RESONANCE AND RESONANCE ENERGY

      It is generally observed that a single valence bond structure of a molecule cannot correctly account for the properties of the molecule. In such cases, the concept of resonance is introduced. According to this concept, if two or more alternate valence bond structures can be written for a molecule, the actual structure is said to be a resonance or mesomeric hybrid of all these alternate structures. For example, carbon dioxide molecule can be represented by the following three structures:

     The calculated values of bond distances between carbon and oxygen in C=O and C≡O are 1.22Å and 1.10Å, respectively but the observed bond distance between carbon and oxygen in carbon dioxide is 1.15Å. Thus, none of the above structures correctly explains the observed bond length.

     It is, thus, said that a hybrid form of these structures can provide the exact explanation. The various structure of which the molecule is a resonance hybrid are know  as canonical forms of mesomeric forms. Actually resonance hybrid does not oscillate between the canonical forms of mixture of these forms but it is a definite form and has a definite form and has a definite structure which cannot be written on paper.

Rules for selecting canonical or mesomeric forms

(i) The relative position of all the atoms of each of the canonical forms must  be the same. They should differ only in the position of electrons.

(ii)The number of unpaired and paired electrons in each of the canonical forms must be same.

(iii) The contributing structures should not differ much in energy.

(iv)The contributing structures should be such that negative charge resides on more electronegative and positive charge on the electropositive. Like charges should not reside on atoms close together in the canonical forms.

Resonance Energy

     It has been observed that the molecule which shows resonance has greater heat of formation than the calculated heat of formation of any one of canonical forms. The difference is represented by 𐤃E and is called resonance energy.

𐤃E= (Experimental value of heat of formation) - (calculated value of heat of formation of most stable canonical form)

     On account of 𐤃E, the hybrid form is more stable than any of the canonical forms.

Some Examples Showing resonance

(i) Nitric oxide molecule :  It is a resonance hybrid of following two structures:

(ii) Nitrogen dioxide molecule : It is a resonance hybrid of the following structures:

     

VALENCE BOND THEORY

      It is the simplest of the three theories and was developed mainly by Pauling. It describes the bonding in terms of hybridized orbitals of the central metal atom or ion. The theory mainly deals withe the geometry and magnetic properties of the complexes. The salient features of the theory are:

(i) The central metal loses a requisite number of electrons to form the ion. The number of electrons lost is the valency of the resulting cation. In some cases, the metal atom does not lose electrons.

(ii) The central metal ion or atom (as the case may be) makes available a number of empty s-, p- and d-atomic orbitals equal to its coordination number. These vacant orbitals hybridize together to form hybrid orbitals which are same in the number as the atomic orbitals hybridizing together. They are vacant, equivalent in energy and have definite geometry.

Some of the common hybridized orbitals met in the coordination compounds are listed below:

Coordination number	Type of hybridization	Geometry	Examples
2	sp	Linear	[Ag(NH3)2]+ ; [Ag(CN)2]-
3	sp2	Trigonal planar	[HgI3]-
4	sp3 dsp2	Tetrahedral Square planar	Ni(CO)4, [Ni(X4]2- [ZnCl4]2-, [CuX4]2- Where X= Cl-, B- , I- [Ni(CN)4]2- , [Cu(NH3)4]2- [Ni(NH3)4]2=
5	dsp3 sp3d	Trigonal bipyramidal Square pyramidal	Fe(CO)5, [CuCl5]3- [SbF5]2-
6	d2sp3 or sp3d2	Octahedral (Inner orbital) (Outer orbital)	[Cr(NH3)6]3+ ; [Fe(CN)6]3- [Fe(H2O)6]2+ ; [Ni(NH3)6]2+ [FeF6]3-

(iii) The non-bonding electrons of the metal occupy the inner orbitals. These ate grouped in accordance with Hund's rule, however, under the influence of some strong ligands, there may be some re-arrangement of electrons in the atomic orbitals the d-orbitals participating in this process of hybridization may be either (n-1)d2sp3 or nsp3d2. The complexes thus formed are referred to as inner or low spin and outer or high spin complexes, respectively.

(iv) The ligands have at least one o-orbital containing a lone pair of electrons. Vacant hybrid orbitals of the metal atom or ion overlap with the o-orbitals containing lone pair or electrons of the ligands to form M<- ligand o-bond.This bond is called coordinate bond and possesses a considerable amount of polarity.

