or dissymmetry.

Dipole moment :
The polarity of a covalent bond can be measured in terms of dipole moment which is defined as \(\mu=q \times 2d\)
Where \(\mu\) is the dipole moment, q is the charge and 2d is the distance between the two charges.
Where p is the dipole moment, q is the charge and 2d is the distance between the two charges. The dipole moment is a vector and the direction of the dipole moment vector points from the negative charge to positive charge.

The unit for dipole moment is columb meter (C m). It is usually expressed in Debye unit (D). The conversion factor is 1 Debye = 3.336 x 10^-30 C m.

Na⁺, Mg²⁺ and Al³⁺ are isoelectronic cations.

The cation with the greater positive charge will have a smaller radius because of the greater attraction of the electrons to the nucleus. Hence the increasing order of ionic radius is, r_Na⁺ > r_Mg²⁺ > r_Al³⁺

According to Boyle's law, at constant temperature for a given mass of gas at constant temperature,
P₁V_I = P₂V₂ = P₃V₃
1 atm x 1 dm³ = 2 atm x V₂ = P₃ x 0.25 dm³
\(\therefore\) 2 atm x V₂ = 1 atm x 1 dm³

\(\boxed{V_2=0.5\ dm^3}\)
and P₃ x 0.25 dm³ = 1 atm x 1 dm³

\(\boxed{P_3=4\ atm}\)

Milk of lime reacts with chlorine to form hypochlorite, a constituent of bleaching powder.
2Ca (OH)₂ + 2Cl₂ ⟶ CaCl₂ + Ca(OCI)₂ + 2H₂O

Erwin Schrodinger expressed the wave nature of electron in terms of a differential equation. This equation determines the change of wave function in space depending on the field of force in which the electron moves. The time independent Schrodinger equation can be expressed as,
\(\overset { \wedge }{ H } \psi =E\psi \)          .........(1)
Where \(\overset { \wedge }{ H } \) is called Hamiltonian operator, \(\psi \) is the wave function and is a function of position coordinates of the particle and is denoted as \(\psi \) (x, y, z) E is the energy of the system
\(\overset { \wedge }{ H } =\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 } } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V \right] \)
can be written as
\(\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 }m } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V\Psi \right] =E\Psi \)
Multiply by \(\frac { 8{ \pi }^{ 2 }m}{ { -h }^{ 2 } } \)and rearranging
\(\frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } +\frac { 8{ \pi }^{ 2 }m }{ { -h }^{ 2 } } (E-V)\Psi =0\)          ........(2)
The above Schrodinger wave equation does not contain time as a variable and is referred to as time independent Schrodinger wave equation. This equation can be solved only for certain values of E, the total energy. i.e. the energy of the system is quantised. The permitted total energy values are called eigen values and corresponding wave functions represent the atomic orbitals.

Let us consider the formation of HI in which, 'a' moles of hydrogen and 'b' moles of iodine gas are allowed to react in a container of volume V. Let 'x' moles of each of H₂ and I₂react together to form 2x moles of HI.
\({ H }_{ 2\left( g \right) }+{ I }_{ 2\left( g \right) }\rightleftharpoons { 2HI }_{ \left( g \right) }\)

Applying law of mass action,
\({ K }_{ C }=\cfrac { { \left[ HI \right] }^{ 2 } }{ { \left[ H \right] }_{ 2 }\left[ { I }_{ 2 } \right] } \)
= \(\cfrac { \left( \cfrac { 2x }{ V } \right) ^{ 2 } }{ \left( \cfrac { a-x }{ V } \right) \left( \cfrac { b-x }{ v } \right) } =\cfrac { { 4x }^{ 2 } }{ \left( a-x \right) \left( b-x \right) } \)
The equilibrium constant Kp can also be calcu•lated as follows:
We know }he \rer~tionship between the K_c and K_p
Here the \(\Delta n_{ g }\)=n_p -n_r = 2 - 2 =0
Hence K = K_c ;\({ K }_{ p }=\cfrac { { 4x }^{ 2 } }{ \left( a-x \right) \left( b-x \right) } \)

