LATTICE ENERGY

         Cation and anion attract each other by electrostatic force of attraction to give a molecule A⁺ B⁻. Since the electrostatic field of a charged particle extend in all directions, a positive ion is surrounded by a  number of negatively charged ions while each negative ion similarly surrounded by a number of positive ions. These cations and anions arrange systematically in an alternating cation-anion pattern. This is called a crystal lattice. This process of clustering ions increases the force of attraction and thus potential energy decreases. The energy released when the requisite number of positive and negative ions are condensed into crystal to form one mole of the compound is called Lattice energy.
       Higher the lattice energy, greater will be the ease of forming an ionic compound. 
       The value of lattice energy depends on the charges present on the two ions and the distance between them. According to coulomb's law, the force of attraction (F) between two oppositely charged ions in air with charges equal to q₁ and  q₂ and separated by a distance d is given by,
                  F= (1/4πε₀K)*q₁q₂/d²
Where d is equal to sum of ionic radii of the two ions and K is dielectric constant of medium.
                 F=(1/4πε₀K)*q₁q₂/(rA⁺+rB⁻)²
       The value of F increase if (i) q₁ and q₂ are high and (ii) (rA⁺+rB⁻) is small.
       The stability of the ionic compound and the strength of the ionic bond depends on the value of F. Higher the value of F, greater shall be the stability of the ionic compound and hence greater shall be the strength of the ionic bond. For example, NaCl is more stable than CsCl as (rNa⁺+rCl⁻) is less than (rCs⁺+rCl⁻). MgO is more stable than NaCl as the product q₁q₂ is four times more in MgO than NaCl.

IONIC BOND AND IONIC COMPOUNDS

     The chemical bond formed between two or more atoms as a result of the transfer of one or more electrons from electropositive to electronegative atom is called electrovalent bond or ionic bond or polar bond.

      The electron transfer results in the formation of cations and anions. The cations are positively charged ions whereas anions are negatively charged ions. Oppositely charged ions are attracted to each other and a bond between them is formed. The bond existing between the oppositively charged ions is ionic bond.
Note: Electrovalent bond is not possible between similar atoms. This type of bonding requires two atoms of different  nature, one atom should have the tendency to lose electron or electrons, i.e., electropositive in nature and the other atom should have the tendency to accept electron or electrons, i.e., electronegative in nature.
Example of electrovalent bond
Potassium chloride : The free potassium atom has one valency electron (electronic configuration 2,8,8,1), i.e., 4s¹ whereas, the chlorine atom has seven valency electrons (electronic configuration 2,8,7), i.e., 3s² 3p⁵. In forming an ionic bond, the potassium atom loses its valency electron which is accepted by chlorine atom. As a result potassium achieves nobel gas configuration of argon (2, 8, 8) and become a positive ion (K⁺). Chlorine achieves nobel gas configuration of argon (2, 8, 8) and acquires a negative charge (Cl⁻). The attraction between potassium ion and chloride ion is an ionic bond.
 It is important to recognize that clean ionic bonding – in which one atom or molecule completely transfers an electron to another cannot exist: all ionic compounds have some degree of covalent bonding, or electron sharing. Thus, the term "ionic bonding" is given when the ionic character is greater than the covalent character – that is, a bond in which a large electronegativity difference exists between the two atoms, causing the bonding to be more polar (ionic) than in covalent bonding where electrons are shared more equally. Bonds with partially ionic and partially covalent character are called polar covalent bonds.
General characteristics of ionic compounds.
(i) Crystalline nature: Ionic compounds are usually crystalline in nature. The constituent units in an ionic crystal are ions and not molecules.
(ii) Melting and boiling points: Due to strong electrostatic forces of attraction, the ions are held tightly in their positions in the crystal lattice. A large amount of energy is needed to dislodge the ions from their positions. Thus, ionic compounds possess high melting and boiling points.
(iii)Hard and brittle: Ionic compounds are hard in nature. The hardness is due to strong forces of attraction between oppositely charged ions which keep them in their alloted positions. The brittleness of the crystals is due to movement of a layer of a crystal on the other layer by application of external force when like ions come infront of each other. The forces of repulsion come into play. The breaking of crystal occurs on account of these forces of these forces or repulsion.
(iv)Solubility: Ionic compounds are fairly soluble in polar solvents and insoluble in polar solvents. The polar solvents have high values of dielectric constants. Water, the solvent, is one of the best solvents as it has a high value of dielectric constant. Due to high value of dielectric constant, the electrostatic force of attraction between the ions decreases and these ions get separated and ultimately solvated by the molecules of the solvent. The non-polar solvents have very low value of dielectric constant and are not capable of dissolving electrovalent compounds.
(v)Electrical conductivity: Ionic solids do not conduct electricity. The reason is that the ions, on account of electrostatic forces of attraction, remain intact occupying fixed positions in the crystal lattice. The ions, thus, do not move where electric current is applied.
(vi)Space isomerism: The electrovalent bond is non-rigid and non-dimensional. Thus, the electrovalent compounds do not show space isimerism or stereo-isomerism.
(vii)Isomorphism: Compounds having same electronic structures are isomorphous to each other. For example, sodium fluride and magnesium oxide are isomorphous to each other.
                 Na⁺F⁻                Mg²⁺O²⁻
              (2,8) (2,8)           (2,8) (2,8)
Potassium sulphide, potassium chloride and calcium chloride are isomorphous to each other.
             K⁺           S²⁻          K⁺            Cl⁻          Ca²⁺         Cl⁻
        (2,8,8)     (2,8,8)     (2,8,8)     (2,8,8)     (2,8,8)     (2,8,8)   
(viii)Ionic reactions: Ionic compounds furnish ions in solution. The chemical reactions are due to the presence of these ions. Such reactions are fast. For example, SO₄²⁻ ions present in Na₂SO₄ solution, from white precipitate of BaSO as soon as BaCl solution is added to it.

