Experiment #3
Enthalpy of Sublimation (∆Hsub) using
Knudsen Method
Sublimation of an element or compound is a transition
from the solid to gas phase with no intermediate liquid
stage. Sublimation is an endothermic phase transition that
occurs at temperatures and pressures below the triple
point.
Introduction
{
{
The Knudsen effusion method is
a dynamic method of vapor
pressure measurement that is
restricted to the measurement of
very low pressures (10-2 and 10-3
mmHg).
The material whose vapor
pressure is to be determined is
placed in a container with a
small hole. The vapor pressure
is related to the amount of
material that escapes through
this hole in a given period of
time. It is assumed that the
temperature is maintained
constant at some known value.
Kinetic Theory
{
{
Low pressure ⇒ ideal gas behaviors
Prediction of the number of molecules
that pass through the hole and out of the
container in a given period of time.
z
z
If the mean free path (λ) of the molecule is
large compared to the diameter and length of
the opening, practically all of the molecules
that hit the opening will pass through to the
other side.
Restricted to low vapor pressures
Mean Free Path
{
The average distance a molecule
travels between collisions
1
RT
λ=
=
2
2 ρπ d
2 N Aπ d 2 P
DABCO
2.615Å
1,4-diazabicyclo[2.2.2]octane
SI units
vx
1 m2
Gas Kinetics
{
The Flux or the number of molecules/sec
colliding with 1 m2 of the container is
J N = ρ vx
{
Maxwell’s distribution of velocities in the x
1/ 2
direction
-mv


m
φ (v x ) = 

π
2
k
T
B 

∞
e
1/ 2
∞
 m 
v x = ∫ v xφ (v x )dv x = ∫ v x 

π
2
k
T
B 

0
0
1/ 2
 kT 
JN = ρ 

 2π m 
2
x
1/ 2
 RT 
= ρ

 2π M 
e
2 k BT
-mv 2x
1/ 2
2 k BT
 kT 
dv x = 

 2π m 
Gas Kinetics
{
For Z defined as the number of moles/sec
colliding with 1 m2 of the wall (i.e., Z = JN)
ρ =n/V
Ideal
gas
laws
{
1/ 2
n  RT 
Z= 

V  2π M 
Z=
P
( 2π MRT )
1/ 2
moles
=
sec m 2
Experimentally, Z can be determined by
monitoring mass loss, from which P can be
evaluated.
Clausing Factor
{
{
The equation is valid only for a hole
with its radius much greater than its
length.
In the present experiment, the hole
length may be larger than the
radius. Thus a correction factor
must be applied to the pressure.
Clausing Factor
At hole length (L) > radius (R), a correction factor
(K) must be applied
L/R
K
L/R
K
0.1
0.952399
6.0
0.275438
0.3
0.869928
7.0
0.247735
0.5
0.801271
8.0
0.225263
0.7
0.743410
9.0
0.206641
0.9
0.694044
10.0
0.190941
1.0
0.671584
20.0
0.109304
1.2
0.632228
30.0
0.076912
1.4
0.597364
40.0
0.059422
1.6
0.566507
50.0
0.048448
1.8
0.538975
60.0
0.040913
2.0
0.514231
70.0
0.035415
3.0
0.420055
80.0
0.031255
4.0
0.356572
90.0
0.027925
5.0
0.310525
100.0
0.025258
Ptrue = Pobs/K
g
1.0
0.8
Chi^2 = 0.00003
a
0.46034
b
0.98441
c
0.00466
±0.00675
±0.00416
±0.00098
0.6
K
{
0.4
y = (b + c(L/R))/(1 + a(L/R))
0.2
0.0
0
20
40
60
L/R
80
100
Clausius-Clapeyron Equation
dP ∆H sub
=
dT T∆V
dP P∆H sub
=
dT
RT 2
∆H sub
dP
1 ∆H sub dT
∫ P = ln P = R ∫ T 2 = − RT + const.
{
{
{
A plot of lnP vs. 1/T gives a value of the molar
enthalpy change of sublimation, ∆Hsub.
If P is measured in bar, the intercept of this plot
will give ∆Ssub.
http://webbook.nist.gov/cgi/cbook.cgi?ID=C2805
79&Mask=200
Key Experimental Steps
{
Clausing Factor
z
{
Mass loss (∆n) at three temperatures
after sublimation for one hour
z
z
{
Radius (r) and length (L) by a
comparator
0 oC, 22 oC, and 35 oC
Z = ∆n/(πr2)(3600) → Pobs → Ptrue → λ
Clausius-Clapeyron Equation
z
∆Hsub, ∆Ssub, and ∆Gsub
Literature Results
54.4
Temperature
(K)
323. - 373.
61.9 ± 3.3
324. - 351.
52.3 ± 3.3
353. - 369.
∆subH (kJ/mol)
Reference
Comment
C
Bondi, 1963
Wada, Kishida,
C
et al., 1960
Wada, Kishida,
C
et al., 1960
http://webbook.nist.gov/cgi/cbook.cgi?ID=C280579&Mask=4#Thermo-Phase
Laboratory Safety
{
{
{
Safety goggles
Liquid nitrogen
Vacuum equipment