Intro/Review of Quantum QM-1

So you might be thinking… I thought I could avoid Quantum Mechanics?!? Well… we will focus on thermodynamics and kinetics, but we will consider this topic with reference to the

molecular basis quantum mechanically

that underlies the laws of thermodynamics. Since molecules behave , we will need to know a few of the

results

that are provided from quantum mechanics. Those interested in more details should take CHE 372 this spring!

Energy is Quantized

Macroscopic

Big things, small relative energy spacings, energy looks

classical

(i.e., continuous) Time

QM-2

Microscopic

Small things, large relative energy spacings, must consider the energy levels to be quantized Time

Energy is Quantized by

h

QM-3

Planck suggests that radiation (light, energy) can only come in quantized packets that are of size

hν.

Planck, 1900 Energy (J)

E



h

 Planck’s constant h = 6.626 × 10 -34 J·s Frequency (s -1 ) Note that we can specify the energy by specifying

any one

of the following: 1. The frequency, n (units: Hz or s -1 ):

E



h

 2. The wavelength, λ, (units: m or cm or mm): Recall:  

c E



hc

 EX-QM1 3. The wavenumber, Recall:  1  (units: cm -1 or m -1 )

E



hc



Where can I put energy?

QM-4

Connecting macroscopic thermodynamics to a molecular understanding requires that we understand how energy is distributed on a molecular level.

ATOMS:

The electrons

: Electronic energy . Increase the energy of one (or more) electrons in the atom.

Nuclear motion

: Translational energy . The atom can move around (translate) in space.

MOLECULES:

The electrons

: Electronic energy . Increase the energy of one (or more) electrons in the molecule.

Nuclear motion

: Translational energy . The entire molecule can translate in space.

Vibrational energy . The nuclei can move relative to one another. Rotational energy . The entire molecule can rotate in space.

Schrödinger Equation QM-5

Schrödinger Erwin Schrödinger formulated an equation used in quantum mechanics to solve for the energy of different systems:

h

  2 2

m

 

x

2 2  (

x

) 

V

(

x

)  (

x

)   (

x

)   2  Kinetic energy Potential energy Total energy  (

x

) is the

wavefunction

. The wavefunction is the most complete possible description of the system. Solving the differential equation (S.E.) gives one set of

wavefunctions

 (

x

) and a set of associated

eigenvalues

(i.e., energies) E.

Interested in solving this problem for specific systems?!?! Take CHE 372 in the spring! Meanwhile, you are required such to be familiar with the

solutions

for the systems we will encounter.

ATOMS I: H atom electronic levels QM-6

Convert J to cm -1 ; Can you?

Electronic Energy Levels: 

el

  2 .

17869  10  18

n

2

J

  109680

cm

 1

n

2

n

must be an integer.

Series limit,

n

(

r

= ∞) + = ∞, the electron and proton are infinitely separated, there is no interaction.

Ground state,

n

= 1, the most probable distance between the electron and proton is r mp = 5.3 × 10 -11 m.

+ EX-QM2

Wavefunctions and Degeneracy

The wavefunctions are the atomic orbitals.

QM-7

3s 2s 1s The number of wavefunctions, or states, with the same energy is called the

degeneracy, g n .

ATOMS II: Translational Energy QM-8

In addition to electronic energy, atoms have translational energy. To find the allowed translational energies we solve the Schrödinger equation for a particle of mass,

m

. 0

a x

In 1D, motion is along the

x

dimension and the particle is constrained to the interval

0 ≤ x ≤ a .



n



n

2

h

2 8

ma

2

n

 1 , 2 , 3 ,...

In 3D…

z



n x

,

n y

,

n z



h

2 8

m n a

2

x

2 

n b

2 2

y



n z

2

c

2

n x n y n z

  1 , 2 , 3 ,...

1 , 2 , 3 ,...

 1 , 2 , 3 ,...

c b a x

These states can be

degenerate

. For example, if a=b=c, then the two

different states

(

n x

=1,

n y

=1,

n z

=2) and (

n x

=2,

n y

=1,

n z

=1) have the

same energy

.

Electronic Energy Levels, Generally QM-9

As we have seen, the electronic energy levels of the hydrogen atom are quantized. However, there is no simple formula for the electronic energy levels of any atom beyond hydrogen. In this case, we will rely on tabulated data.

For the electronic energy levels, there is a large gap from the ground state to the first excited state. As a result, we seldom need to consider any states above the ground state at the typical energies that we will be working with.

