Course: B.Tech.

Subject: Engineering Chemistry

Unit: I

Imagine a particle strictly confined between two

``walls'' by a potential energy that is shown in

the figure below.

confines the particle to the region .

1

Mathematically, the potential energy is expressed as

If the potential is infinite outside the box, then there is

zero probability that the particle will be found there.

Thus, we require outside the box, and this can

only happen if at the boundaries

and .

Since outside the box, we only need to

integrate between 0 and L , so that

Since we already know that outside the box,

we only need to solve the Schrödinger inside the box

where .

In this case, the Schrödinger looks like the free-particle

equation we already wrote down

Rearranging gives

We have already seen that the solutions are sin or cos

functions.

In fact, we can easily show that the solution of this

differential equation must be either

Where A and B are arbitrary constants. But since we need , the

only solution that is consistent with the boundary condition is the

sin solution. Thus, we take

It's a simple matter to verify that this function satisfies the differential

equation:

At this point, all we have is a general solution to the

equation, but we still do not know the allowed values of E .

But, there is also still one boundary condition we have

not yet enforced. We need . This requires

In order to satisfy this, we can take any point where the

sin function vanishes. We know that for

any value of n . Thus, our boundary condition is

equivalent to

Solving this equation for E, gives us the allowed

energies:

Angular momentum, moment of momentum, or

rotational momentum is a measure of the amount of

rotation an object has, taking into account its mass,

shape and speed.

It is a vector quantity that represents the product of a

body's rotational inertia and rotational velocity about a

particular axis.

The angular momentum of a system of particles (e.g. a

rigid body) is the sum of angular momenta of the

individual particles.

For a rigid body rotating around an axis of symmetry

(e.g. the wings of a ceiling fan)

The angular momentum can be expressed as the

product of the body's moment of inertia, I,

(i.e., a measure of an object's resistance to changes in

its rotation velocity) and its angular velocity, ω.

In quantum mechanics, angular momentum is

quantized – that is, it cannot vary continuously, but

only in "quantum leaps" between certain allowed

values.

For any system, the following restrictions on

measurement results apply, where is the reduced Planck

constant and is any direction vector such as x, y, or z:

The orbital angular momentum is quantized according

to the relationship:

It is a characteristic of angular momenta in quantum

mechanics that the magnitude of the angular

momentum in terms of the orbital quantum number is

of the form

that the z-component of the angular momentum in

terms of the magnetic quantum number takes the form

This general form applies to orbital angular

momentum, spin angular momentum, and the total

angular momentum for an atomic system.

The relationship between the magnitude of the angular

momentum and its projection along any direction is

space is often visualized in terms of a vector model.

The orbital angular momentum of electrons in atoms

associated with a given quantum state is found to be

quantized in the form

This is the result of applying quantum theory to the

orbit of the electron.

The solution of the Schrodinger equation yields the

angular momentum quantum number. Even in the case

of the classical angular momentum of a particle in

orbit,

the angular momentum is conserved. The Bohr theory

proposed the quantization of the angular momentum in

the form

The spectroscopic notation used for characterizing energy

levels of atomic electrons is based upon the orbital quantum

number.

When the orbital angular momentum and spin angular

momentum are coupled, the total angular momentum is

of the general form for quantized angular momentum

where the total angular momentum quantum number is

This gives a z-component of angular momentum

As long as external interactions are not extremely

strong, the total angular momentum of an electron can

be considered to be conserved and j is said to be a

"good quantum number".

Consider the hydrogen atom as a proton fixed at the

origin, orbited by an electron of reduced mass .

The potential due to electrostatic attraction is

___________(1)

in SI units. The kinetic energy term in the Hamiltonian

is

____________(2)

so we write out the Schrödinger equation in spherical polar coordinates as

_(3)

It happens that we can factor into , where are again the spherical harmonics. The radial part then can be shown to obey the equation

which is called the radial equation for the hydrogen

atom. Its (messy) solutions are

Where , and is the Bohr radius , The

functions are the associated Laguerre

functions. The hydrogen atom eigenvalues are

Atomic orbitals are (energy) states or wave forms of

electrons in the atom.

If we insist on the particle nature of electrons, then the

probability of finding an electron in an atomic orbital is

proportional to the square of the wave function.

The values of the wave function can be either positive

or negative, but the probability is always a positive

value.

The 1s, 2s, and 3s orbitals

First Energy Level: ( n = 1 )

Number of Orbitals: n2 = 12 = 1

First Orbital is Designated as " s "

So, First Energy Level is called the "1s" Orbital.

•Second Energy Level: ( n = 2 )

•Number of Orbitals: n2 = 22 = 4 ;[ 2s, 2px, 2py, 2pz ]

• "p" Orbitals shapes look like "dumbbells" which lie along each axis (x,y,z)

An Orbital DOES NOT describe a "house" in which an electron roams about. The orbital is

a description of an electron wave.

