Joule-Thomson Coefficient

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Joule-Thomson Coefficient

• The Measurement of Non-ideal Behavior of Gases

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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 ?E 0. The same is
true for the enthalpy for such a process ?H
0. Thus
These are non-zero for a non-ideal gas.
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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 ?E - (P2V2 - P1V1) So
E2 P2V2 E1 P1V2 And therefore
H2 H1 For an isenthalpic
process
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This can be rearranged to give
which is zero for an ideal gas. For a non-ideal
gas dH TdS VdP and at constant temperature
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Using the two relationships
So
which upon substitution gives the Joule-Thomson
coefficient for a non-ideal gas
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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
which can be differentiated after neglect of the
smallest magnitude term to give
And using the approximation
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leads to
So the Van der Waals form for the JT coefficient
is
with an inversion temperature of
If the neglected term is included, the exact
solution is
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Another equation that describes non-ideality is
the Beattie- Bridgeman equation
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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
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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.
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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.
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• 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

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• 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.

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• 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.

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• The pressure reduction should take at least 5
  minutes.
• Record ?P and ?T values until a steady state is
  obtained at this new setting ( around 20 min. )
• Repeat the procedure to obtain data at ?P 450,
  300 and 150 Torr.

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Data Analysis

• For each gas plot ?P against ?T 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.