(v) It is  possible to predict the magnetic properties of the complex if the geometry of the complex ion is known. If the complex contains unpaired electrons, it is paramagnetic in nature whereas if it does not contain unpaired electrons all re paired, the complex is diamagnetic in nature.
the number of unpaired electrons and the geometries of the complex ions having central metal ion with configurations d1 to d9 are related to each other as shown below:

dx Configuration	Number of unpaired electrons for different geometries
				Octahedral
				Inner orbitals (d2sp3)	Outer orbitals (sp3d2)
d1	1		1	1	1
d2	2		2	2	2
d3	3		3	3	3
d4	4		4	2	4
d5	5		3	1	5
d6	4		2	0	4
d7	3		1	1 (Shifted to higher orbits)	3
d8	2		0	0 (2 electrons shifted)	2
d9	1		1 (Shifted)	1 (3 electrons shifted)	1

           

        Limitation of valence bond theory:

The valence bond theory was fairly successful in explaining qualitatively the geometry and magnetic properties of complexes. However, it has a number of limitations.

(i) The theory does not offer any explanation about the spectra of complex (why most of the complexes are coloured).

(ii) Sometimes the same metal ion assumes different geometry when formation of complex ion takes place.The theory is unable to explain why at one time the electrons are rearranged against the Hund's rule while at other times the electronic configuration is not disturbed.

(iii) The theory does not offer an explanation for the existence of inner-orbital and outer-orbital complexes.

(iv) The theory does not explain why certain complexes are labile while others are inert.

(v) In the formation of [Cu(NH3)4]2+, one electron is shifted from 3d to 4p-orbital. The theory is silent about the energy availability for shifting such an electron. Such an electron can be easily lost, then why [Cu(NH3)4]2+ complex does not show reducing properties.

(vi) The changes in energies of the metal orbitals on formation of complex are difficult to calculate mathematically. The properties of the complexes are more satisfactorily explained by other two theories. The approach of ligands towards a metal or ion creates a field which is responsible for the splitting of d-orbitals into different energy levels. The extent of splitting depends on the nature and number of ligands which surround the metal atom or ion and explains the magnetic and spectroscopic properties of the complex. More details about these theories are dealt in higher classes. 

BORN-HABER CYCLE

     There are several important concept to understand before the Born-Haber Cycle can be applied to determine the lattice energy of an ionic solid; ionization energy, electron affinity, dissociation energy, sublimation energy, heat of formation, and Hess's Law.

Ionization Energy is the energy required to remove an electron from a neutral atom or an ion. This process always requires an input of energy, and thus will always have a positive value. In general, ionization energy increases across the periodic table from left to right, and decreases from top to bottom. There are some excepts, usually due to the stability of half-filled and completely filled orbitals.

 Electron Affinity is the energy released when an electron is added to a neutral atom or an ion. Usually, energy released would have a negative value, but due to the definition of electron affinity, it is written as a positive value in most tables. Therefore, when used in calculating the lattice energy, we must remember to subtract the electron affinity, not add it. In general, electron affinity increases from left to right across the periodic table and decreases from top to bottom.

 Dissociation energy is the energy required to break apart a compound. The dissociation of a compound is always an endothermic process, meaning it will always require an input of energy. Therefore, the change in energy is always positive. The magnitude of the dissociation energy depends on the electronegativity of the atoms involved.

Sublimation energy is the energy required to cause a change of phase from solid to gas, bypassing the liquid phase. This is an input of energy, and thus has a positive value. It may also be referred to as the energy of atomization.

The heat of formation is the change in energy when forming a compound from its elements. This may be positive or negative, depending on the atoms involved and how they interact.

 Hess's Law states that the overall change in energy of a process can be determined by breaking the process down into steps, then adding the changes in energy of each step. The Born-Haber Cycle is essentially Hess's Law applied to an ionic solid.

 USING THE BORN-HABER CYCLE

 The values used in the Born-Haber Cycle are all predetermined changes in enthalpy for the processes described in the section above. Hess' Law allows us to add or subtract these values, which allows us to determine the lattice energy.