The 2p orbitals

The 3d orbitals

•Third Energy Level: ( n = 3 )

•Number of Orbitals: n2 = 32 = 9 ; [ 3s, 3px, 3py, 3pz, & 5 - 3d ] (there are five "3d" orbitals)

The 4fxyz orbital, one of the seven 4f orbitals

3

•Fourth Energy Level: ( n = 4 )

•Number of Orbitals: n2 = 42 = 16 ; [ 4s, 4px, 4py, 4pz, 5 - 4d, 7 - 4f ]

3

Electron density is the measure of the probability of

an electron being present at a specific location.

In molecules, regions of electron density are usually

found around the atom, and its bonds. In de-localized

or conjugated systems, such as phenol, benzene and

compounds such as hemoglobin and chlorophyll, the

electron density covers an entire region, i.e., in benzene

they are found above and below the planar ring.

This is sometimes shown diagrammatically as a series

of alternating single and double bonds. In the case of

phenol and benzene, a circle inside a hexagon shows

the de-localized nature of the compound.

In quantum chemical calculations, the electron density,

ρ(r), is a function of the coordinates r, defined so

ρ(r)dr is the number of electrons in a small volume dr.

For closed-shell molecules, can be written in terms of a

sum of products of basis functions, φ:

A molecular orbital diagram, or MO diagram, is a

qualitative descriptive tool explaining chemical

bonding in molecules in terms of molecular orbital

theory in general and the linear combination of atomic

orbitals (LCAO) molecular orbital method in particular.

A fundamental principle of these theories is that as

atoms bond to form molecules, a certain number of

atomic orbitals combine to form the same number of

molecular orbitals, although the electrons involved may

be redistributed among the orbitals.

Molecular orbital diagrams are diagrams of molecular orbital (MO)

energy levels, shown as short horizontal lines in the center, flanked

by constituent atomic orbital (AO) energy levels for comparison,

with the energy levels increasing from the bottom to the top.

Lines, often dashed diagonal lines, connect MO levels with their

constituent AO levels. Degenerate energy levels are commonly

shown side by side. Appropriate AO and MO levels are filled with

electrons symbolized by small vertical arrows, whose directions

indicate the electron spins.

For a diatomic molecule, an MO diagram effectively shows the

energetics of the bond between the two atoms, whose AO unbonded

energies are shown on the sides.

For simple polyatomic molecules with a "central atom" such as

methane (CH4) or carbon dioxide (CO2), a MO diagram may show

one of the identical bonds to the central atom.

The smallest molecule, hydrogen gas exists as

dihydrogen (H-H) with a single covalent bond between

two hydrogen atoms.

As each hydrogen atom has a single 1s atomic orbital

for its electron, the bond forms by overlap of these two

atomic orbitals.

In above figure the two atomic orbitals are depicted on

the left and on the right.

The vertical axis always represents the orbital energies.

2

Molecular orbitals of Hydrogen Fluoride are mostly

derived from the Fluorine 2s and 2p atomic orbitals

respectively.

Qualitatively, the high fluorine character of these

orbitals is a consequence of the high electronegativity

of fluorine as compared to hydrogen.

The 1 π orbitals are non-bonding and Fluorine 2p in

character. Finally, the anti-bonding "3-sigma" orbital is

primarily H 1s in character. The use of nodes to give a

general idea of the energy ordering still works. "1-

sigma" has no nodes, "2-sigma" and "1 π" have one

node, and "3-sigma" has two nodes.

The type of chemical bond developed between the two

combining atoms depends upon the way these atoms

acquire a stable noble gas configuration.

Elements may combine through any one of the following

ways to form stable compounds.

i. By the transfer of electrons from the atom of an

element to the atom or atoms of another. This gives

rise to an ionic (or electrovalent) bond.

ii. By mutually sharing the electrons. This gives rise to a

covalent bond.

iii. By one-sided sharing of electrons. This gives rise to a

coordinate bond.

This type of bond is established by transfer of

an electron from one atom to another.

In an ionic bond, one atom loses an electron

to another atom, forming a cation and anion,

respectively. And, they attract towards each

other ,by electrostatic attraction they combined

togather.

An IONIC BOND is an electrostatic interaction

that holds together a positively charged ion (cation)

and a negatively charged ion (anion).

A has one electron in excess and B has one electron short than the stable

octet.

Therefore, A transfers an electron to B and in this transaction both atoms

acquire stable electronic configuration.

And they held togather by electrostatic attraction.

In table salt, for example, a valence electron from a sodium

atom is transferred to a chlorine atom, forming Na+ and Cl-.

Because the ions have opposite charges, they are attracted

to each other. The loss of a valence electron and the

attraction to the atom that took it happen simultaneously.

