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Physical Chemistry: Introduction to Waves and Vibrations in Chemistry, Study notes of Physical Chemistry

An excerpt from a university chemistry course, specifically unit i of chem 3510 from fall 2006. It introduces the main subdivisions of physical chemistry, focusing on waves and vibrations in chemistry. Topics covered include the history of classical physics, the wave equation, and the solutions of standing waves. Useful for students studying physical chemistry, particularly those in the fields of quantum mechanics, spectroscopy, and thermodynamics.

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CHEM 3510 Fall 2006
Unit I
Introduction
A. Introduction to Physical Chemistry
1. Physical Chemistry is the part of chemistry dealing with application of
physical methods to investigate chemistry.
2. Physical Chemistry main subdivisions are:
a. Quantum Mechanics
deals with structure and properties of molecules
b. Spectroscopy
deals with the interaction between light and matter
c. Computational Chemistry
deals with modeling chemical properties of reactions using
computers
d. Statistical Mechanics
deals with how knowledge about molecular energy levels (or
microscopic world) transforms into properties of the bulk (or
macroscopic world)
e. Thermodynamics
deals with properties of systems and their temperature dependence
and with energetics of chemical reactions
f. Electrochemistry
deals with processes in with electrons are either a reactant or a
product of a reaction
g. Chemical Kinetics
deals with the rates of chemical reactions or physical processes
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Unit I

Introduction

A. Introduction to Physical Chemistry

1. Physical Chemistry is the part of chemistry dealing with application of

physical methods to investigate chemistry.

2. Physical Chemistry main subdivisions are:

a. Quantum Mechanics

□ deals with structure and properties of molecules

b. Spectroscopy

□ deals with the interaction between light and matter

c. Computational Chemistry

□ deals with modeling chemical properties of reactions using

computers

d. Statistical Mechanics

□ deals with how knowledge about molecular energy levels (or

microscopic world) transforms into properties of the bulk (or

macroscopic world)

e. Thermodynamics

□ deals with properties of systems and their temperature dependence

and with energetics of chemical reactions

f. Electrochemistry

□ deals with processes in with electrons are either a reactant or a

product of a reaction

g. Chemical Kinetics

□ deals with the rates of chemical reactions or physical processes

B. Classical Physics Review

1. Classical Physics was introduced in the 17

th

century by Isaac Newton.

2. At the end of 19

th

century, classical physics (mechanics, thermodynamics,

kinetic theory, electromagnetic theory) was fully developed and was

divided into:

a. the corpuscular side or particle domain (the matter)

b. the undulatory side or wave domain (the light)

3. Some useful classical physics equations:

a. Total energy E :

E = K + V

○ K is

○ V is

b. Kinetic energy K :

m

p

K m

v

○ m is the mass

○ v is the velocity (or speed)

○ p is the momentum

c. Frequency ν (Greek letter nu):

π

ω ν λ

ν

= = c =

c

○ λ is

○ c is

○ ν

is

○ ω is

4. Classical mechanics was successful in explaining the motion of everyday

objects but fails when applied to very small particles. These failures led to

the development of Quantum Mechanics.

b. These conditions are called because

c. To solve the differential equation, we assume that u ( x , t ) factors into a

function of x times a function of t :

d. This technique (or method) is called

e. Solving further the equation:

  • Substituting u ( x , t )in the equation above:

2

2

2 2

2 () ( ) v

dt

d T t X x dx

d X x T t =

  • Dividing by u ( x , t )= X ( x ) T ( t ):
⇒ K

dt

d T t

dx Tt

d X x

X x

2

2

2 2

2 ()

v

  • In order for this equation to be true for every x and t , each side should be equal to a

constant K called the separation constant.

