AE 3051, Lab #15
Investigation of the Ideal Gas State Equation
By: George P. Burdell
Group E3
Summer Semester 2000
Introduction
An important relationship often used to relate gas properties is the ideal gas equation of state.
This report describes experiments intended to investigate the validity of this state equation,
which can be written as p  RT , where p is the absolute pressure, T is the absolute temperature
and R is the specific gas constant. In order to accomplish this, the pressure was measured for a
fixed amount of gas confined in a heated, fixed volume container. The container and the gas
inside were heated to a fixed temperature using an electrical resistance heater. The temperature of
the gas was measured with a type-K thermocouple, and the pressure was monitored with a
mercury manometer. Three gases, helium (He), nitrogen (N2) and argon (Ar), were tested. A
vacuum pump was used to empty the container before it was filled with the test gas. In addition,
the volume of the container was determined by filling it with water and measuring the amount
with a graduated cylinder.
Results
Raw Data
The first measurement was determination of the test cell’s volume. By filling the test cell
with water, and then measuring the volume of water by pouring it into a graduated cylinder, the
lab group determined its volume to be 83 ml, or equivalently, 83 cm3. Next, the lab group
measured the atmospheric pressure using the digital thermometer/barometer located in the wind
tunnel control room. The measured pressure was 29.66 in. Hg. Finally, the gauge pressure of the
various gases as a function of temperature was measured. The results are listed in Table I for all
three gases (He, N2 and Ar), with 6 measurements for each gas.
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Reduced Data
The temperatures and pressures can be converted to absolute values using the following:
 
T K   T  C  273

(1)
Pa 25.4mmHg
101325
760mmHg 1in .Hg
pPa   p gage in .Hg   patm in .Hg 
.
(2)
With the measured atmospheric pressure in the test room, Equation (2) can be simplified to:


pPa   3386 p gage in.Hg   29.66
(3)
The raw data listed in Table I were converted to absolute values using Equations (1) and (3).
They are shown plotted in Figure 1. In order to compare the results to the ideal gas law, it is more
appropriate to plot the ratio p/T , or even better, p/RT, with R given by:
RR
M
(4)
where R is the universal gas constant (8314 J/kmol K) and M is the molecular weight of the gas
(see Table II). The results for p/RT as a function of temperature are plotted in Figure 2, and the
actual values are listed in Table III.
Brief Discussion
Supplement Questions
Since the gas inside the container is trapped and the container volume is fixed (assuming
negligible change in the size of the container due to thermal expansion during heating), the
measurements for each gas sample are at the same density. According to the ideal gas state
equation, we would expect the ratio p/RT to remain constant here as the gas temperature is
changed. This is essentially the behavior displayed in Table III and Figure 2. Thus, we can
conclude that the ratio, p/T, is proportional to density as suggested by the ideal gas law. With the
current measurements, however, the proportionality constant cannot be proven to be R M .
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In order to test the remainder of the ideal gas law, one would have to determine the mass of
gas trapped inside the cylinder with some independent measurement. One possibility would be to
weigh the test cell twice: once with a vacuum inside and once when it is loaded with the test gas.
The difference would be the weight of the test gas, which could be converted to mass using the
standard value for the gravitational acceleration on Earth. We could then compare this to the
mass determined from combining: 1) the p and T measurements, 2) the ideal gas law, and 3) the
volume of the test cell, i.e.,
m
p
V
RT
(5)
However, getting an accurate value with Equation (5) would be a very difficult task. Assuming
the ideal gas law is accurate, our density results from this experiment for the heaviest gas, Ar,
were approximately 0.82 kg/m3. Since the volume of the test cell was measured to be 83 cm3, the
mass of the Ar inside the cell was only 68 mg. It would be difficult to measure this small change
in the mass of the test cell, which was approximately 2000g. In other words, an error of 0.1% in
the measurement of the “empty” test cell would be ~2g. To increase the mass of gas to the level
of our expected uncertainty in the test cell’s mass, the initial pressure in the cell would need to be
increased from ~80 kPa (see Figure 1) to roughly 2.4 MPa or almost 350 psi. In addition, it
would probably require a stronger and heavier chamber to withstand this high pressure, making
the measurement even more difficult.
Additional Discussion
In the current experiment, we kept the density of the gas constant and changed its
temperature. An alternate approach would be to use a piston-cylinder arrangement to change the
volume and, therefore, the density of the gas and measure the pressure and temperature. The
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drawback to this approach is that the volume would need to change very rapidly and the
temperature measurement would need to be quick enough to prevent errors due to heat exchange
with the cylinder walls. In other words, if one were to rapidly compress the gas, its temperature
would rise above the temperature of the cylinder walls. Very quickly, however, the gas would
start losing heat to the walls and the temperature would drop back towards its original value.
Tables and Figures
Table I. Pressures and temperatures for 3 gases in a heated, fixed volume container.
Gas
T (C)
pgage (in. Hg)
He
He
He
He
He
He
N2
N2
N2
N2
N2
N2
Ar
Ar
Ar
Ar
Ar
Ar
22
29
38
50
65
85
23
28
40
49
63
81
22
30
39
51
63
84
-20.82
-20.57
-20.34
-19.96
-19.46
-18.93
-17.88
-17.62
-17.20
-16.83
-16.23
-15.55
-14.95
-14.48
-14.01
-13.39
-12.82
-11.84
Table II. Molecular weights of the gases measured.
Gas
M (kg/kmol)
He
N2
Ar
4
28
40
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Table III. Measurements of the ratio p/RT; also listed are the average value of the ratio for
each gas.
Gas
T (K)
p/RT (kg/m3)
He
He
He
He
He
He
He
N2
N2
N2
N2
N2
N2
N2
Ar
Ar
Ar
Ar
Ar
Ar
Ar
295
302
311
323
338
358
0.0477
0.0490
0.0488
0.0499
0.0482
0.0488
Average
296
301
313
322
336
354
0.0487
0.454
0.456
0.454
0.454
0.463
0.455
Average
295
303
312
324
336
357
0.456
0.826
0.816
0.817
0.833
0.817
0.813
Average
0.820
5
80000
p (Pa)
60000
Ar
N2
40000
He
20000
0
250
300
350
400
T (K)
Figure 1. Absolute pressure dependence on absolute temperature for three fixed density
gases.
0.8
3
p/RT (kg/m )
1
0.6
Ar
N2
He
0.4
0.2
0
250
300
350
400
T (K)
Figure 2. Variation in the normalized pressure/temperature ratio for three gases of
constant mass in a fixed volume container.
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