Joule-Thomson effect
TEP
Related Topics
Real gas; intrinsic energy; Gay-Lussac theory; throttling; Van der Waals equation; Van der Waals force;
inverse Joule- Thomson effect; inversion temperature.
Principle
A stream of gas is fed to a throttling point, where the gas (CO 2 or N 2 ) undergoes adiabatic expansion.
The differences in temperature established between the two sides of the throttle point are measured at
various pressures and the Joule- Thomson coefficients of the gases in question are calculated.
Equipment
1 Joule-Thomson apparatus
1 Temperature meter digital, 4-2
2 Temperature probe, immers. Type
1 Reducing valve for CO 2 / He
1 Reducing valve f. nitrogen
1 Wrench for steel cylinders
1 Steel cylinder rack, mobile
1 Steel cylinder, CO 2 , 10 l, full
1 Steel cylinder, nitrogen, 10 l, full
04361-00
13617-93
11759-01
33481-00
33483-00
40322-00
54042-00
41761-00
41763-00
Fig. 1: Experimental set-up: Joule-Thomson effect.
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Joule-Thomson effect
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Tasks
1. Determination of the Joule-Thomson coefficient of CO 2 .
2. Determination of the Joule-Thomson coefficient of N 2 .
Set-up and procedure
The set-up of the experiment is as in Fig. 1.
If necessary, screw the reducing valves onto the steel cylinders and check the tightness of the main
valves. Secure the steel cylinders in their location. Attach the PVC-tubing between the reducing valve
and the Joule-Thomson apparatus with hose tube clips.
On each side of the glass cylinder, introduce a temperature probe up to a few millimetres from the frit
and attach with the union nut. Connect the temperature probe on the pressure side to inlet 1 and the
temperature probe on the unpressurised side to inlet 2 of the temperature measurement apparatus.
Important:
The experimenting room and the experimental apparatus must be in a thermal equilibrium at the start of
the measurement. The experimental apparatus should be kept out of direct sunlight and other sources of
heating or cooling.
Set the temperature measurement apparatus at temperature difference measurement. Temperature
meter should be switched on at least 30 min before performing the experiment to avoid thermal drift.
Read operating instructions for further explanations of the temperature meter. Open the valves in the
following order: steel cylinder valve, operating valve, reducing valve, so that an initial pressure of 100
kPa is established. Reduce the pressure to zero in stages, in each case reading off the temperature
difference one minute after the particular pressure has been established.
Perform the measurement for both gases, and determine the atmospheric pressure and ambient
temperature.
Theory and evaluation
The state for real gases is given by the van der Waals equation.
a 

 p  2   V  b   RT
V 

where p is the pressure, V the molar volume and T the temperature of the gas. R is the universal gas
constant, a and b are the characteristic Van der Waals coefficients of the gas. The additional pressure by
intermolecular forces of attraction is described by a, b represents the volume of molecules.
In real gases the intrinsic energy U is composed of a thermocinetic and a potentional component. The
total change of the intrinsic energy U of a real gas therefore depends not only on the temperature and
the molar heat C V of a gas but also on the volume. The potential energy –a/V in the gas is given by the
work against the intermolecular forces.
U  CV  T 
2
a
V
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the total differential of the intrinsic energy is given by
 U 
 U 
dU  
  dT  
  dV
 T V
 V T
with
 U 

  CV
 T 
for
V  0
a
 U 

 2
V
 V 
for
T  0
and
Fig. 2: Throttling and the Joule-Thomson effect.
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Joule-Thomson effect
TEP
Fig. 3: Temperature differences measured at various ram pressures.
if the total differential is substituted through differences, you get
U  CV  T 
with
and
a
 V
V2
U
  CV
T
for
V  0
U
a
 2
V
V
for
T  0
The expansion of the gas at the throttle-point is adiabatic (ΔQ = 0). If external heat losses and friction
during the flow of the gas are excluded, the total energy H of the process is constant.
H  U  p  V  U1  p1  V1  U 2  p2  V2
For further calculations pressure p can be substituted by using van der Waals equation.
H  CV  T 
a  R T
a 

 2  V
V V  b V 
The relationship between ΔT an ΔV is given by the total differential of this equation
H  CV  T 
2a
R V
R  T V
R T
 V 
 T 
V 
 V
2
2
V b
V b
V
V  b 
 2a R  T  b 
R V 

