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.