MOLECULES I: Vibrational QM-10

We model the vibrational motion as a harmonic oscillator, two masses attached by a spring.

Solving the Schr ödinger equation for the harmonic oscillator you find the following quantized energy levels: 

v v

  nu and vee!

h

 (

v

 1 2 ) 0 , 1 , 2 ,...

The energy levels The level are non-degenerate, that is

g v

=1 for all values of

v

.

The energy levels are equally spaced by

h

n .

R e R The energy of the lowest state is NOT zero. This is called the zero-point energy.

 0  1 2

h



MOLECULES II: Rotational

Moment of inertia:

I

QM-11



m

1

R

1 2 

m

2

R

2 2  4  10  2

I J

=4 Treating a diatomic molecule as a rigid rotor, and solving the Schrödinger equation, you find the following quantized energy levels… 

J

  2

I

2

J

(

J

 1 )

J

 0 , 1 , 2 ,...

 3  6  2

I

 2 

J

 1   2  0

I

 3  2

I

The degeneracy of these energy levels is:

g J

 2

J

 1

J

=3

J

=2

J

=1

J

=0

EX-QM3

Dissociation Energy QM-12

The dissociation energy and the electronic energy of a diatomic molecule are related by the zero point energy. Negative of the electronic energy

D e



D

0 

h

 2 Dissociation energy For H 2 …

D e

= 458 kJ ·mol -1

D 0

= 432 kJ·mol -1  = 4401 cm -1 (=52 kJ ·mol -1 )

Polyatomic Molecules I: Vibrations QM-13

For polyatomic molecules we can consider each of the

n vib

vibrational degrees of freedom as independent harmonic oscillators. We refer to the characteristic independent vibrational modes as

normal modes

.

For example, water has 3 normal modes: Since the normal modes are independent, the total energy is just the sum: 

vib



n vib j

  1

h



j



v j

 1 2   ~  1595

cm

 1 Bending Mode EX-QM4  3686

cm

 1 Symmetric Stretch  3725

cm

 1 Asymmetric Stretch

Polyatomic Molecules II: Rotations QM-14

Linear molecules

: The same as diatomics with the moment of inertia defined for more than 2 nuclei: 

J

  2

I

2

J

(

J

 1 )

J

 0 , 1 , 2 ,...

g J

 2

J

 1

I



j n

  1

m j

(

x j



x cm

) 2

Nonlinear molecules

: There is one moment of inertia for each of the 3 rotational axes. This leads to three ways to define polyatomic rotors:

Spherical top

(baseball, CH 4 ):

I A = I B = I C Symmetric top

(American football, NH 3 ):

I A = I B ≠ I C Asymmetric top

(Boomerang, H 2 0):

I A ≠ I B ≠ I C

Degrees of Freedom QM-15

To specify the position of a molecule with

n

require 3

n

nuclei in space we coordinates, this is 3 Cartesian coordinates for each nucleus. We say there are

3 n degrees of freedom

. We can divide these into translational, rotational, and vibrational degrees of freedom: Degrees of Freedom Translation: (3

n

in total) Motion of the center of mass 3 Rotation (Orientation about COM): Linear Molecule Non-Linear Molecule Vibration (position of

n

nuclei): Linear Molecule Non-Linear Molecule 3 3 2 3

n n

-5 -6 EX-QM5

Total Energy QM-16

The total energy is the energy of each degree of freedom:   

trans

 

rot

 

vib

 

elec



n x

,

n y

,

n z



h

2 8

m n x

2

a

2 

n b

2 2

y



n z

2

c

2

n x n y n z

  1 , 2 , 3 ,...

1 , 2 , 3 ,...

 1 , 2 , 3 ,...



v

 

J J

  2

J

(

J

2

I

 0 , 1 , 2 ,...

 1 ) For linear molecules.

h

 (

v

 1 2 ) For each vib. DOF Look up values in a table (i.e.,

D e

).

Relative Energy Spacings QM-17

The general trend in energy spacing: Electronic > Vibrations > Rotations > > Translations

J=2 J=1 J=0

EX-QM6

Population: Boltzmann Distribution QM-18

The Boltzmann distribution determines the relative population of quantum energy states.

Ludwig Boltzmann Probability that a randomly chosen system will be in state j with E j

p j



e



E j k B T



i e



E i k B T

Partition function This equation is the key equation in

statistical mechanics

, the topic of the next few sections of this class. Statistical mechanics is used to comprehend ‘macroscopic’ thermodynamics in terms of a ‘microscopic’ molecular basis.