  • The problem of finding transformed into two problems of finding X ( x ) and T ( t ) by

solving the following linear differential equations with constant coefficient (they are

ordinary differential equations):

u ( x , t )

2

2

KX x = dx

d X x

v () 0

2

2

K Tt = dt

d Tt

f. Solving for X ( x ): ⇒

l

n x

X x B

π

( )= sin

  • Trivial solution is obtained (that is X ( x ) = 0) if K ≥ 0.
  • If K < 0, set ( β is real):

2

K =− β

2

2

  • X x = dx

d X x β

  • The general solution for this equation is:

i x i x X x ce c e

  • β − β ( )= 1 + 2
  • Considering Euler equation ( e x i x ):

ix =cos ± sin

±

X ( x )= A cos β x + B sin β x

  • This solution of X ( x ) should verify the boundary conditions:

X ( 0 )= 0 ⇒ A = 0

X ( l )= 0 ⇒ B sin β l = 0 ⇒ β l = n π where n = 1, 2, ...

g. Look more closely to the solutions:

Number of

wavelength

that fits in 2 l :

n = 1

n = 2

n = 3

n = 4

Number of

wavelength

that fits in 2 l :

n = 1

n = 2

n = 3

n = 4

□ By generalization:

n

l

n

λ =

○ This is called

□ The solutions are a set a functions called

x B x

l

n

X x B

n

n n n

λ

π 2 π

( )= sin = sin

□ Also

2 v v

2 ω

π

λ

π ω πν n

l

n

n

n n

= = = = (where

l

v

π ω = ) are called

k. Solutions:

□ First term is

l

x

t

l

A

π φ

π

)sin

v

cos(

○ First term is called

○ The frequency is: v/ 2 l

ν =

□ Second term is

l

x

t

l

A

π φ

π 2

)sin

2 v

cos(

○ Second term is called

○ The frequency is: v/ l

ν =

○ The midpoint has a zero displacement at all times, and it is

called

□ Third term is

l

x

t

l

A

π φ

π 3

)sin

3 v

cos(

○ Third term is called

○ The frequency is: 3 v/ 2 l

ν =

○ This term has

□ Fourth term is

l

x

t

l

A

π φ

π 4

)sin

4 v

cos(

l. Let’s consider now the case of:

l

x

t

l

x

u x t t

π π ω

π ω

)sin

cos(

( , ) cos( )sin

4

π ω 1 t =

2

π

4

3 π (^0) π 4

π ω 1 t =

2

π

4

3 π (^0) π

□ This is an example of a sum of standing waves yielding

m. Thinking backwards, any general wave function can be decomposed

into

n. The number of allowed standing waves on a string of length l :

□ increases as the wavelength decreases ⇒ the possible high-

frequency oscillations outnumber the low-frequency ones.

n

l n

  • Consider that l >> λso we can approximate the set of integers n by a continuous

function n ( λ).

d

l dn

l n n

2

  • The negative sign indicates that the number of standing waves decreases as λ

increases.

o. The number of standing waves in an enclosure of volume V (use c not

v for the speed):

d

V

dn 4

= − but v

c

λ = and

c

v = ; λ

d

c dv 2

= − ⇒ dv c

d

2

v dv c

V

dv c

V

dn

2 3

2

4

c. Again, the general function is a superposition of normal modes

u ( x , y , t ) but in this case one obtains

nm

d. Examples:

m = 1

n = 2

m =

n =

m =

n =

m

n =

e. The case of a square membrane ( a = b ), the frequencies of the normal

modes are given by:

v 22

n m

a

nm

π ω

□ For the cases of n = 1, m = 2 and n = 2, m = 1 one can see that:

a

π ω ω

5 v

although ( , , ) ( , , )

u x y t ≠ u x y t

f. This is an example of

□ The frequency

ω = ω is

□ This phenomenon appears because of the symmetry ( a = b ).

D. Unit Review

1. Important Terminology

frequency

wavelength

wavenumber

angular frequency

independent variables

boundary conditions

separation of variables

eigenfunctions

eigenvalues

stationary wave

traveling wave

node

degeneracy