(i) They are covalent compounds of carbon and generally insoluble in water and readily soluble in organic solvent such as benzene, toluene, ether, chloroform, etc ...
(ii) Many of the organic compounds are inflammable  (except CCI₄).They possess low boiling and melting points due to their covalent nature.
(iii) Organic compounds are characterised by functional groups. A functional group is an atom or a specific combination of bonded atoms that react in a characteristic way, irrespective of the organic molecule in which it is present. In
almost all the cases, the reaction of an organic compound takes place at the functional group. They exhibit isomerism which is a unique phenomenon.
(iv) Homologous series: A series of organic compounds each containing a characteristic functional group and the successive members differ from each other in molecular formula by a CH₂ group is called homologous series.
Alkanes: Methane (CH₄) Ethane (C₂H₆)  Propane (C₃H₈) etc.

Homologous series: A series of organic compounds each containing a characteric functional group and the successive' members differ from each other in molecular formula by a CH₂ group is called homologous series. Eg.
Alkanes : Methane (CH₄), Ethane (C₂H₆), Propane (C₃Hg) etc .
Alcohols: Methanol (CH₃OH), Ethanol (C₂H₅OH) Propanol (C₃H₇OH) etc ..)
Compounds of the homologous series are represented by a general formula Alkanes CnH_2n+2' Alkenes C_nH_2n, Alkynes C_nH_2n-2 and can be prepared by general methods. They show regular gradation in physical properties but have almost similar chemical property.

Molecular orbital diagram of nitrogen molecule (N₂) :
Electronic configuration of N atom 1s² 2s² 2p³
Electronic configuration of N, molecule
\(\sigma^{2}_{1s},\sigma^{*2}_{1s},\sigma^{2}_{2s},\sigma^{*2}_{2s}\)\(\pi^{2}_{2py},\pi^{2}_{2pz},\sigma^2_{2px}\)
Bond order = \({N_b-N_a\over2}={10-4\over2}=3\)
Molecule has no unpaired electrons hence it is diamagnetic.

Electronic Configuration of Aluminium
\(\underbrace { { Al }^{ 13 }{ 1s }^{ 2 } }_{ (n-2) } \underbrace { 2s^{ 2 }{ 2p }^{ 2 } }_{ (n-1) } \underbrace { { 3s }^{ 2 }{ 3p }^{ 1 } }_{ n } \)

∴ Effective nuclear charge = Z - S = 13 - 9.5
(Z_eff)_Al =3.5
Electronic Configuration of chlorine
\(\underbrace { { 1s }^{ 2 } }_{ (n-2) } \underbrace { 2s^{ 2 }{ 2p }^{ 2 } }_{ (n-1) } \underbrace { { 3s }^{ 2 }{ 3p }^{ 5 } }_{ n } \)

∴ Effective nuclear charge = Z - S = 17 - 10.9
(Z_eff)_cl = 6.1
(Z_eff)_cl > (Z_eff)_Al and hence r_clAl

