properties of gases - university of oxfordwallace.chem.ox.ac.uk/teaching/8_slides.pdf · properties...
TRANSCRIPT
• 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⇥
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