4

A covalent bond is formed between two atoms (similar or dissimilar) by a mutual sharing of electrons. The shared pairs of electrons are counted towards the stability of both the participating atoms.

A covalent bond is defined as the force of attraction arising due to mutual sharing of electrons between the two atoms.

The combining atoms may share one, two or three pairs of electrons.

When the two atoms combine by mutual sharing of electrons,

then each of the atoms acquires stable configuration of the nearest

noble gas. The compounds formed due to covalent bonding are

called covalent compounds.

Covalency

The number of electrons which an atom contributes towards

mutual sharing during the formation of a chemical bond is called

its covalency in that compound.

Thus, the covalency of hydrogen in H2 (H - H) is one; that of

oxygen in O2 is two (O = O), and that of nitrogen in N2 is three (N

N).

4

If a hydrogen atom is bonded to a highly electronegative element such as fluorine, oxygen, nitrogen, then the shared pair of electrons lies more towards the electronegative element. This leads to a polarity in the bond in such a way that a slight positive charge gets developed on H-atom, viz.,

H+ d : O- d H+ d : F- d H+ d : N- d

This positive charge on hydrogen can exert electrostatic attraction on the negatively charged electronegative atom of the same or the other molecule forming a bridge-like structure such as

Xd - - Hd+ × × × × × × Yd- - Hd+

where X and Y are the atoms of strongly electronegative elements. The bond between the hydrogen atom of one molecule and a more electronegative atom of the same or another molecule is called hydrogen bond.

Hydrogen fluoride (HF).

Water (H2O).

The relatively weak attractive forces that act on neutral atoms

and that arise because of the electric polarization induced in

each of the particles by the presence of other particles.

Inter molecular: between molecules (not a bond)

Intra molecular: bonds within molecules(stronger)

1) dipole-dipole

2) dipole-induced dipole

3) dispersion

-Two polar molecules align so that d+ and d- are

matched (electrostatic attraction)

Ex: ethane (C2H6) vs. fluromethane (CH3F)

Fluoromethane (CH3F)

H H

H C F H C F

H H

d-d-d+ d+

Ethane (C2H6)

H H H H

H C C H H C C H

H H H H

Dipole-Dipole

NOT Dipole-Dipole

A dipole can a temporary dipole to form in a non-polar molecule

The molecules then line up to match d+ and d- charges.

H Cld+ d- Are-

e-e-

e-

e-

e-

e-

e- e-

e-

e- e-

e-

e-

e-

e-e-

e-

A DIPOLE

(it’s polar)

non-polarINDUCED

DIPOLE

d-d+

Dipole – Induced Dipole

(weak and short-lived)

A temporary dipole forms in a

non-polar molecule…

which leads to…

a temporary dipole to form in ANOTHERnon-polar molecule

Dispersion is the ONLY intermolecular attraction that occurs between non-polar molecules

Cl-Cle-

e-

e-

e-

e-

e-

e- e-e-

e-

e-e-

e-

e-e-

e-e-

e-

non-polarINDUCED

DIPOLE

d-d+

TEMPORARY

DIPOLEnon-polar

Cl-Cle-

e-e-

e-

e-

e-e- e-

e-

e-

e-e-

e-e-

e-

e- e-

d-d+

Dispersion

(weakest and very short-lived)

5

Electronegativity is a measure of the tendency of an

atom to attract a bonding pair of electrons.

Electronegativity, symbol χ, is a chemical property

that describes the tendency of an atom or a functional

group to attract electrons (or electron density) towards

itself.

An atom's electronegativity is affected by both its

atomic number and the distance at which its valence

electrons reside from the charged nucleus.

The higher the associated electronegativity number, the

more an element or compound attracts electrons

towards it.

2 bonded pairs

0 lone pairs

Bond angle of 180

Examples:

BeCl2,

CO2,

HCN,

C2H2

3 bonding pairs

0lone pairs

Bond angle of 120

Examples:

BF3

SO3

NO3-

CO32-

4 bonding pairs

0 lone pairs

Bond angle of 109.5

Examples

NH4+

SO42-

PO43-

Ni(CO)4

CH4

3 bonding pairs

1 lone pair

Bond angle of 107

Examples

CO3

PH3

SO32-

NH3

6

6 bonding pairs

0 lone pair

Bond angle of 90

Example

SF6

7

1.http://www.chemguide.co.uk/atoms/properties/atomorbs.html

2.http://www.chem.latech.edu/~upali/chem281/notes/Ch2-MO-Theory.pdf

3. http://www.avon-chemistry.com/electron_lecture.html

4.http://www.wpclipart.com/energy/atom/atomic_structure.png.html

5.Essentials of Physical chemistry by Bahl & Tuli

6.http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Molecular_Geometry

7.http://www.elmhurst.edu/~chm/vchembook/222octahedral.html