Bryn Mawr College
Department of Physics
Undergraduate Laboratories
PVT in Carbon Dioxide
Introduction
Bryn Mawr College's truly magnificent CO2 isotherm apparatus makes it possible
for you to measure the pressure P and volume V of a sample of CO2 at various temperatures T for a fixed amount n of gas. The relationship between these parameters is
reasonably well approximated by the van der Waals equation of state (P + a/v2)(v - b) =
RT where R is the gas constant, v =V/n is the molar volume (n is the number of moles of
CO2) and a and b are parameters to be investigated. They are related to the long-range
and the short-range intermolecular potentials, respectively. At high enough
temperatures, or low enough densities, this equation of state behaves approximately like
an ideal gas, PV = nRT. In this experiment you can investigate all these regimes, you can
determine the critical point of CO2 and you can determine the van der Waals parameters
for CO2. A good starting place for the physics is chapter 11 in Zemansky's "Heat and
Thermodynamics" which is to found with the experiment. This gives a "physical
chemistry" approach to the subject. F. Reif, "Fundamentals of Statistical Mechanics and
Thermodynamics," Mc Graw Hill, New York, 1965 gives a thorough "chemical physics"
approach.
This is the "millennium experiment" and is being brought back to life after being
dormant for many years.
It's a good idea to read through this entire write-up before actually doing any-thing.
This way you will get a feel for the big picture. As can be seen, the apparatus was not
built yesterday. The goal is to simultaneously measure P, V, and T for a fixed amount of
carbon dioxide gas and then determine the amount of gas n and the Van der Waal
constants a and b. In the process one should try to understand the relation-ship between
the constants a and b and the underlying physics.
The Apparatus and the Experiment
We first get an overview of the apparatus. There is a capillary tube containing a
fixed amount of carbon dioxide gas. It can be compressed hydraulically with the double
circuit oil-mercury pump operated by the lever as indicated schematically in the
accompanying figure 1. The pumping action compresses the oil which moves the
PVT in carbon dioxide
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mercury. The mercury, in turn, compresses the gas. It is interesting that the mercury
will not allow either the gas or the oil to pass through it, even under high pressure.
There is a cooling jacket around the CO2 capillary tube to keep the temperature of
the system constant by dissipating heat produced when work is done on the gas. In the
jacket, water flows from bottom to top so that no air is left in the jacket and bubbling is
reduced. (This part of the apparatus, including the outer glass jacket and all the hosing,
was last cleaned and replaced November 1999, in time for the new millennium.)
Perturbation of the system due to water falling on the capillary tube is minimized with
this setup. This also minimizes damage to the capillary tube in the long term. The
cooling jacket is connected to the water reservoir (the large old vat) through an electric
water pump. The water vat contains a large quantity of water which acts as a thermal
reservoir. There are three electric a.c. heaters supplied by a variable voltage source.
Water hoses take water from the vat to the pump, then to the assembly, then back to the
vat. There is a very high precision manual thermo-meter in the assembly. There is also a
regular thermometer in the vat and there may be a digital thermometer for easy reading.
One can check for temperature differences between water jacket and reservoir.
There are two pressure controls. The smaller of the two wheels on the side of the
apparatus (the wheel on your left as you face the apparatus from the end with the
handle) is the pressure control valve. Closing this valve by turning it all the way
clockwise makes it possible to apply pressure to the system. Turning it counterclockwise releases the pressure. You don't have to turn very far. The larger wheel is the fine
control. It allows for small adjustments of the pressure. To operate, open the control
valve, let some oil flow by moving the lever up and down, then close the valve and
pump until the pressure increases. When you observe, on the large pressure gauge, that
the pressure is starting to increase rather than just oscillate, keep one eye on the capillary
to watch for the mercury to appear. When it does, proceed cautiously. Do not exceed
about 100 Kg/cm2. When all the CO2 in the capillary is in the liquid state, the pressure
increases very rapidly and can break the capillary. Also, do not decrease the pressure
abruptly, as this may cause the tube containing the gas to rupture. Notice that kg/cm2 is
an unusual combination of cgs and SI units. The kg on the dial means kg-weight. So, the
readings are kg/cm2 times g. Note the following:
1
kg
cm2
kg
g = 1 2 9.8 m
s2
cm
2
10+4 cm
m2
PVT in carbon dioxide
3
= 9.8 x 10+4 Pascal x
1 atm
1.01 x 105 Pascal
= (0.97) atm
So to within 3% the dial reads atmospheres! The pressure you read on the pressure
gauge is gauge pressure, i.e., the difference between the pressure applied to the sample
and barometric pressure. As you will need absolute pressures in your calculations, read
and record the barometric pressure before and after the experiment. This will give you
an idea of the uncertainty with which you know the atmospheric pressure throughout
the experiment.
There are several ways to actually do the experiment. With three people, one
person can view the meniscus of the mercury through the telescope (as outlined in detail
below) and adjust the vernier. A second person can read the vernier and a third person
can read the pressure. If there are fewer than three people doing the experiment, you can
perhaps commandeer people at the time of measurement. It is important to tap the
pressure gauge constantly when making readings. With practice, this gauge can be read
much more accurately than one might think on first inspec-tion. Because of various leaks
in the oil circuit, the pressure will not stay constant unless the fine control is continuously
adjusted. The pressure increases when the fine control valve is turned counterclockwise.
Watch your hands, there are sharp edges in the vicinity of the valve. On the other hand,
with three people, you can allow the pressure to slowly decrease and then make
continuous readings. You have to find the way that works best for you and your team.
