Lecture 14Maxwell-Bolztmann distribution.
Heat capacities.Phase diagrams.
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
Maxwell-Boltzmann distribution
2
3/ 2
2 / (2 )( ) 42
mv kTmf v v e
kT
Maxwell-Boltzmanndistribution
higher T
higher speeds are more probable
Distribution = probability density
= probability of finding a molecule with speed between v and v + dv ( )f v dv
Normalization:
0( ) 1f v dv
2
11 2( ) probability of fi nding molecule with speeds between and
v
vf v dv v v
= area under the curve
Most probable speed, average speed, rms speed
2mp
kTv
m
Most probable speed (where f(v) is maximum)
0
0
0
( ) 8( ) ...
( )
vf v dv kTv vf v dv
mf v dv
Average speed
2 2
0
3( ) ...
kTv v f v dv
m
Average squared
speed
2rms
3kTv v
m rms speed
Molar heat capacity
How much heat is needed to change by ΔT the temperature of n moles of a certain substance?
m nMn = number of moles
M = mass of one mole (molar mass)
Q mc T nMc T C Mc
Q nC T molar heat capacityC
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
VC
For liquids and solids, the system is usually kept at constant pressure (1 atm)
PC
You can define both for any system. You need to know what’s in the table!
Heat capacity for a monoatomic ideal gas
Average total kinetic energy total
3 32 2
K NkT nRT
total
32
d K nRdT
From the macroscopic point of view, this is the heat entering or leaving the system:
VdQ nC dT
32 VnRdT nC dT 3
2VC RMolar heat capacity at constant volume for monoatomic ideal gas
Point-like particles
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.
Diatomic ideal gas
tr rot
3 2
2 25
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 52 2
K NkT nRT Average total energy
VdQ nC dT
52VC R
Molar heat capacity at constant volume for diatomic ideal gas
Including rotationDEMO: Mono and diatomic “molecules”
Monoatomic solid
Simple model of a solid crystal: atoms held together by springs.
1
32
K kT Vibrations in 3 directions
VdQ nC dT
1 1PE K
But we also have potential energy!
(for any harmonic oscillator)
1 1
3
3
NU N KE N PE
NkT
nRT
For N atoms:
3d U nRdT
Molar heat capacity at constant volume for monoatomic solid
3VC R
pT diagram
Critical point
Sublimation curve (gas/solid transition)
Melting curve (solid/liquid transition)
solid liquid
gasTriple point
Vapor pressure curve (gas/liquid transition)
pT diagram for water
1 atm
T
p
solid
liquid
gas
Critical point
0°C 100°C
DEMO: Boiling water
with ice
demo
In-class example: pT diagram for CO2
Which of the following states is NOT possible for CO2 at 100 atm?
A. Liquid
B. Boiling liquid
C. Melting solid
D. Solid
E. All of the above are possible.
solid
Melting (at ~ -50°C)
liquidFor a boiling transition, pressure must be lower:.
Boiling (at ~ -5°C)
At normal atmospheric pressure (1 atm), CO2 can only be solid or gas.
Triple point for CO2 has a pressure > 1 atm.
Sublimation at T = -78.51°C
pT diagram for N2
T
p
solid
liquid
gas
Triple point
Triple point for N2: p = 0.011 atm, T = 63 K
DEMO: N2 snow
At 1 atm, Tboiling = 77 K Tmelting = 63 K
1 atm
77 K63 K
demo
pV diagrams
Expansion at constant pressure
(isobaric process)
Convenient tool to represent states and transitions from one state to another.
V
p
B
VA VB
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/ BA/ B
pVT
nR
Example: helium in balloon expanding in the room and warming up
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
pA
pB
A
In either case, volume is constant and pressure is decreasing.
In case A, becauseT decreases.
In case B, because n decreases.
(isochoric process)
Isothermal curves
For an ideal gas, nRT
pV
(For constant n, a hyperbola for each T )
1 2 3 4T T T T Each point in a pV diagram is a possible state (p, V, T )
Isothermal curve = all states with the same T
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
Final state has larger V, lower p
During the rapid expansion, the gas does NOT uniformly fillV at a uniform p hence it is not in a thermal state. hence no “states” during process 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
Beyond the ideal gas
When a real gas is compressed, it eventually becomes a liquid…
Decrease volume at constant temperatureT2:
• 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.
The critical temperature
For T >> Tc, 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
pVT diagram: Ideal gas
States are points on this surface.
pVT diagram: Water
Phase transitions appear as angles.