Lecture 6 Notes: Finite Equations
Goal: Derive Virial Eqn. of State
p
A
lnQ
kT
V N,T
V
pressure
2mkT
3N/2
3N
N!h
Q
lnQ ln
N,T
2mkT
Z N,V,T
2mkT
3N /2
N lnV
3N
N!h
Plugging ln Q into equation for p
3N/2
N
3N
N N
V exp
N!h
N N
2V
2 V
2
constants N l
Lecture 2 Notes: Quantum Descriptions
QUANTUM DESCRIPTION Calculate x average kinetic energy in x-direction
x0xP xdx
1/2
1/2
PxkT x e x
x
x
1 1/2
kT
0
1/2e x kTd
1 1/2 kT
kT
x
kT
1/2
2
1
kT
x
kT
2
1
This is not an accident. It is our first glimpse of equi
Lecture 3 Notes: Vibration Degrees
A molecule with n atoms has 3n "degrees of freedom" or 3n coordinates to
describe its position and therefore has 3n ways of incorporating energy due to nuclear
motion where n is the number of atoms in the molecule.
For a
Lecture 5 Notes: Classical NKT Values
E E0
Nkvib
Nkvib
1vib T
1
vib
e
x2 x3 1
x
e 1 x for x < 1 2! 3!
vib
NkT
T
This quantum result yields the same value for (EE0)vib as the classical approach when T vib
or vib kT. Classical equipartition principle say
Lecture 4 Notes: Graphical Representations
lim Erot lim 6Nkre
T0
T0
rot
lim
lim CV
T0
12Nkr
T
T0
2
2 r
2
e
T
0
2r T
0
1.0
rot
C
V
nR
1.0
Low T Limit
CVrot 12r2 e
E
rot
nR
Note maximum in
2 T
r
6re
v
T/rot
Cvrot 1
nR
2r T
C
nR
rot
High T Limit
T2
nR
E
E
Lecture 7 Notes: Statistical Mechanics
aA bB cC dD A,B,C,D are all ideal gases iT,pi
o
i (T) RTlnpi
o
means standard state, usually p = 1 bar.
at equilibrium,
cC + dD aA bB = 0
plug in equations for each i(T,pi)
o
o
o
o
C
D
A
B
0 c (T) d (T) a (T) b
(T)
Lecture 10 Notes: Diffusion Property
We begin by considering the important case of diffusion. Diffusion is a very
important transport property for chemists because it describes the mass transport
necessary to bring molecules into sufficiently close proxim
Lecture 9 Notes: Free Path
The mean free path. The mean free path is the average distance a particle traverses
before it experiences a collision. In Lecture #31 we determined the average collision
frequency for a particle, Z. The mean time between collisi
Lecture 8 Notes: Distance Pairs
Goal: For U( q ) 0, calculate Z to obtain corrections for non-ideal contributions to the
equation of state.
Z
~
dq
where
U(q)/kT
3N
e
~
U( q ) = Total Interaction Potential Energy
Simplifications based on form of U( q ):
1.
Lecture 1 Notes: Partition Function
The macroscopic thermodynamic properties are written in terms of Q. Q is
related to the single-molecule partition function q, which is the sum over the molecular
energy levels or states. Atoms and molecules have differe