• Ideal & real gases !

• Kinetic theory of gases!

• Maxwell-Boltzmann distribution!

• Applications of kinetic theory

Properties of gases

Collision Frequency, z

• Number of collisions for one particle = volume × number density

• Collision frequency for one particle = number of collisions in unit time

• Volume swept-out by a single particle

⇡d2 hvreli t

Collision Density, Z

• Collision density is the total number of collisions occurring per unit volume

Mean free path� =

�v⇥z

• Effusion!

• Diffusion!

• Thermal Conductivity!

• Viscosity

Applications of kinetic theory

Effusion

• Gas escapes into a vacuum through a small hole!

• No collisions occur after the particles pass through the hole.!

• The rate of effusion will be proportional to the rate at which particles strike the area defining the hole!

• The diameter of the hole is smaller than the mean free path in the gas!

• No collisions occur as the particles pass through the hole

Effusion Collisions with container walls

⇥V ⇤ =⇤ �

0V P (vx)dvx

= A�t

⇤ �

0vx

⌅m

2�kBTexp

��mv2

x

2kBT

⇥dvx

P (vx)dvx =�

m

2�kBT

⇥1/2

e�mv2

x2kBT dvx

⇥V ⇤ =⇤ �

0V P (vx)dvx

= A�t

⇤ �

0vx

⌅m

2�kBTexp

��mv2

x

2kBT

⇥dvx

Collision frequency with wall (per unit area, per unit time)

⇢N =N

V=

p

kBT

Rate =dN

dt= zwalla =

pa

(2�mkBT )1/2

dp

dt=

d

dt

�NkBT

V

⇥=

kBT

V

dN

dt

p = p0 exp�t/⇥ where ⇥ =�

2�m

kBT

⇥1/2 V

a

Effusion Collisions with aperture

t

p

Effusionp = p0 exp�t/⇥ where ⇥ =

�2�m

kBT

⇥1/2 V

a

t

p• Graham’s law of effusion!

• Rate of effusion of a gas is inversely proportional to the square root of the mass of its particles

Rate =dN

dt= zwalla =

pa

(2�mkBT )1/2

Transport Properties

PropertyTransported

Quantity Kinetic Theory Units

Diffusion Matter m2s-1

Thermal Conductivity Energy J K-1 m-1 s-1

Viscosity Momentum kg m-1 s-1

� =13⇥ �v⇥CV [X]

D =13� �v⇥

Flux Describes the amount of matter, energy, charge... passing through a unit

area per unit time

Jz �d�N

dzwhere ρN is the number density. For matter flux,

Fick’s first law of diffusion

Jz = �Dd�N

dz

D =13� �v⇥

⇥N (��) = ⇥N (0)� �

�d⇥N

dz

⇥

0

⇥N (+�) = ⇥N (0) + �

�d⇥N

dz

⇥

0

Jleft =⇥v⇤ ⇥N (��)

4=

⇥v⇤4

�⇥N (0) � �

�d⇥N

dz

⇥

0

⇥Jright =

�v⇥ ⇥N (+�)4

=�v⇥4

�⇥N (0) + �

�d⇥N

dz

⇥

0

⇥

Jz = Jleft � Jright = �12

�d⇥N

dz

⇥

0

� ⇥v⇤

D =12� �v⇥Jz �

d�N

dzwhere ρN is the number density. For matter flux,

zwall = ⇢N hV i = N

V

✓kBT

2⇡m

◆1/2

= ⇢Nhvi4

Diffusion in a Gas

D =13� �v⇥

⇥N (��) = ⇥N (0)� �

�d⇥N

dz

⇥

0

⇥N (+�) = ⇥N (0) + �

�d⇥N

dz

⇥

0

Jleft =⇥v⇤ ⇥N (��)

4=

⇥v⇤4

�⇥N (0) � �

�d⇥N

dz

⇥

0

⇥Jright =

�v⇥ ⇥N (+�)4

=�v⇥4

�⇥N (0) + �

�d⇥N

dz

⇥

0

⇥

Jz = Jleft � Jright = �12

�d⇥N

dz

⇥

0

� ⇥v⇤

zwall =N

V

�kBT

2�m

⇥1/2

=N

V

�v⇥4

D =12� �v⇥Jz �

d�N

dzwhere ρN is the number density. For matter flux,

Diffusion

Jz = �Dd�N

dz

D =13� �v⇥

Diffusion in a Gas

Thermal Conductivity in a Gas

T

Jz = Jleft � Jright = �12

�dT

dz

⇥

0

�kB⇤N⇥ ⇥v⇤

⇥ =13�kB⌅N⇤ �v⇥ � =

13⇥ �v⇥CV [X]

zwall =N

V

�kBT

2�m

⇥1/2

=N

V

�v⇥4

T

⇥N (��) = ⇥N (0)� �

�d⇥N

dz

⇥

0

⇥N (+�) = ⇥N (0) + �

�d⇥N

dz

⇥

0Previously

Now

Transport Properties

PropertyTransported

Quantity Kinetic Theory Units

Diffusion Matter m2s-1

Thermal Conductivity Energy J K-1 m-1 s-1

Viscosity Momentum kg m-1 s-1

� =13⇥ �v⇥CV [X]

D =13� �v⇥

Transport Properties

PropertyTransported

Quantity Kinetic Theory Units

Diffusion Matter m2s-1

Thermal Conductivity Energy J K-1 m-1 s-1

Viscosity Momentum kg m-1 s-1

� =13⇥ �v⇥CV [X]

D =13� �v⇥

Classical Mechanics Summary

• Translation!

• Newton’s laws!

• Equations of motion!

• Linear momentum!

• Frames of reference & reduced mass!

• Rotation!

• Angular momentum!

• Moments of inertia!

• Vibration!

• Simple harmonic motion!

• The wave equation!

• Work & Energy!

• Elastic & inelastic Collisions!

• Centre of Mass

Properties of Gases Summary

• Ideal & real gases!

• pV=nRT!

• Virial expansion!

• Compression factor!

• Kinetic theory of gases!

• Derivation!

• Classical equipartition!

• Predicting Cv!

!

• Maxwell Boltzmann!

• (Derivation)!

• Collision frequencies!

• Mean free path!

• Applications!

• Effusion!

• Diffusion