CH107/ D1 CH107/ D1 Physical Chemistry Physical Chemistry G. - - PowerPoint PPT Presentation

CH107/ D1 CH107/ D1 Physical Chemistry Physical Chemistry

• G. Naresh Patwari
• G. Naresh Patwari

Room No. 215; Department of Chemistry naresh@chem.iitb.ac.in 2576 7182 Charine Astrid (TA) charine@chem.iitb.ac.in 2576 4159

Contents: Physical Chemistry Contents: Physical Chemistry

• Atomic and Molecular Structure:
• Intermolecular Forces & Rates of Chemical Reactions
• Forces to Equilibrium

Why should Chemistry interest you? Why should Chemistry interest you?

Chemistry plays major role in 1.Daily use materials Plastics, LCD displays 2.Medicine Aspirin, Vitamin supplements 3.Energy Li-ion Batteries, Photovoltaics 4.Atmospheric Science Green-house gasses, Ozone depletion 5.Biotechnology Insulin, Botox 6.Molecular electronics Transport junctions, DNA wires Chemistry plays major role in 1.Daily use materials Plastics, LCD displays 2.Medicine Aspirin, Vitamin supplements 3.Energy Li-ion Batteries, Photovoltaics 4.Atmospheric Science Green-house gasses, Ozone depletion 5.Biotechnology Insulin, Botox 6.Molecular electronics Transport junctions, DNA wires

Haber Process

Haber Process

The Haber process remains largest chemical and economic

• venture. Sustains third of

worlds population

Haber Process

The Haber process remains largest chemical and economic

• venture. Sustains third of

worlds population Quantum theory is necessary for the understanding and the development of chemical processes and molecular devices Quantum theory is necessary for the understanding and the development of chemical processes and molecular devices

Atomic Spectra Atomic Spectra

Balmer Series 410.1 nm 434.0 nm 486.1 nm 656.2 nm Balmer Series 410.1 nm 434.0 nm 486.1 nm 656.2 nm

λ

∞ − ∞

  = −     =

7 1

R n n R 1 9678 x 1 m

2 2 1 2

1 1 1 .0

“R∞ is the most accurately measured fundamental physical constant” “R∞ is the most accurately measured fundamental physical constant” The Rydberg-Ritz Combination Principle states that the spectral lines

• f any element include frequencies

that are either the sum or the difference of the frequencies of two

• ther lines.

The Rydberg-Ritz Combination Principle states that the spectral lines

• f any element include frequencies

that are either the sum or the difference of the frequencies of two

• ther lines.

Bohr Phenomenological Model of Atom Bohr Phenomenological Model of Atom

Electrons rotate in circular orbits around a central (massive) nucleus, and

• beys the laws of classical mechanics.

Allowed orbits are those for which the electron’s angular momentum equals an integral multiple of h/2π i.e. mevr = nh/2π Energy of H-atom can only take certain discrete values: “Stationary States” The Atom in a stationary state does not emit electromagnetic radiation When an atom makes a transition from one stationary state of energy Ea to another of energy Eb, it emits or absorbs a photon of light: Ea – Eb = hv Electrons rotate in circular orbits around a central (massive) nucleus, and

• beys the laws of classical mechanics.

Allowed orbits are those for which the electron’s angular momentum equals an integral multiple of h/2π i.e. mevr = nh/2π Energy of H-atom can only take certain discrete values: “Stationary States” The Atom in a stationary state does not emit electromagnetic radiation When an atom makes a transition from one stationary state of energy Ea to another of energy Eb, it emits or absorbs a photon of light: Ea – Eb = hv

Rutherford Model of Atom Rutherford Model of Atom

Planetary model of atoms with central positively charged nucleus and electrons going around Planetary model of atoms with central positively charged nucleus and electrons going around Classical electrodynamics predicts that such an arrangement emits radiation continuously and is unstable Classical electrodynamics predicts that such an arrangement emits radiation continuously and is unstable