(i) \({ K }_{ 2 }\overset { +6 }{ \underset { \underset { 2\times { 3e }^{ - } }{ \uparrow } }{ Cr_{ 2 } } } { O }_{ 7 }+K\overset { -1 }{ \underset { { 1e }^{ - } }{ \underset { \downarrow }{ I } } } +{ H }_{ 2 }{ SO }_{ 4 }\longrightarrow { K }_{ 2 }{ SO }_{ 4 }+{ \overset { +3 }{ Cr } }_{ 2 }({ SO }_{ 4 })_{ 3 }+\overset { 0 }{ I } _{ 2 }+{ H }_{ 2 }O\)
K₂Cr₂O₇ + 6KI + H₂SO₄ \(\longrightarrow \)  K₂SO₄ + Cr₂(SO₄)₃ + I₂ + H₂O
K₂Cr₂O₇ + 6KI + H₂SO₄ \(\longrightarrow \)  K₂SO₄ + Cr₂(SO₄)₃ + 3I₂ + H₂O
K₂Cr₂O₇ + 6KI + 7H₂SO₄  \(\longrightarrow \)  4k₂SO₄ + Cr₂(SO₄)₃ + ₃I₂ + 7H₂
ii) \({ K }Mno_{ 4 }+{ Na }_{ 2 }{ So }_{ 3 }\longrightarrow { MnO }_{ 2 }+{ Na }_{ 2 }{ So }_{ 4 }+KOH\)
 \({ K }\overset { +7 }{ \underset { \underset { 3e^{ - } }{ \uparrow } }{ M } } n{ O }_{ 4 }+{ Na }_{ 2 }\overset { +4 }{ \underset { { 2e }^{ - } }{ \underset { \downarrow }{ S } } } { O }_{ 3 }\longrightarrow \overset { +4 }{ M } { nO }_{ 2 }+{ Na }_{ 2 }\overset { +6 }{ s } { O }_{ 4 }+KOH\)
\(\Rightarrow\) 2KMnO₄ + 3Na₂SO₃ \(\longrightarrow \)   MnO₂ + Na₂ SO₄ + KOH
\(\Rightarrow\) 2KMnO₄ + 3Na₂SO₃ \(\longrightarrow \)  2MnO₂ + 3Na₂SO₄ + KOH
\(\Rightarrow\) 2KMNO₄ + 3NaSO₃ + H₂O \(\longrightarrow \) 2MnO₂ + ₃Na₂ SO₄ + 2KOH
iii) \(Cu+{ HNO }_{ 3 }\longrightarrow Cu\left( { No }_{ 3 } \right) _{ 2 }+{ No }_{ 2 }+{ H }_{ 2 }O\)
\(\overset { 0 }{ \underset { \underset { 2e^{ - } }{ \downarrow } }{ Cu } } { O }_{ 7 }+H\overset { +5 }{ \underset { { 1e }^{ - } }{ \underset { \uparrow }{ N } } } { O }_{ 3 }\longrightarrow \overset { +2 }{ Cu } \left( { No }_{ 3 } \right) _{ 2 }+\overset { +4 }{ N } { O }_{ 2 }+{ H }_{ 2 }O\)
Cu +2HNO₃ \(\longrightarrow \)  Cu(NO₃)₂ + NO₂ + H₂O
Cu + 2HNO₃ + 2HNO₃ \(\longrightarrow \) Cu(NO₃)₂ + 2NO₂ + 2H₂O
Cu + 4HNO₃ \(\longrightarrow \) Cu (NO₃)₂ + 2No₂ + 2H₂O
iv) \({ KMn }O_{ 4 }+{ H }_{ 2 }{ C }_{ 2 }{ O }_{ 4 }+{ H }_{ 2 }{ SO }_{ 4 }\longrightarrow { K }_{ 2 }{ SO }_{ 4 }+{ MnSO }_{ 4 }+{ CO }_{ 2 }+{ H }_{ 2 }O\)
\({ K }\overset { +7 }{ \underset { \underset { 2\times { 3e }^{ - } }{ \downarrow } }{ M } } n{ O }_{ 4 }+{ H }_{ 2 }\overset { -1 }{ \underset { { 1e }^{ - } }{ \underset { \uparrow }{ C_{ 2 } } } } { O }_{ 4 }+{ H }_{ 2 }{ SO }_{ 4 }\longrightarrow { K }_{ 2 }{ SO }_{ 4 }+\overset { +2 }{ M } n{ SO }_{ 4 }+\overset { +4 }{ C } { O }_{ 2 }+{ H }_{ 2 }O\)
2KMnO₄ + 5 H₂C2O₄ + H₂S0₄ \(\longrightarrow \) Mn0₂ + Na₂S0₄ + KOH
2KMnO₄+ 5 H₂C₂O₄ + H₂S0₄ \(\longrightarrow \) K₂SO₄ + 2MnSO₄ + 10CO₂ + H₂O
2KMnO₄ + 5 H₂C₂O₄ + 3H₂S0_{4 \(\longrightarrow \)}  K₂S0₄ + 2MnS0₄ + 10C0₂ + 8 H₂0

