Lecture 3

Question 1

What is the molecular partition function?

Answer 1

Sum(i)e^-Bie

Question 2

What formula describes the energy from each type of energy system? (comes from the partition function)

Answer 2

Ei = ei(translational) + ei (rotational) + ei(vibrational) + ei(electronic)

Question 3

How are the different energy states separated?

Answer 3

-Assumption that electronic states are always in the ground state
-Based on the born-Oppenheimer approx
-The rot is independent of the vib state

Question 4

How is the formula of energies put into the partition function?

Answer 4

q = Sum(i)e^-Bie
ei = eiT + eiR + eV + eiE
Gives over all
q = qTqRqVqE

Question 5

What is the partition function for translational for a molecule moving only 1 direction?

Answer 5

-Approximate that its only moving a single direction (X) in a box with fixed volume
The energy levels are
En = n^2h^2/8mX^2

When q is subbed into this
qx = (2pim)/h^2B))^1/2 all times by X (the direction)

Question 6

What is the MPF for a molecule moving in all directions for translational?

Answer 6

In all directions (X,Y and Z) the total energy is
ei = eXn1 + eYn2 + xZn3

This is subbed into q and since e^(a+b+c) is equal to e^a x e^b x e^c then it becomes q = qxqyqz (where x,y and z are a,b and c)

The formula is therefor
qx = (2pim/h^2B)^3/2 x XYZ
XYZ = the volume of the box and simplifies to V

So q simplifies too
q = V/TW^3

TW = h/SQRT(2pikmT) and has units of m

m = kg
h = J s
V = m3
k = J K-1

Question 7

How is cm3 converted too m3?

Answer 7

By dividing by E6

Question 8

What are the units of thermal wavelength and therefor q?

Answer 8

TW = m
Volume = m3
q = V/TM^3
q = unitless

Question 9

What are the main conclusions on translational partition function?

Answer 9

-The gap between trans energy levels in small (due to q being large)
-At room temp a large amount of states are occupied (as q is large)
-A continuum at room temp

Question 10

How do we get the vibrational partition function?

Answer 10

Use the harmonic oscillator approx, where e = hcv

c = SOF
v = frequency of wavenumebers

As the parition function (q) is equal to 1/1-e^-Bev for HO and using equation for e above

qV = 1/1-e-Bhcv

To find B = hcv/kT

Question 11

What does a small q mean for vib?

Answer 11

That only ground and first excited states are occupied at that temperature

Question 12

How is speed of light converted to fit the equation?

Answer 12

Times by 100 to give cm

Question 13

What is the equation for rotational energy levels?

Answer 13

ej = hc”B”J(J+1)

Where this B is equal to a rot constant.
J = rot quantum number

Question 14

How are rot energy levels separated?
(Degeneracy increases by 2 each time)

Answer 14

By 2J + 1
The first level has 1 energy state, then the next has 3 and so on.

Question 15

What is the PF for rot?

Answer 15

The degenracy, gj = 2J + 1, and the energy equation, ej = hc”B”J(J+1) are subsitiuted into

Sum(gj)e-Be

gives qR = Sum(i)(2J+1)e^-Bhc”B”J(J+1)

Question 16

What trends does the rot partition function show?

Answer 16

That q for increasing energy levels will increase then drop rapidly. This is due to the Boltzmann factor contribution (due to the exponential), that energy levels that get higher aren’t favoured at RT as there’s not enough energy.

Question 17

How do we use the rot PF?

Answer 17

Each energy level is calculated separately then added up (to give total qR), it will keep decreasing as the level goes up, so eventually the no will be so small that it doesn’t contribute.

Question 18

How is the MPF done for a rotational energy levels of a LINEAR molecule? (This is a more updated version that above, both are for linear)

Answer 18

qR = kT/sigma x hc”B”

Approximation only right is T&raquo_space; hc”B”/k

Question 19

What is the symmetry number for symmetrical and non-sym linear molecules?

Answer 19

Sym is 2
Non sym is 1

Question 20

What is the rot MPF for non-linear molecules?

Answer 20

qR = (1/sigma)x(kT/hc)^3/2 x (pi/ABC*)^1/2

A, B and C are rot constants
sigma is the sym number (from tells stuff) - how many time it can turn to look the same

Question 21

What is the MPF of electronic states?

Answer 21

Ground electronic states usually have a lot less energy that excited states, and its assumed that only the GS is occupied. So, qE = g0E, the degeneracy of the ground state.

Question 22

What can q be used for? (when considering weak interactions!!)

Answer 22

-To find the mean energy, which can be used to find other thermodynamic processes

Question 23

How is the MPF of mean energy derived?

Answer 23

Using the MPF (from boltzman) and the fact that the E = sum(i)niei

[e] = E/N = 1/q sum(i)energyi e^-Bei

Using derivatives, this simplifies to -1/q x dp/dB

Question 24

What is the mean energy of a two-state system?

Answer 24

MPF is q = 1 + e^-Be
So [e] = e(energy state)/1+e^Benergy

Question 25

How does mean energy change with temp?

Answer 25

As T goes to 0, ME goes to 0.
As T goes to infinity, ME is equal half the energy of the system.

Question 26

How do we correct for the real mean energy?

Answer 26

As assumed, e0 = 0 so we need to add the energy of the ground state to the ME calculated to get the real one.

Question 27

How else should this ME be expressed?

Answer 27

Usually depends on something, for example volume.

[e] = -1/q (delq/del B) V
[e] = - (del ln(q)/delB) V

Question 28

What is the mean trans energy?

Answer 28

Given same as above except a T is in it.
Sub with the MPF to give
[eT]=3/2kT

Question 29

What is the mean vib energy?

Answer 29

Given same as above but with a v, and using the MPF of vib energy.
[eV] =hcv/(e^Bhcv -1)