Energy expression Energy expression

Bohr Model of Atom Bohr Model of Atom

Angular momentum quantized Angular momentum quantized

n=1,2,3,... 2 (2 ) π π λ = = nh mvr r n

4 2 2 2

1 . 8ε = −

e n

m e E h n

Spectral lines Spectral lines

4 2 2 2 2

1 1 , 1,2,3,... 8

ν ε   ∆ = − = =  ÷  ÷  

e i f i f

m e E h n n h n n Explains Rydberg formula Ionization potential of H atom 13.6 eV Explains Rydberg formula Ionization potential of H atom 13.6 eV

4 2 1 2 2

1.09678 x 10 nm 8ε

− − ∞ =

=

e

m e R h

Bohr Model of Atom Bohr Model of Atom

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics

Photoelectric Effect: Wave –Particle Duality Photoelectric Effect: Wave –Particle Duality

Electromagnetic Radiation Wave energy is related to Intensity I ∝ E2

0 and is independent of ω

Electromagnetic Radiation Wave energy is related to Intensity I ∝ E2

0 and is independent of ω

( ) ω = − E E Sin kx t

Einstein borrowed Planck’s idea that ΔE=hν and proposed that radiation itself existed as small packets

• f energy (Quanta)now known as PHOTONS

φ = Energy required to remove electron from surface Einstein borrowed Planck’s idea that ΔE=hν and proposed that radiation itself existed as small packets

• f energy (Quanta)now known as PHOTONS

φ = Energy required to remove electron from surface

2

1 2 φ φ = = + = +

P M

E hv KE mv

Diffraction of Electrons : Wave –Particle Duality Diffraction of Electrons : Wave –Particle Duality Davisson-Germer Experiment A beam of electrons is directed onto the surface of a nickel crystal. Electrons are scattered, and are detected by means of a detector that can be rotated through an angle θ. When the Bragg condition m = 2dsin λ θ was satisfied (d is the distance between the nickel atom, and m an integer) constructive interference produced peaks of high intensity Davisson-Germer Experiment A beam of electrons is directed onto the surface of a nickel crystal. Electrons are scattered, and are detected by means of a detector that can be rotated through an angle θ. When the Bragg condition m = 2dsin λ θ was satisfied (d is the distance between the nickel atom, and m an integer) constructive interference produced peaks of high intensity

Diffraction of Electrons : Wave –Particle Duality Diffraction of Electrons : Wave –Particle Duality

• G. P. Thomson Experiment

Electrons from an electron source were accelerated towards a positive electrode into which was drilled a small hole. The resulting narrow beam of electrons was directed towards a thin film of nickel. The lattice of nickel atoms acted as a diffraction grating, producing a typical diffraction pattern on a screen

• G. P. Thomson Experiment

Electrons from an electron source were accelerated towards a positive electrode into which was drilled a small hole. The resulting narrow beam of electrons was directed towards a thin film of nickel. The lattice of nickel atoms acted as a diffraction grating, producing a typical diffraction pattern on a screen

de Broglie Hypothesis: Mater waves de Broglie Hypothesis: Mater waves

Since Nature likes symmetry, Particles also should have wave-like nature Since Nature likes symmetry, Particles also should have wave-like nature De Broglie wavelength De Broglie wavelength

λ = = h h p mv

Electron moving @ 106 m/s Electron moving @ 106 m/s

• 34

10

• 31

6

6.6x10 J s 7 10 9.1x10 Kg 1x10 m/s λ

−

= = = × × h m mv

He-atom scattering Diffraction pattern of He atoms at the speed 2347 m s-1 on a silicon nitride transmission grating with 1000 lines per millimeter. Calculated de Broglie wavelength 42.5x10-12 m de Broglie wavelength too small for macroscopic objects He-atom scattering Diffraction pattern of He atoms at the speed 2347 m s-1 on a silicon nitride transmission grating with 1000 lines per millimeter. Calculated de Broglie wavelength 42.5x10-12 m de Broglie wavelength too small for macroscopic objects