When compounds containing hydrogen are treated with D₂O, hydrogen undergoes an exchange with deuterium
2NaOH + D₂O ➝ 2NaOD + HOD
HCl + D₂O ➝ DCl + HOD
NH₄Cl + 4D₂O ➝ ND₄Cl + 4HOD
These exchange reactions are useful in determining the number of ionic hydrogens present in a given compound.
For example, when D₂O is treated with of hypophosphorous acid only one hydrogen atom is exchanged with deuterium. It indicates that, it is a monobasic acid.
H₃PO₂ + D₂O ➝ H₂DPO₂ + HDO

The van der equation for a real gas is
\(\left( P+{{{am}^{2}}\over{{V}^{2}}} \right)(V-nb)=nRT\)
Pressure, Correction:
The pressure of a gas is directly proportional to the force created by the bombardment of molecules on the walls of the container. The speed of a molecule moving towards the wall of the container is reduced by the attractive forces exerted by its neighbours. Hence, the measured gas pressure is lower than the ideal pressure of the gas. Hence, van der Waals introduced a correction term to this effect.
Van der Waals found out the forces of attraction experienced by a molecule near the wall are directly proportional to the square of the density of the gas.
\(P^{\prime} \propto \rho^{2} ; \quad \rho=\frac{n}{v}\)
where n is the number of moles of gas and
V is the volume of the container
\( \Rightarrow p^{\prime} \alpha \frac{n^{2}}{V^{2}} \)
\(\Rightarrow p^{\prime}=a \frac{n^{2}}{V^{2}}\)
where a is proportionality constant and depends on the nature of gas
Therefore \(P_{\text {ideal }}=P+\frac{\operatorname{an}^{2}}{V^{2}}\)

Volume Correction
As every individual molecule of a gas occupies a certain volume, the actual volume is less than the volume of the container,
V. Van der Waals introduced a correction factor V' to this effect. Let us calculate the correction term by considering gas molecules as spheres.
V = excluded volume
Excluded volume for two molecules
\(=\frac{4}{3} \pi(2 r)^{3}=8\left(\frac{4}{3} \pi r^{3}\right)=8 V_{m}\)
Where V_m it a volume of a single molecule
Excluded volume for single molecule = \(\frac{8 \mathrm{~V}_{\mathrm{m}}}{2}=4 \mathrm{~V}_{\mathrm{m}}\)
Excluded volume for n molecule = n(4V_m) = nb
Where b is van der waals constant which is equal to 4V_m
\( \Rightarrow V^{\prime}=n b \)
\(V_{\text {ideal }}=V-n b\)
Replacing the corrected pressure and volume in the ideal gas equation PV = nRT we get the Van der Waals equation of state for real gases as below,
\(\left(p+\frac{a^{2}}{V^{2}}\right)(V-n b)=n R T\)
The constants a and b are van der Waals constants and their values vary with the nature of the gas. It is an approximate formula for the non-ideal gas.

Statement : "No two electrons in an atom can have the same set of values of all four quantum numbers"
Explanation : It means that, each electron must have unique values for the four quantum numbers (n, l, m and s).
For the lone electron present in hydrogen atom, the four quantum numbers are: n = 1; l = 0; m = 0 and s = +1/2. For the two electrons present in helium, one electron has the quantum numbers same as the electron of hydrogen atom, n = 1.
l = 0, m = 0 and s = +1/2. For other electron, the fourth quantum number is different i.e., n = 1, l = 0, m = 0 and s = -1/2.
As we know that the spin quantum number can have only two values +1/2 and - 1/2, only two electrons can be accommodated in a given orbital in accordance with pauli exclusion principle.