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lecture18.pdf
1. Nils Walter: Chem 260
σ
σ
σ
σ = collision cross-
section (target area)
= π
π
π
πd2
Molecular collisions
r.m.s. speed z
z
c λ
λ
=
=
1
Mean free path
between two collisions
Time of flight
between two collisions
z = collision frequency
RT
pc
N
z
p
N
RT A
A
σ
σ
λ
2
;
2
=
=
⇒
⇒
⇒
⇒
2. Nils Walter: Chem 260
In reality: Gases have attractive and
repulsive forces
Lennard-Jones
6-12 potential
⇒
⇒
⇒
⇒
e.g.,
CO2
At high T: perfect
gas isotherms
At low T:
liquefaction
3. Nils Walter: Chem 260
The critical point: Gas and liquid
density become equal
⇒
⇒
⇒
⇒ Application:
Extraction of caffeine
from coffee with
supercritical CO2
At critical point
(for water 373oC
@ 218 atm!) the
boundary is lost
Heating a liquid in a container
4. Nils Walter: Chem 260
Describing the deviation from the perfect gas
Introducing the
compression factor Z:
molar volume of real gas
molar volume of perfect gas
Z=
RT
pV
p
RT
V
V
V
Z m
m
perfect
m
m
=
=
=
Z = 1 ⇒ perfect gas
Z < 1 ⇒ molecules cluster, attractive
forces are dominant
Z > 1 ⇒ molecules repel each other,
repulsive forces are dominant
5. Nils Walter: Chem 260
The virial equation of state
Empirically: ...
1 2
+
+
+
=
m
m V
C
V
B
Z
virial coefficients
B > 0 ⇒ Z > 1, e.g., H2
B < 0 ⇒ Z < 1, e.g., CH4, NH3
C > 0 ⇒ Z > 1 at high pressure (Vm small)
+
+
+
= ...
1 2
2
V
C
n
V
nB
V
nRT
p
RT
pV
Z m
=
⇒
⇒
⇒
⇒
and very accurate
6. Nils Walter: Chem 260
Physically more palpable:
The van der Waals equation
[Johannes van der Waals 1873]
( ) nRT
nb
V
V
an
p =
−
+ 2
2
⇒
⇒
⇒
⇒
molecules have a
non-zero volume
⇒
⇒
⇒
⇒ additional volume
needed: nb
molecules have
attractive forces
⇒
⇒
⇒
⇒ reduction in exerted
pressure: a(n/V)2
[molecules strike less
frequently and with
reduced force]
Lennard-Jones
6-12 potential
7. Nils Walter: Chem 260
Plotting the van der Waals equation:
In reasonable agreement with reality
only the van
der Waals
loops are
unrealistic
in 3D
p
V
T
8. Nils Walter: Chem 260
Liquefaction of real gases:
The Joule-Thomson effect
Real gases have attractive
forces
Linde refrigerator
⇒
⇒
⇒
⇒ if they are allowed to
expand through a throttle
without outside heat
entering (“adiabatic”
process) they will use their
kinetic (heat) energy to
escape each other’s
attraction
⇒
⇒
⇒
⇒ they will cool down