Diffraction of Electrons : Wave –Particle Duality Diffraction of Electrons : Wave –Particle Duality

The wavelength of the electrons was calculated, and found to be in close agreement with that expected from the De Broglie equation The wavelength of the electrons was calculated, and found to be in close agreement with that expected from the De Broglie equation

Wave –Particle Duality Wave –Particle Duality

Light can be Waves or Particles. NEWTON was RIGHT! Electron (matter) can be Particles or Waves Electrons and Photons show both wave and particle nature “WAVICLE” Best suited to be called a form of “Energy” Light can be Waves or Particles. NEWTON was RIGHT! Electron (matter) can be Particles or Waves Electrons and Photons show both wave and particle nature “WAVICLE” Best suited to be called a form of “Energy”

Wave –Particle Duality Wave –Particle Duality

Uncertainty Principle Uncertainty Principle

Uncertainty principle Uncertainty principle

. 4π ∆ ∆ ≥

x

h x p

Uncertainty Principle Uncertainty Principle

Schrodinger’s philosophy Schrodinger’s philosophy

PARTICLES can be WAVES and WAVES can be PARTICLES PARTICLES can be WAVES and WAVES can be PARTICLES New theory is required to explain the behavior of electrons, atoms and molecules Should be Probabilistic, not deterministic (non-Newtonian) in nature Wavelike equation for describing sub/atomic systems New theory is required to explain the behavior of electrons, atoms and molecules Should be Probabilistic, not deterministic (non-Newtonian) in nature Wavelike equation for describing sub/atomic systems

Schrodinger’s philosophy Schrodinger’s philosophy

PARTICLES can be WAVES and WAVES can be PARTICLES PARTICLES can be WAVES and WAVES can be PARTICLES A concoction of A concoction of

2 2

1 2 2 Wave is Particle Particle is Wave ν ω λ = + = + = + = = = = h h p E T V mv V V m E h h k p

let me start with classical wave equation

Do I need to know any Math? Do I need to know any Math?

Algebra Trigonometry Differentiation Integration Differential equations Algebra Trigonometry Differentiation Integration Differential equations

[ ]

1 1 2 2 1 1 2 2

( ) ( ) ( ) ( ) + = + A c f x c f x c Af x c Af x ( ) ( ) ikx Sin kx Cos kx e

2 2 2 2

∂ ∂ ∂ ∂ d d dx dx x x ( )

∫ ∫

b ikx a

e dx f x dx

2 2 2 2

( ) ( ) ( ) ( ) ( ) ( )

+ +

∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂

f f f f m

x y x y f x nf y k x y x y

Remember!

∂ Ψ ∂ Ψ = ∂ ∂ Ψ =   Ψ = = −  ÷   = = = = × − ×   = − =  ÷   h h

2 2 2 2 2

( , ) 1 ( , ) Classical Wave Equation ( , ) Amplitude ( , ) ; Where 2 is the phase 2 2

i

x t x t x c t x t x x t Ce t E h h p k x x p E t t

α

α π ν λ ν ω π λ α π ν λ Schrodinger’s philosophy Schrodinger’s philosophy

i

x t E iCe i x t i x t t t t ( , ) ( , ) ( , )

α

α α ∂Ψ ∂ ∂ −   = × = × Ψ × = × Ψ ×  ÷ ∂ ∂ ∂   h Schrodinger’s philosophy Schrodinger’s philosophy x t E x t i t ( , ) ( , ) − ∂Ψ = × Ψ ∂ h

i

x p E t x t Ce ( , ) and

α

α × − × Ψ = = h

Schrodinger’s philosophy Schrodinger’s philosophy ∂Ψ = × Ψ ∂ h ( , ) ( , )

x

x t p x t i x

i

x p E t x t Ce ( , ) and

α

α × − × Ψ = = h

α

α α ∂Ψ ∂ ∂   = × = × Ψ × = × Ψ ×  ÷ ∂ ∂ ∂   h ( , ) ( , ) ( , )

i x

p x t iCe i x t i x t x x x

i

x t E iCe i x t i x t t t t ( , ) ( , ) ( , )

