Lecture 14 maxwell-boltzmann distribution. heat capacities

Lecture 14 maxwell-boltzmann distribution. heat capacities

1. Lecture 14
Maxwell-Bolztmann distribution.
Heat capacities.
Phase diagrams.

2. Molecular speeds
Not all molecules have the same speed.
If we have N molecules, the number of molecules with
speeds between v and v + dv is:
( )dN Nf v dv=
( )f v= distribution function
= probability of finding a molecule with speed
between v and v + dv
( )f v dv

3. Maxwell-Boltzmann distribution
2
3/2
2 /(2 )
( ) 4
2
mv kTm
f v v e
kT
p
p
-æ ö
=ç ¸
è ø
Maxwell-Boltzmann
distribution
higher T
higher speeds are
more probable

4. Distribution = probability density
= probability of finding a molecule with speed between v and v + dv ( )f v dv
Normalization:
0
( ) 1f v dv
¥
=ò
2
1
1 2
( ) probability of finding molecule with speeds between and
v
v
f v dv v v=ò
= area under the curve

5. Most probable speed, average speed,
rms speed
2
mp
kT
v
m
=
Most probable speed
(where f(v) is maximum)
()
0
0
0
( )
8
( ) ...
( )
vf v dv
kT
v vf v dv
m
f v dv
p
¥
¥
¥
= = = =
ò
ò
ò
Average speed
()
2 2
0
3
( ) ...
kT
v v f v dv
m
¥
= = =ò
Average squared speed
2
rms
3kT
v v
m
= = rms speed

6. Molar heat capacity
How much heat is needed to change by ΔT the
temperature of n moles of a certain substance?
=m nM
n = number of moles
M = mass of one mole (molar mass)
Q mc T nMc T= D = D C Mc=
Q nC T= D molar heat capacityC=

7. Constant volume or constant pressure?
The tables of data for specific heats (or molar capacities) come
from some experiment.
For gases, the system is usually
kept at constant volume V
C
For liquids and solids, the
system is usually kept at
constant pressure (1 atm)
P
C
You can define both for any system.
You need to know what’s in the table!

8. Heat capacity for a monoatomic ideal gas
Average total kinetic energy
total
3 3
2 2
K NkT nRT= =
total
3
2
d K nRdT=
From the macroscopic point of view, this is
the heat entering or leaving the system:
V
dQ nC dT=
3
2
V
nRdT nC dT=
3
2
V
C R=
Molar heat capacity at
constant volume for
monoatomic ideal gas
Point-like particles

9. Beyond the monoatomic ideal gas
Until now, this microscopic model is only valid for monoatomic
molecules.
Monoatomic molecules (points) have 3 degrees of freedom
(translational)
Diatomic molecules (points) have 5 degrees of freedom: 3
translational + 2 rotational)
Principle of equipartition of energy: each velocity
component (radial or angular) has, on average,
associated energy of ½kT
The equipartition principle is very general.

10. Diatomic ideal gas
tr rot
3 2

2 2
5

2
K K K
kT kT
kT
= +
= +
=
Average energy
per molecule
The same temperature involves more
energy per molecule for a diatomic
gas than for a monoatomic gas..
total
5 5
2 2
K NkT nRT= =Average total energy
V
dQ nC dT=
5
2
V
C R=
Molar heat capacity at
constant volume for
diatomic ideal gas
Including rotation
DEMO:
Mono and diatomic
“molecules”

11. Monoatomic solid
Simple model of a solid crystal: atoms held together by springs.
1
3
2
K kT= Vibrations in 3 directions
V
dQ nC dT=
1 1
PE K=
But we also have potential energy!
(for any harmonic oscillator)
1 1
3
3
N
U N KE N PE
NkT
nRT
= +
=
=
For N atoms:
3d U nRdT=
Molar heat capacity at
constant volume for
monoatomic solid
3
V
C R=

12. pT diagram
Critical
point
Sublimation curve
(gas/solid transition)
Melting curve (solid/liquid transition)
solid liquid
gas
Triple
point
Vapor pressure curve
(gas/liquid transition)

13. pT diagram for water
1 atm
T
p
solid
liquid
gas
Critical point
0°C
100°C
DEMO:
Boiling water
with ice
demo

14. In-class example: pT diagram for CO
2
Which of the following states is NOT possible for CO
2
at 100 atm?
A.Liquid
B.Boiling
liquid
C.Melting
solid
D.Solid
E.All of the
above are
possible.

16. solid
Melting (at ~ -50°C)
liquid
For a boiling transition,
pressure must be lower:.
Boiling (at ~ -5°C)

17. At normal atmospheric pressure (1 atm), CO
2
can only be solid or gas.
Triple point for CO
2
has a pressure > 1 atm.
Sublimation at T = -78.51°C

18. pT diagram for N
2
T
p
solid liquid
gas
Triple
point
Triple point for N
2
: p = 0.011 atm, T = 63 K
DEMO:
N
2
snow
At 1 atm, T
boiling
= 77 K T
melting
= 63 K
1 atm
77 K63 K
demo

19. pV diagrams
Expansion at constant
pressure
(isobaric process)
Convenient tool to represent states and transitions from one
state to another.
V
p
B
V
A
V
B
A
states
process
DEMO:
Helium balloon
If we treat the helium in the
balloon is an ideal gas, we can
predict T for each state:
A/B
A/B
pV
T
nR
=
Example: helium in balloon expanding in the room and warming up

20. ACT: Constant volume
This pV diagram can describe:
A.A tightly closed container cooling
down.
B.A pump slowly creates a vacuum
inside a closed container.
C.Either of the two processes.
V
p
B
p
A
p
B
A
In either case, volume is constant and
pressure is decreasing.
In case A, becauseT decreases.
In case B, because n decreases.
(isochoric process)

21. Isothermal curves
For an ideal gas,
nRT
p
V
= (For constant n, a hyperbola for each T )
1 2 3 4
T T T T< < <
Each point in a pV
diagram is a possible
state (p, V, T )
Isothermal curve = all states with the same T

22. ACT: Free expansion
A container is divided in two by a thin wall. One side contains an ideal gas, the
other has vacuum. The thin wall is punctured and disintegrates. Which of the
following is the correct pV diagram for this process?
Initial state
Final state
2
Initial state
Final state
Initial state
Final state
3
Initial state
Final state
4
1A B
C D

23. Final state has larger V, lower p
During the rapid expansion, the gas does NOT uniformly fillV at a
uniform p
Þ hence it is not in a thermal state.
Þ hence no “states” during process
Þ hence this process is not represented by line
Initial state
Final state
2
Initial state
Final state
Initial state
Final state
3
Initial state
Final state
4
1A B
C D

24. Beyond the ideal gas
When a real gas is compressed, it eventually becomes a liquid…
Decrease volume at constant
temperatureT
2
:
• At point “a”, vapor begins to condense
into liquid.
• Between a and b: Pressure and T
remain constant as volume decreases,
more of vapor converted into liquid.
• At point “b”, all is liquid. A further
decrease in volume will required large
increase in p.

25. The critical temperature
For T >> T
c
, ideal gas.
critical temperature= highest temperature where a
phase transition happens.
T
p
solid
liquid
gas
Triple
point
Critical point
Supercritical fluid
Critical point for water: 647K and 218 atm

26. pVT diagram: Ideal gas
States are points on
this surface.

27. pVT diagram: Water
Phase transitions
appear as angles.