Download Graham`s Law of Effusion

Graham’s Law of Effusion
Effusion is the movement of gases
through small passages
(e.g., passing through a plug of fine sand)
where essentially all collisions are between gas and sand
This feels like something where the faster you go, the faster you get
through the maze, so we might expect an effusion rate  velocity.
Graham observed that for gases A and B
Rate(A)/Rate(B) = [M(B)/M(A)]1/2
The rate of effusion of a gas is inversely proportional
to the square root of its molar mass (M)
This is not the kind of motion that characterizes
an ideal gas confined in a large box
Real Gases
(ideal gas eq of state now an approximation)
PV
PV
Z

nRT RT
 Compressibility, Z
Z=1
Z<1
Z>1
Ideal Gas behavior
PV less than expected
PV greater than expected
Z
Attractive forces
Repulsive forces
Ar
Ideal Gas
1
P
Real Gases – data!
 Compressibility
PV
Z
nRT
Boyle Temperature, TB
Temperature of greatest extent
of near-ideal behavior.
We can determine TB analytically.
Z T , P 
 f T 
P
lim p  0
df
 0 at T  TB , the Boyle temperature
dT
van der Waals equation of state
Physically-motivated corrections to Ideal Gas EoS.
For a real gas, both attractive and repulsive intermolecular
forces are present. Empirical terms were developed to help account
for both.
1. Repulsive forces: make pressure higher than ideal gas
Excluded volume
P
nRT
V  nb
Volume of one molecule of radius r is Vm = (4/3) r3
Closest approach of two molecules with radius r is 2r.
ConcepTest #5
Excluded volume
nRT
P
V  nb
The volume of one molecule of radius r is Vm = 4/3 r3
The closest approach of two molecules with radius r is 2r.
What is the excluded volume for the two molecules?
A. 2Vm
B. 4Vm
C. 8Vm
D. 16 Vm
van der Waals equation of state
Physically-motivated corrections to Ideal Gas EoS.
For a real gas, both attractive and repulsive intermolecular
forces are present. Empirical terms were developed to help account
for both.
1. Repulsive forces: make pressure higher than ideal gas
(or, equivalently, make the volume smaller)
Excluded volume
P
nRT
V  nb
Volume of one molecule of radius r is Vm = 4/3 r3
Closest approach of two molecules with radius r is 2r.
The excluded volume Vexc is 23 Vm = 8Vm for two molecules.
So we might estimate that b  4VmNA
This assumes binary collisions only. Always true? NO!
van der Waals equation of state
Physically-motivated corrections to Ideal Gas EoS.
For a real gas, both attractive and repulsive intermolecular
forces are present. Empirical terms were developed to help account
for both.
2. Attractive forces: make pressure lower than ideal gas
Pressure derives from molecules colliding with the walls,
both the frequency and the force of the collisions.
Both scale as n/V, so we expect a pressure correction of the
form –a(n/V)2, giving the van der Waals Equation of State
nRT
an
RT
a
 2 
 2
P
V  nb V
V b V
2
Let’s plot this function P(V,T)
3D van der Waals eqn of state
T= T/Tc
Real Gases
CO2
Look at 20 C isotherm.
ABC Compression
At C, liquid condensation begins
D – liquid- vapor mixture at
Pvap(20 C)
E – last vapor condenses
F – Steep rise in pressure
A liquid or solid is much less
compressible than a gas
For T >Tc, there is a single phase,
with no liquid formed.
van der Waals Isotherms near Tc
v d W “loops” are
not physical. Why?
Patch up with Maxwell
construction
van der Waals Isotherms, T/Tc