α

α α ∂Ψ ∂ ∂ −   = × = × Ψ × = × Ψ ×  ÷ ∂ ∂ ∂   h Schrodinger’s philosophy Schrodinger’s philosophy x t E x t i t ( , ) ( , ) − ∂Ψ = × Ψ ∂ h

i x

p x t iCe i x t i x t x x x ( , ) ( , ) ( , )

α

α α ∂Ψ ∂ ∂   = × = × Ψ × = × Ψ ×  ÷ ∂ ∂ ∂   h

x

x t p x t i x ( , ) ( , ) ∂Ψ = × Ψ ∂ h

i

x p E t x t Ce ( , ) and

α

α × − × Ψ = = h

µ

µ

− ∂ ∂ ∂ ∂ = = = − = ∂ ∂ ∂ ∂ h h h h Operators

x

i E i p i t t i x x Operators Operators

x

x t x t E x t p x t i t i x ( , ) ( , ) ( , ) ( , ) − ∂Ψ ∂Ψ = × Ψ = × Ψ ∂ ∂ h h Operator A symbol that tells you to do something to whatever follows it Operators can be real or complex, Operators can also be represented as matrices Operator A symbol that tells you to do something to whatever follows it Operators can be real or complex, Operators can also be represented as matrices

x

x t E x t x t p x t i t i x ( , ) ( , ) ( , ) ( , ) − ∂ ∂ Ψ = × Ψ Ψ = × Ψ ∂ ∂ h h

Operators and Eigenvalues Operators and Eigenvalues Operator operating on a function results in re-generating the same function multiplied by a number The function f(x) is eigenfunction of operator  and a its eigenvalue Operator operating on a function results in re-generating the same function multiplied by a number The function f(x) is eigenfunction of operator  and a its eigenvalue

( )

( ) α = f x Sin x

( )

( ) α α = × d f x Cos x dx

( ) ( )

2 2 2 2

( ) ( ) α α α α α = × = − × = − ×     d d f x Cos x Sin x f x dx dx is an eigenfunction of

• perator and

is its eigenvalue is an eigenfunction of

• perator and

is its eigenvalue

( )

α Sin x

2 2

d dx

2

α −

µ

( ) ( ) Eigen Value Equation A f x a f x × = ×

The mathematical description of quantum mechanics is built upon the concept of an operator The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator. The average value of the observable corresponding to

• perator  is

The state of a system is completely specified by the wavefunction (x,y,z,t) Ψ which evolves according to time-dependent Schrodinger equation The mathematical description of quantum mechanics is built upon the concept of an operator The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator. The average value of the observable corresponding to

• perator  is

The state of a system is completely specified by the wavefunction (x,y,z,t) Ψ which evolves according to time-dependent Schrodinger equation Laws of Quantum Mechanics Laws of Quantum Mechanics ˆ * υ = Ψ Ψ

∫

a A d

Probability Distribution Probability Distribution Average Values Average Values

2 2 1 1

( ) and ( )

n n j j j j j j j j

x x P x x x P x

= =

= =

∑ ∑

Let us consider Maxwell distribution of speeds The mean speed is calculated by taking the product of each speed with the fraction of molecules with that particular speed and summing up all the products. However, when the distribution of speeds is continuous, summation is replaced with an integral Let us consider Maxwell distribution of speeds The mean speed is calculated by taking the product of each speed with the fraction of molecules with that particular speed and summing up all the products. However, when the distribution of speeds is continuous, summation is replaced with an integral RT v vf v dv M

12

8 ( ) π

∞

  = =  ÷  

∫

Mv RT

M f v v e RT

2

3 2 2 2

( ) 4 2 π π

−

  =  ÷   Probability Distribution Probability Distribution

Born Interpretation Born Interpretation In the classical wave equation (x,t) Ψ is the Amplitude and | (x,t)| Ψ