                        Na₂SO₄    2Na⁺+ SO₄²⁻

PARTICLES IN ONE DIMENSIONAL BOX

      Consider an electron move in one dimensional box with width L.

Let potential v=0 inside the box, v=& outside the box
Take Schrodinger time independent wave equation
d²ψ/dx² +2m(E-V)ψ/ħ²=0 . . . (1)
 Particles is in one dimensional box so, v=0
d²ψ/dx² +2mψE/ħ²=0 . . . . (2)
Let Ψ(x)=A sin kx+B cos kx . . . . (3)
be the general solution of eq(2)
boundary condition are x=0 at ψ(x)=0
apply boundary condition in eq(3)
0=A sink(0)+ B cos k(0)
0=0+B      .·.B=0
second boundary condition x=L at ψ(x)=0,B=0
apply the above condition to eq(3)
0=A sin kL+ B cos kL
 =A sin kL + 0 cos kL
 =A sin kL=0   A≠ 0
Asin kL =0
kL=±nπ, where n= 1 2 3 4 ..
k=nπ/L . . . . (4)
put eq(4)in eq (3) and B=0
ψ(x)=A sin nπ/L*x . . . (5)
now differentiate eq(5) with respect to x
(dψ/dx)x=A cos nπ/L*x *nπ/L
again differentiate with respect to x we get,
d²ψ/dx²= -nπ/L A sin nπ/L*x*nπ/L
d²ψ/dx² =-n²π²/L² A sin nπ/L*x
d²ψ/dx²=( -n²π²/L²)ψ
d²ψ/dx²+n²π²ψ/L²=0 . . . (6)
compare eq(6) with eq(2) we get
2mE/ħ²=n²π²/L²
E=n²π²ħ²/L²2m
E=n²π²h²/L²2m4π²
E=n²h²/L²8m . . . . (7)
where n is integral
           h is planks constant
           m is mass
           L is width

when n=1; E₁= h²/8mL²
          n=2; E₂=4h²/8mL²
          n=3; E₂=9h²/8mL²
Now find the constant value of A of eq(5) by normalization method
∫|ψ(x)|² dx=1                                                (we suppose the probability of finding electron is 1)
∫A²sin²(nπx/L)dx=1                                                                                     
A²∫1-cos(2nπx/L)/2 dx=1
A²/2∫[1-cos(2nπx/L)] dx=1
A²/2 x-sin(2nπx/L)/(2nπ/L)=1
A²/2[x-(L/2nπ) sin2nπx/L]=1   put x=L
A²/2 [L-0=1]
A²L/2 =1
A²=2/L  .·.A=√(2/L)
ψ(x)=√(2/L)sin nπx/L.