  V
H   CV 
  T   2 
2 
V b
V


V

b



ΔH = 0 because H is constant.
For the next step b is neclected against V.
The molar heat at constant volume C V should be replaced by the molar heat at constant pressure C p .
This could be done approximately by the equation for ideal gases.
C p  CV  R
Then
T R  T  b  2a

V
V 2 Cp
At the throttle point a pressure gradient p 1 – p 2 and a temperature gradient T 1 – T 2 are established.
This effect is named the Joule Thomson effect and is described by the coefficient

4
T1  T2
T

p1  p2 p
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To get this coefficient, the difference ΔV has to be changed into Δp . This could be done approximately
by the equation for ideal gases.
V
R T
p
by differentiation an substitute total differentials by differences again you get
V 
R T
p
p2
T 
V
  R  T  b  2a 
V Cp
T  
2
R  T  p
p2
R  T  b  2a


2
2
2
Cp
p
R T
The Joule Thomson coefficient is then

T
1  2a



 b
p C p  RT

In real gases, the intrinsic energy U is composed of a thermokinetic content and a potential energy
content: the potential of the intermolecular forces of attraction. This is negative and tends towards zero
as the molecular distance increases. In real gases, the intrinsic energy is therefore a function of the
volume, and:
U
0
V
During adiabatic expansion ( Q = 0), during which also no external work is done, the overall intrinsic
energy remains unchanged, with the result that the potential energy increases at the expense of the
thermokinetic content and the gas cools.
At the throttle point, the effect named after Joule-Thomson is a quasi-stationary process.
A stationary pressure gradient p 2 – p 1 is established at the throttle point. If external heat losses and
friction during the flow of the gas are excluded, then for the total energy H, which consists of the intrinsic
energy U and displacement work pV:
H1  U1  p1V1  U 2  p2V2  H 2
In this equation, p 1 V 1 or p 2 V 2 is the work performed by an imaginary piston during the flow of a small
amount of gas by a change in position from position 1 to 2 or position 3 to 4 (see Figure 2). In real gases,
the displacement work p 1 V 1 does not equal the displacement work p 2 V 2 ; in this case:
p1 V1 < p 2 V 2
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Joule-Thomson effect
TEP
This means that, from the molecular interaction potential, displacement work is permanently done and
removed:
U1  U 2
or
T1  T2
The Joule-Thomson effect is described quantitatively by the coefficients

T1  T2
p1  p 2
For a change in the volume of a Van der Waals gas, the change in intrinsic energy is
U 
a
 V
V2
and the Joule-Thomson coefficient is thus
 2a
 1
 b 
 RT
 cp
VDW  
In this equation, c p is the specific heat under constant pressure, and a and b are the Van der Waals
coefficients.
If the expansion coefficients

1 V

V 0 T
(p = const)
are inserted, then
VdW 
VT 
1
   .
cp 
T
The measurement values in Fig. 3 give the straight line gradients
 CO  1.084  0.050  10 5
2
K
Pa
and
 N   0.253  0.030   10 5
2
K
Pa
The two temperature probes may give different absolute values for the same temperature. This is no
problem, as only the temperature difference is important for the determination Joule-Thomson
coefficients.
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Joule-Thomson effect
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The literature values are
 CO  1.16  10 5
2
K
Pa
at 20°C and 10 5 Pa,
 air  0.23  10 5
at 20°C and 10 5 Pa.
For CO 2 , with
a = 3.60 Pa m 6 /mol 2
b = 42.7 cm 3 /mol
c p = 366.1 J/mol K
the Van der Waals equation gives the coefficient
VdW ,CO  0.795  10 5
2
K
Pa
For air, with
a
= 1.40 Pa m 6 /mol 2
b
= 39.1 cm 3 /mol
c p = 288.9 J/mol K
the Van der Waals equation gives the coefficient
VdW ,air  0.387  10 5
Remarks
The formula for the Joule Thomson coefficient gives the condition for a cooling process:
Ti 
2a
R b
is the inversion temperature.
If Δp < 0 the temperature T has to be lower then Ti to cool gas with the process:
Δp < 0 and
T < Ti
then
ΔT < 0
For air (N 2 ) or CO 2 cooling is observed at room temperatur. H 2 or He have a positive coefficient at
room temperature, for H 2 this can be hazardous because self ignition is possible.
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