2 is the Intensity

In the classical wave equation (x,t) Ψ is the Amplitude and | (x,t)| Ψ

2 is the Intensity

The state of a quantum mechanical system is completely specified by a wavefunction (x,t) Ψ ,which can be complex All possible information can be derived from (x,t) Ψ From the analogy of classical wave equation, Intensity is replaced by Probability. The probability is proportional to the square of the of the wavefunction | (x,t) Ψ |2 , known as probability density P(x) The state of a quantum mechanical system is completely specified by a wavefunction (x,t) Ψ ,which can be complex All possible information can be derived from (x,t) Ψ From the analogy of classical wave equation, Intensity is replaced by Probability. The probability is proportional to the square of the of the wavefunction | (x,t) Ψ |2 , known as probability density P(x)

Born Interpretation Born Interpretation

P x x t x t x t

2

( ) ( , ) ( , ) ( , )

∗

= Ψ = Ψ × Ψ

Probability density Probability

a a a a

P x x x dx x t dx x t x t dx

2

( ) ( , ) ( , ) ( , )

∗

≤ ≤ + = Ψ = Ψ × Ψ

Probability in 3-dimensions

* 2

P( , , )

( , , , '). ( , , , ') ( , , , ') τ

≤ ≤ + ≤ ≤ + ≤ ≤ + = Ψ

Ψ = Ψ

a a a a a a

a a a a a a a a a

x x x dx y y y dy z z z dz

x y z t x y z t dxdydz x y z t d

Laws of Quantum Mechanics Laws of Quantum Mechanics

$

µ

x x x x

x x d d p mv p i i dx dx p T m

2

Position, Momentum, Kinetic Energy, 2 = = = − = h h

µ µ

µ

x y x z

d T m dx p p p T T m m m m x y z V x V x

2 2 2 2 2 2 2 2 2 2 2 2 2

2 Kinetic Energy, + 2 2 2 2 Potential Energy, ( ) ( ) − =   − ∂ ∂ ∂ = + = + +  ÷ ∂ ∂ ∂   h h

Classical Variable QM Operator

The mathematical description of QM mechanics is built upon the concept of an operator The mathematical description of QM mechanics is built upon the concept of an operator

Normalization of Wavefunction Normalization of Wavefunction

∞ ∞ x  Ψ ∞ ∞ x  Ψ

Unacceptable wavefunction

Since Ψ*Ψdτ is the probability, the total probability of finding the particle somewhere in space has to be unity If integration diverges, i.e.  : ∞ Ψ can not be normalized, and therefore is NOT an acceptable wave function. However, a constant value C 1 is perfectly ≠ acceptable. Since Ψ*Ψdτ is the probability, the total probability of finding the particle somewhere in space has to be unity If integration diverges, i.e.  : ∞ Ψ can not be normalized, and therefore is NOT an acceptable wave function. However, a constant value C 1 is perfectly ≠ acceptable.

* *

( , , ). ( , , ) 1 τ Ψ Ψ = Ψ Ψ = Ψ Ψ =

∫∫∫ ∫

all space all space

x y z x y z dxdydz d Ψ must vanish at ± , or more appropriately at the boundaries ∞ and Ψ must be finite Ψ must vanish at ± , or more appropriately at the boundaries ∞ and Ψ must be finite

Laws of Quantum Mechanics Laws of Quantum Mechanics The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator In any measurement of observable associated with

• perator Â, the only values that will be ever observed are

the eigenvalues an, which satisfy the eigenvalue equation: Ψn are the eigenfunctions of the system and an are corresponding eigenvalues If the system is in state Ψ k , a measurement on the system will yield an eigenvalue ak In any measurement of observable associated with

• perator Â, the only values that will be ever observed are

the eigenvalues an, which satisfy the eigenvalue equation: Ψn are the eigenfunctions of the system and an are corresponding eigenvalues If the system is in state Ψk , a measurement on the system will yield an eigenvalue ak

µ ×

Ψ = × Ψ

n n n

A a