Joule-Thomson Coefficient
The Measurement of Non-ideal
Behavior of Gases
Background
The objective of this experiment is to quantitatively measure
the non-ideality of gases using the Joule-Thomson
coefficient and relating it to the coefficients of equations for
non-ideality and the Lennard-Jones potential.
For an ideal gas, the internal energy is only a function of the
absolute temperature so in an isothermal process ΔE = 0.
The same is true for the enthalpy for such a process: ΔH = 0.
Thus:
 H   H 

 
 0
 V T  P T
These are non-zero for a non-ideal gas.
Work done by the gas on the first piston:
W1 = P1V1
and the work by the gas on the second piston: W2 = P2V2
The change in the internal energy is: ΔE = - (P2V2 - P1V1)
So:
And therefore:
E2 – P2V2 = E1 – P1V2
H2 = H1
For an isenthalpic process:
 H 
 H 
dH  
 dP  
 dT  0
 P T
 T  P
This can be rearranged to give:
 H 


1  H 
P T
 T 



 

   JT
C P  P T
 H 
 P  H


 T  P
which is zero for an ideal gas. For a non-ideal gas:
dH = TdS + VdP
and at constant temperature:
 H 
 S 

  T   V
 P T
 P T
Using the two relationships:
 G 

 V
 P T
So:
 G 

  S
 T  P
  2G   V 
 S 

  



 
 P T
 P T   T  P
which upon substitution gives the Joule-Thomson
coefficient for a non-ideal gas:

1   V 
 T 
    JT 
 V 
T 
CP   T  P
 P  H

From known equations of state, the sign and magnitude of
the Joule-Thomson coefficient can be calculated.
The Van der Waals Equation serves as an example:

an2 
 P  2 V  nb  nRT
V 

which can be differentiated after neglect of the smallest
magnitude term to give:
nR na
 V 


 
2

T
P
RT

P
And using the approximation:
 nR  V  nb na

 
2
P
T
RT
 
leads to:
V  nb 2na
 V 


 
T
RT 2
 T  P
So the Van der Waals form for the JT coefficient is:

2na / RT   nb
 JT 
CP
with an inversion temperature of:
Ti 
2a
Rb
If the neglected term is included, the exact solution is:
2a
3a
Ti 
Ti  2  0
Rb
R
2
Another equation that describes non-ideality is the BeattieBridgeman equation:
This gives another expression for the JT coefficient:
where:
The most general equation is the virial equation :
Which yields as an expression for the JT coefficient:
The virial coefficient, B2, can be found from statistical
mechanics:
Differentiation with respect to temperature gives a theoretical
expression for the JT coefficient that makes it now a function
of the potential energy of interaction of the molecules, U(r). A
common representation of this potential is the Lennard-Jones
potential:
Thus, the JT coefficient can be related directly to a
theoretical model.
Procedure
A simple apparatus for the JT experiment is set up
where the temperature is measured only on one side
of the porous plug. On the other side of the plug is a
coil of copper tubing submerged in a constant
temperature bath. It is assumed that the gas that
passes through the long coil of tubing reaches the
same temperature as the bath thus eliminating the
necessity of measurement of the temperature of the
gas on the inlet side of the porous plug.
• Three gases JT coefficients are measured in the
order: helium, nitrogen and carbon dioxide.
• Check that all connections are securely wired and
clamped.
• Pressure is applied by opening the appropriate gas
cylinder valve VERY SLOWLY.
– Too fast a fluid will be blown out of manometer
– Too fast and porous plug will freeze up and end the
experiment for the day.
• It will take 20 – 30 minutes to achieve a steady
temperature and pressure.
• Thermocouple calibrations are posted on wall
• Initial equilibration: close needle valve and adjust
main gas supply until regulator reads about 40 psi.
• VERY SLOWLY open the needle valve until the
manometer indicates pressure is increasing by
about 50 Torr/min.
• Continue to adjust needle valve until the
temperature differential is about 750 Torr (this
should take at least 15 min).
• A steady state should be achieved in about 40 min.
• The thermocouple reference junction is embedded
in wax or oil at the bottom of a test tube immersed
in the bath.
• Thermocouple wires to measure the exit
temperature are loosely coiled inside the tube
through which the gas vents from the porous plug.
• Adjust so that the thermocouple junction is in the
center of the tube about 5 to 10 mm above the
porous plug. Stickey tape may be used.
• Record voltage of thermocouple, the pressure
differential from the manometer and the time until
there is no significant change of temperature over
a 10 – 15 minute interval.
• Then VERY SLOWLY close the needle valve until
the pressure differential is about 600 Torr.
• The pressure reduction should take at least 5
minutes.
• Record ΔP and ΔT values until a steady
state is obtained at this new setting ( around
20 min. )
• Repeat the procedure to obtain data at ΔP =
450, 300 and 150 Torr.
Data Analysis
• For each gas plot ΔP against ΔT and obtain the
best straight line fit.
• The line should pass through the origin
• Use the slope to obtain the JT coefficient
• Calculate the JT coefficient for the gases from the
van der Walls and Beattie-Bridgeman constants
and compare to your value.
• Plot the Lennard-Jones potentials for each gas and
obtain the µJT by numerical integration.
– Compare to the other calculated and your measured
values.