STEP 1

 Determine the energy of the metal and nonmetal in their elemental forms. (Elements in their natural state have an energy level of zero.) Subtract from this the heat of formation of the ionic solid that would be formed from combining these elements in the appropriate ration. This is the energy of the ionic solid, and will be used at the end of the process to determine the lattice energy.

 STEP 2

 The Born-Haber Cycle requires that the elements involved in the reaction are in their gaseous forms. Add the changes in enthalpy to turn one of the elements into its gaseous state, and then do the same for the other element.

 STEP 3

 Metals exist in nature as single atoms and thus no dissociation energy needs to be added for this element. However, many nonmetals will exist as polyatomic species. For example, Cl exists as Cl in its elemental state. The energy required to change Cl into 2Cl atoms must be added to the value obtained in Step 2. 

STEP 4

 Both the metal and nonmetal now need to be changed into their ionic forms, as they would exist in the ionic solid. To do this, the ionization energy of the metal will be added to the value from Step 3. Next, the electron affinity of the nonmetal will be subtracted from the previous value. It is subtracted because it is a release of energy associated with the addition of an electron. *This is a common error due to confusion caused by the definition of electron affinity, so be careful when doing this calculation. 

STEP 5

 Now the metal and nonmetal will be combined to form the ionic solid. This will cause a release of energy, which is called the lattice energy. The value for the lattice energy is the difference between the value from Step 1 and the value from Step 4.

The diagram below is another representation of the Born-Haber Cycle.

EQUATION

The Born-Haber Cycle can be reduced to a single equation: Heat of formation= Heat of atomization+ Dissociation energy+ (sum of Ionization energies)+ (sum of Electron affinities)+ Lattice energy

*Note: In this general equation, the electron affinity is added. However, when plugging in a value, determine whether energy is released (exothermic reaction) or absorbed (endothermic reaction) for each electron affinity. If energy is released, put a negative sign in front of the value; if energy is absorbed, the value should be positive. Rearrangement to solve for lattice energy gives the equation: Lattice energy= Heat of formation- Heat of atomization- Dissociation energy- (sum of Ionization energies)- (sum of Electron Affinities)

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.

SCHRODINGER WAVE EQUATION (Time Independent)

     Let us suppose an electron moving with velocity of light, it undergoes wave motion.

So, if we want to describe  the motion of the electron, we have to use  quantum mechanics and quantum mechanics principle.

     Here, electron moving like a wave there must be a vibration, frequency and amplitude there must be phase.

 Let ψ = ψ₀ sin(ωt-kx). . . .(1)

 it is simple harmonic wave equation.

 where, ψ= wave function,

             ω= angular velocity,

              t = time,

              k= wave vector and

              x= displacement

 Diff. eq(1) with respect to x 

we get,

        dψ/dx = ψ₀ cos(ωt-kx)(-k)

Diff. again with respect to x

we get,

         d²ψ/dx²= -ψ₀ sin(ωt-kx)(-k)(-k)

                       = -k²ψ₀ sin(ωt-kx)

                       = -k²ψ  [ψ = ψ₀ sin(ωt-kx). . . .(1)]                                                                                     d²ψ/dx² +k²ψ =0 . . . .(2)

we know k=2π/λ

putting the value of k in eq(2)

we get,

         d²ψ/dx² + 4π²/(λ²ψ) =0 . . . .(3)

   according to de-broglie hypothesis

                λ=h/(mv)

similarly, λ²= h²/(m²v²)

putting the value of λ² in eq(3)

we get,

         d²ψ/dx² + 4π²m²v²ψ/h²=0 . . . .(4)

         E=K+V

         E=1/2 (mv²) + v

         (E-V) = 1/2 (mv²)

        2(E-V)= mv²

multiply m both side,

we get,

 2m(E-V) = m²v²

putting the value of m²v² in eq(4)

  d²ψ/dx² + 4π²/h² 2m(E-V)ψ=0

according to bohr

  ħ=h/2π or ħ²=h²/4π²

 d²ψ/dx² +2m(E-V)ψ/ħ²=0

This equation represent time independent Schrodinger wave in one dimension.

In three dimension it may be written in the following way

d²ψ/dx² + d²ψ/dy² + d²ψ/dz² + 2m(E-V)ψ/ħ² =0

∆²ψ+ 2m(E-V)ψ/ħ²=0

where ∆=id/dx + jd/dy + kd/dz.