The three heaters in the vat are plugged into a terminal strip which, in turn, is
plugged into a variable transformer. You will need to experimentally determine the
relationship between temperature and heater voltage since the relationship will depend
on the amount of water in the vat. As the water in the vat evaporates you will need to
add more and this happens quite fast at elevated temperatures. As a crude measure, (T
[in C], V [in volts ac]) is approximately (20, 0), (28, 25), (35, 50), (58, 100). It takes about
twelve hours to reach steady state after a significant temperature change. If you are
going to add water, do so at the time you change temperature. The measures of the route
to steady state are (1) the temperature difference between the vat and the assembly, and
(2) how this temperature difference changes with time. When the apparatus is at a
steady state temperature, a set of measurements will prob-ably take between fifteen
PVT in carbon dioxide
4
minutes and an hour. Ideally, you would make measure-ments every 24 hours or so.
In order to measure the volume occupied by the CO2 in gaseous or liquid form you
will use a telescope which runs on a vertically mounted scale. You must use the telescope
backwards! The reading is taken on the scale with the help of a slider, like a vernier
caliper, which helps in obtaining a better estimate of the height of the capil-lary occupied
by the CO2. The image of the capillary can be focused by moving the telescope closer or
further away from the tube with the help of two screws, one of which loosens the rod
holding the telescope and the other of which regulates its height. The position of the
telescope on the vertical rod has to be accurately chosen before beginning the
experiment, because a change in this position during the measurements would irreparably
corrupt the entire experiment. On the side of the eye the lens holder can be rotated to adjust
the position of the scales engraved in a support internal to the telescope itself. The scales
serve a double purpose: they replace the crosshair not present in this kind of telescope
and they allow one to measure the diameter of the capillary as discussed below. Be
careful to use the scale always in the same position when you take measurements, as this
may introduce reading errors which differ from measurement to measurement, therefore
making the measurement error larger.
Measure the volumes occupied by the CO2 above the mercury level for different
values of the pressure to familiarize yourself with the apparatus. Notice that the system
has some thermal inertia, therefore you should wait for equilibrium to be reached before
taking the reading.
Measure the diameter of the capillary in the following way. Calibrate the scale that
you read in the telescope comparing it with the divisions (mm) that you read off a ruler
positioned at the same distance as the capillary in front of the telescope. (The scales are
calibrated themselves, but the length that you measure changes depending on the
distance object-telescope.) Now point the telescope again at the capillary and measure its
internal diameter, checking whether it looks uniform along its length (important
caution). Using the paper by Walter Michels published on the American Physics Teacher
(1939), correct your reading to obtain the actual internal diameter of the capillary. Take a
reading, or more than one, of the position on your vertical scale, of the top of the
capillary tube. (Is it easy to determine? Is there any uncertainty due to the shape of the
capillary? Does the amount of uncertainty so introduced contri-bute significantly to the
uncertainty in the volume?). Taking the difference between this reading and the one that
you will take for each position of the mercury column you will be able to estimate the
PVT in carbon dioxide
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actual volume of the capillary (and its uncertainty).
Take measurements of pressure and volume along different isotherms, i.e. fix the
temperature, change the pressure and measure the volume by reading the height of the
mercury column with the telescope.
In this way you can establish the general shape of the isotherms. Try to define the
boundaries of the coexistence region, between liquid and gas phase, as well as you can.
Note that by repeatedly increasing the pressure in the CO2 column you can slowly
heat up the water in the cooling jacket. If your set of measurements is taken too quickly
the water flow might not be fast enough to keep the temperature around the capillary at
the same value as that of the reservoir. To avoid this problem one can either take the set
of measurements slowly enough, so that there is no tempera-ture difference between
water jacket and reservoir (compare the two thermometer readings, one of which is in the
water reservoir), or one can take into account the temperature difference which occurs in
the measurements as an experimental uncertainty (probably you need both!).
Remember to monitor the water temperature in the cooling jacket periodically,
writing down the reading (with uncertainty).
To get a large coexistence region between liquid and gas phase, you can lower the
temperature by adding ice to the cooling water. The disadvantages of this are that water
condenses on the outer part of the glass ampule, the temperature decreases more and
more slowly, the smaller the temperature difference between ice and water (can you
explain that?), and the water temperature can increase much more in the jacket because
of gas compression, before circulation can have its effect. Nonetheless it can be an
interesting and useful measurement.
Whenever one takes a measurement with a telescope one uses the crosshairs (in
your telescope, the scales) as a reference, moving the telescope in such a way as to arrive
to the level (mercury meniscus in this case) to measure always from the same direction.
E.g. once you have decided to take the measurement "from above" bring the telescope
above the meniscus and lower it until you can line up the crosshair with the mercury
surface. In the next measurements do not reach the alignment between crosshair and
meniscus by raising the telescope because this could introduce a reading error which
arises from two factors: i) human bias: the width of the cross-hair can influence the point
at which you stop when you move the telescope upwards, instead of downwards, once
your eye has "pictured" the position at which to stop; ii) every gear has some play
(otherwise it could not move), and part of the turn of the screw does not correspond to an
PVT in carbon dioxide
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actual movement when you invert direc-tion. This precaution is particularly important
when the screw has the scale incor-porated. As long as you take differences between
positions, the above procedure keeps the results free of those systematic errors discussed
above. It does not eliminate errors such as errors in the scale calibration.
expt13_PVTofCO2.docx 2013A